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NumPy Beginner's Guide
Third Edition
Build efficient, high-speed programs using the
high-performance NumPy mathematical library
Ivan Idris
BIRMINGHAM - MUMBAI
NumPy Beginner's Guide
Third Edition
Copyright © 2015 Packt Publishing
All rights reserved. No part of this book may be reproduced, stored in a retrieval system,
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Every effort has been made in the preparation of this book to ensure the accuracy of the
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and distributors will be held liable for any damages caused or alleged to be caused directly
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However, Packt Publishing cannot guarantee the accuracy of this information.
First published: November 2011
Second edition: April 2013
Third edition: June 2015
Production reference: 1160615
Published by Packt Publishing Ltd.
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ISBN 978-1-78528-196-9
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Credits
Author
Ivan Idris
Reviewers
Alexandre Devert
Project Coordinator
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Proofreader
Safis Editing
Davide Fiacconi
Ardo Illaste
Commissioning Editor
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Acquisition Editors
Indexer
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Graphics
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Content Development Editor
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Copy Editors
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Cover Work
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About the Author
Ivan Idris has an MSc in experimental physics. His graduation thesis had a strong emphasis
on applied computer science. After graduating, he worked for several companies as a Java
developer, data warehouse developer, and QA Analyst. His main professional interests are
business intelligence, big data, and cloud computing. Ivan enjoys writing clean, testable
code and interesting technical articles. He is the author of NumPy Beginner's Guide, NumPy
Cookbook, Learning NumPy Array, and Python Data Analysis. You can find more information
about him and a blog with a few examples of NumPy at http://ivanidris.net/
wordpress/.
I would like to take this opportunity to thank the reviewers and the team
at Packt Publishing for making this book possible. Also thanks go to my
teachers, professors, colleagues, Wikipedia contributors, Stack Overflow
contributors, and other authors who taught me science and programming.
Last but not least, I would like to acknowledge my parents, family, and
friends for their support.
About the Reviewers
Davide Fiacconi is completing his PhD in theoretical astrophysics from the Institute for
Computational Science at the University of Zurich. He did his undergraduate and graduate
studies at the University of Milan-Bicocca, studying the evolution of collisional ring galaxies
using hydrodynamic numerical simulations. Davide's research now focuses on the formation
and coevolution of supermassive black holes and galaxies, using both massively parallel
simulations and analytical techniques. In particular, his interests include the formation of the
first supermassive black hole seeds, the dynamics of binary black holes, and the evolution of
high-redshift galaxies.
Ardo Illaste is a data scientist. He wants to provide everyone with easy access to data for
making major life and career decisions. He completed his PhD in computational biophysics,
prior to fully delving into data mining and machine learning. Ardo has worked and studied in
Estonia, the USA, and Switzerland.
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I dedicate this book to my aunt Lies who recently passed away. Rest in peace.
Table of Contents
Preface
Chapter 1: NumPy Quick Start
Python
Time for action – installing Python on different operating systems
The Python help system
Time for action – using the Python help system
Basic arithmetic and variable assignment
Time for action – using Python as a calculator
Time for action – assigning values to variables
The print() function
Time for action – printing with the print() function
Code comments
Time for action – commenting code
The if statement
Time for action – deciding with the if statement
The for loop
Time for action – repeating instructions with loops
Python functions
Time for action – defining functions
Python modules
Time for action – importing modules
NumPy on Windows
Time for action – installing NumPy, matplotlib, SciPy, and IPython on Windows
NumPy on Linux
Time for action – installing NumPy, matplotlib, SciPy, and IPython on Linux
NumPy on Mac OS X
Time for action – installing NumPy, SciPy, matplotlib, and IPython with
MacPorts or Fink
[i]
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Table of Contents
Building from source
Arrays
Time for action – adding vectors
IPython – an interactive shell
Online resources and help
Summary
16
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25
26
Chapter 2: Beginning with NumPy Fundamentals
27
Chapter 3: Getting Familiar with Commonly Used Functions
53
NumPy array object
Time for action – creating a multidimensional array
Selecting elements
NumPy numerical types
Data type objects
Character codes
The dtype constructors
The dtype attributes
Time for action – creating a record data type
One-dimensional slicing and indexing
Time for action – slicing and indexing multidimensional arrays
Time for action – manipulating array shapes
Time for action – stacking arrays
Time for action – splitting arrays
Time for action – converting arrays
Summary
File I/O
Time for action – reading and writing files
Comma-seperated value files
Time for action – loading from CSV files
Volume Weighted Average Price
Time for action – calculating Volume Weighted Average Price
The mean() function
Time-weighted average price
Value range
Time for action – finding highest and lowest values
Statistics
Time for action – performing simple statistics
Stock returns
Time for action – analyzing stock returns
Dates
[ ii ]
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Table of Contents
Time for action – dealing with dates
Time for action – using the datetime64 data type
Weekly summary
Time for action – summarizing data
Average True Range
Time for action – calculating Average True Range
Simple Moving Average
Time for action – computing the Simple Moving Average
Exponential Moving Average
Time for action – calculating the Exponential Moving Average
Bollinger Bands
Time for action – enveloping with Bollinger Bands
Linear model
Time for action – predicting price with a linear model
Trend lines
Time for action – drawing trend lines
Methods of ndarray
Time for action – clipping and compressing arrays
Factorial
Time for action – calculating the factorial
Missing values and Jackknife resampling
Time for action – handling NaNs with the nanmean(), nanvar(),
and nanstd() functions
Summary
Chapter 4: Convenience Functions for Your Convenience
Correlation
Time for action – trading correlated pairs
Polynomials
Time for action – fitting to polynomials
On-balance volume
Time for action – balancing volume
Simulation
Time for action – avoiding loops with vectorize()
Smoothing
Time for action – smoothing with the hanning() function
Initialization
Time for action – creating value initialized arrays with the full() and
full_like() functions
Summary
[ iii ]
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Table of Contents
Chapter 5: Working with Matrices and ufuncs
121
Chapter 6: Moving Further with NumPy Modules
145
Matrices
Time for action – creating matrices
Creating a matrix from other matrices
Time for action – creating a matrix from other matrices
Universal functions
Time for action – creating universal functions
Universal function methods
Time for action – applying the ufunc methods to the add function
Arithmetic functions
Time for action – dividing arrays
Modulo operation
Time for action – computing the modulo
Fibonacci numbers
Time for action – computing Fibonacci numbers
Lissajous curves
Time for action – drawing Lissajous curves
Square waves
Time for action – drawing a square wave
Sawtooth and triangle waves
Time for action – drawing sawtooth and triangle waves
Bitwise and comparison functions
Time for action – twiddling bits
Fancy indexing
Time for action – fancy indexing in-place for ufuncs with the at() method
Summary
Linear algebra
Time for action – inverting matrices
Solving linear systems
Time for action – solving a linear system
Finding eigenvalues and eigenvectors
Time for action – determining eigenvalues and eigenvectors
Singular value decomposition
Time for action – decomposing a matrix
Pseudo inverse
Time for action – computing the pseudo inverse of a matrix
Determinants
Time for action – calculating the determinant of a matrix
Fast Fourier transform
[ iv ]
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Table of Contents
Time for action – calculating the Fourier transform
Shifting
Time for action – shifting frequencies
Random numbers
Time for action – gambling with the binomial
Hypergeometric distribution
Time for action – simulating a game show
Continuous distributions
Time for action – drawing a normal distribution
Lognormal distribution
Time for action – drawing the lognormal distribution
Bootstrapping in statistics
Time for action – sampling with numpy.random.choice()
Summary
Chapter 7: Peeking into Special Routines
Sorting
Time for action – sorting lexically
Time for action – partial sorting via selection for a fast median
with the partition() function
Complex numbers
Time for action – sorting complex numbers
Searching
Time for action – using searchsorted
Array elements extraction
Time for action – extracting elements from an array
Financial functions
Time for action – determining the future value
Present value
Time for action – getting the present value
Net present value
Time for action – calculating the net present value
Internal rate of return
Time for action – determining the internal rate of return
Periodic payments
Time for action – calculating the periodic payments
Number of payments
Time for action – determining the number of periodic payments
Interest rate
Time for action – figuring out the rate
Window functions
[v]
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Table of Contents
Time for action – plotting the Bartlett window
Blackman window
Time for action – smoothing stock prices with the Blackman window
Hamming window
Time for action – plotting the Hamming window
Kaiser window
Time for action – plotting the Kaiser window
Special mathematical functions
Time for action – plotting the modified Bessel function
sinc
Time for action – plotting the sinc function
Summary
Chapter 8: Assuring Quality with Testing
Assert functions
Time for action – asserting almost equal
Approximately equal arrays
Time for action – asserting approximately equal
Almost equal arrays
Time for action – asserting arrays almost equal
Equal arrays
Time for action – comparing arrays
Ordering arrays
Time for action – checking the array order
Object comparison
Time for action – comparing objects
String comparison
Time for action – comparing strings
Floating-point comparisons
Time for action – comparing with assert_array_almost_equal_nulp
Comparison of floats with more ULPs
Time for action – comparing using maxulp of 2
Unit tests
Time for action – writing a unit test
Nose test decorators
Time for action – decorating tests
Docstrings
Time for action – executing doctests
Summary
[ vi ]
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Table of Contents
Chapter 9: Plotting with matplotlib
217
Chapter 10: When NumPy Is Not Enough – SciPy and Beyond
245
Simple plots
Time for action – plotting a polynomial function
Plot format string
Time for action – plotting a polynomial and its derivatives
Subplots
Time for action – plotting a polynomial and its derivatives
Finance
Time for action – plotting a year's worth of stock quotes
Histograms
Time for action – charting stock price distributions
Logarithmic plots
Time for action – plotting stock volume
Scatter plots
Time for action – plotting price and volume returns with a scatter plot
Fill between
Time for action – shading plot regions based on a condition
Legend and annotations
Time for action – using a legend and annotations
Three-dimensional plots
Time for action – plotting in three dimensions
Contour plots
Time for action – drawing a filled contour plot
Animation
Time for action – animating plots
Summary
MATLAB and Octave
Time for action – saving and loading a .mat file
Statistics
Time for action – analyzing random values
Sample comparison and SciKits
Time for action – comparing stock log returns
Signal processing
Time for action – detecting a trend in QQQ
Fourier analysis
Time for action – filtering a detrended signal
Mathematical optimization
Time for action – fitting to a sine
Numerical integration
[ vii ]
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Table of Contents
Time for action – calculating the Gaussian integral
Interpolation
Time for action – interpolating in one dimension
Image processing
Time for action – manipulating Lena
Audio processing
Time for action – replaying audio clips
Summary
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270
Chapter 11: Playing with Pygame
271
Appendix A: Pop Quiz Answers
Appendix B: Additional Online Resources
295
299
Appendix C: NumPy Functions' References
Index
301
307
Pygame
Time for action – installing Pygame
Hello World
Time for action – creating a simple game
Animation
Time for action – animating objects with NumPy and Pygame
matplotlib
Time for Action – using matplotlib in Pygame
Surface pixels
Time for Action – accessing surface pixel data with NumPy
Artificial Intelligence
Time for Action – clustering points
OpenGL and Pygame
Time for Action – drawing the Sierpinski gasket
Simulation game with Pygame
Time for Action – simulating life
Summary
Python
Mathematics and statistics
[ viii ]
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Preface
Scientists, engineers, and quantitative data analysts face many challenges nowadays. Data
scientists want to be able to perform numerical analysis on large datasets with minimal
programming effort. They also want to write readable, efficient, and fast code that is as close
as possible to the mathematical language they are used to. A number of accepted solutions
are available in the scientific computing world.
The C, C++, and Fortran programming languages have their benefits, but they are not
interactive and considered too complex by many. The common commercial alternatives,
such as MATLAB, Maple, and Mathematica, provide powerful scripting languages that are
even more limited than any general-purpose programming language. Other open source
tools similar to MATLAB exist, such as R, GNU Octave, and Scilab. Obviously, they too lack
the power of a language such as Python.
Python is a popular general-purpose programming language that is widely used in the
scientific community. You can access legacy C, Fortran, or R code easily from Python. It
is object-oriented and considered to be of a higher level than C or Fortran. It allows you
to write readable and clean code with minimal fuss. However, it lacks an out-of-the-box
MATLAB equivalent. That's where NumPy comes in. This book is about NumPy and related
Python libraries, such as SciPy and matplotlib.
What is NumPy?
NumPy (short for numerical Python) is an open source Python library for scientific
computing. It lets you work with arrays and matrices in a natural way. The library contains
a long list of useful mathematical functions, including some functions for linear algebra,
Fourier transformation, and random number generation routines. LAPACK, a linear algebra
library, is used by the NumPy linear algebra module if you have it installed on your system.
Otherwise, NumPy provides its own implementation. LAPACK is a well-known library,
originally written in Fortran, on which MATLAB relies as well. In a way, NumPy replaces some
of the functionality of MATLAB and Mathematica, allowing rapid interactive prototyping.
Preface
We will not be discussing NumPy from a developing contributor's perspective, but from more
of a user's perspective. NumPy is a very active project and has a lot of contributors. Maybe,
one day you will be one of them!
History
NumPy is based on its predecessor Numeric. Numeric was first released in 1995 and has
deprecated status now. Neither Numeric nor NumPy made it into the standard Python library
for various reasons. However, you can install NumPy separately, which will be explained in
Chapter 1, NumPy Quick Start.
In 2001, a number of people inspired by Numeric created SciPy, an open source scientific
computing Python library that provides functionality similar to that of MATLAB, Maple, and
Mathematica. Around this time, people were growing increasingly unhappy with Numeric.
Numarray was created as an alternative to Numeric. That is also deprecated now. It was
better in some areas than Numeric, but worked very differently. For that reason, SciPy kept
on depending on the Numeric philosophy and the Numeric array object. As is customary
with new latest and greatest software, the arrival of Numarray led to the development of
an entire ecosystem around it, with a range of useful tools.
In 2005, Travis Oliphant, an early contributor to SciPy, decided to do something about this
situation. He tried to integrate some of Numarray's features into Numeric. A complete
rewrite took place, and it culminated in the release of NumPy 1.0 in 2006. At that time,
NumPy had all the features of Numeric and Numarray, and more. Tools were available to
facilitate the upgrade from Numeric and Numarray. The upgrade is recommended since
Numeric and Numarray are not actively supported any more.
Originally, the NumPy code was a part of SciPy. It was later separated and is now used by
SciPy for array and matrix processing.
Why use NumPy?
NumPy code is much cleaner than straight Python code and it tries to accomplish the
same tasks. There are fewer loops required because operations work directly on arrays
and matrices. The many convenience and mathematical functions make life easier as well.
The underlying algorithms have stood the test of time and have been designed with high
performance in mind.
NumPy's arrays are stored more efficiently than an equivalent data structure in base Python,
such as a list of lists. Array IO is significantly faster too. The improvement in performance
scales with the number of elements of the array. For large arrays, it really pays off to use
NumPy. Files as large as several terabytes can be memory-mapped to arrays, leading to
optimal reading and writing of data.
[x]
Preface
The drawback of NumPy arrays is that they are more specialized than plain lists. Outside the
context of numerical computations, NumPy arrays are less useful. The technical details of
NumPy arrays will be discussed in later chapters.
Large portions of NumPy are written in C. This makes NumPy faster than pure Python code.
A NumPy C API exists as well, and it allows further extension of functionality with the help
of the C language. The C API falls outside the scope of the book. Finally, since NumPy is open
source, you get all the related advantages. The price is the lowest possible—as free as a
beer. You don't have to worry about licenses every time somebody joins your team or you
need an upgrade of the software. The source code is available for everyone. This of course is
beneficial to code quality.
Limitations of NumPy
If you are a Java programmer, you might be interested in Jython, the Java implementation of
Python. In that case, I have bad news for you. Unfortunately, Jython runs on the Java Virtual
Machine and cannot access NumPy because NumPy's modules are mostly written in C. You
could say that Jython and Python are two totally different worlds, though they do implement
the same specifications. There are some workarounds for this discussed in NumPy Cookbook
- Second Edition, Packt Publishing, written by Ivan Idris.
What this book covers
Chapter 1, NumPy Quick Start, guides you through the steps needed to install NumPy on
your system and create a basic NumPy application.
Chapter 2, Beginning with NumPy Fundamentals, introduces NumPy arrays and
fundamentals.
Chapter 3, Getting Familiar with Commonly Used Functions, teaches you the most commonly
used NumPy functions—the basic mathematical and statistical functions.
Chapter 4, Convenience Functions for Your Convenience, tells you about functions that
make working with NumPy easier. This includes functions that select certain parts of your
arrays, for instance, based on a Boolean condition. You also learn about polynomials and
manipulating the shapes of NumPy objects.
Chapter 5, Working with Matrices and ufuncs, covers matrices and universal functions.
Matrices are well-known in mathematics and have their representation in NumPy as well.
Universal functions (ufuncs) work on arrays element by element, or on scalars. ufuncs expect
a set of scalars as the input and produce a set of scalars as the output.
[ xi ]
Preface
Chapter 6, Moving Further with NumPy Modules, discusses a number of basic modules
of universal functions. These functions can typically be mapped to their mathematical
counterparts, such as addition, subtraction, division, and multiplication.
Chapter 7, Peeking into Special Routines, describes some of the more specialized NumPy
functions. As NumPy users, we sometimes find ourselves having special requirements.
Fortunately, NumPy satisfies most of our needs.
Chapter 8, Assuring Quality with Testing, teaches you how to write NumPy unit tests.
Chapter 9, Plotting with matplotlib, covers matplotlib in depth, a very useful Python plotting
library. NumPy cannot be used on its own to create graphs and plots. matplotlib integrates
nicely with NumPy and has plotting capabilities comparable to MATLAB.
Chapter 10, When NumPy Is Not Enough – SciPy and Beyond, covers more details about
SciPy. We know that SciPy and NumPy are historically related. SciPy, as mentioned in the
History section, is a high-level Python scientific computing framework built on top of NumPy.
It can be used in conjunction with NumPy.
Chapter 11, Playing with Pygame, is the dessert of this book. You learn how to create fun
games with NumPy and Pygame. You also get a taste of artificial intelligence in this chapter.
Appendix A, Pop Quiz Answers, has the answers to all the pop quiz questions within
the chapters.
Appendix B, Additional Online Resources, contains links to Python, mathematics, and
statistics websites.
Appendix C, NumPy Functions' References, lists some useful NumPy functions and
their descriptions.
What you need for this book
To try out the code samples in this book, you will need a recent build of NumPy. This means
that you will need one of the Python versions supported by NumPy as well. Some code
samples make use of matplotlib for illustration purposes. matplotlib is not strictly required
to follow the examples, but it is recommended that you install it too. The last chapter is
about SciPy and has one example involving SciKits.
Here is a list of the software used to develop and test the code examples:
Python 2.7
NumPy 1.9
SciPy 0.13
[ xii ]
Preface
matplotlib 1.3.1
Pygame 1.9.1
IPython 2.4.1
Needless to say, you don't need exactly this software and these versions on your computer.
Python and NumPy constitute the absolute minimum you will need.
Who this book is for
This book is for the scientists, engineers, programmers, or analysts looking for a high-quality,
open source mathematical library. Knowledge of Python is assumed. Also, some affinity, or
at least interest, in mathematics and statistics is required. However, I have provided brief
explanations and pointers to learning resources.
Sections
In this book, you will find several headings that appear frequently (Time for action, What just
happened?, Have a go hero, and Pop quiz).
To give clear instructions on how to complete a procedure or task, we use the following
sections.
Time for action – heading
1.
2.
3.
Action 1
Action 2
Action 3
Instructions often need some extra explanation to ensure that they make sense, so they are
followed by these sections.
What just happened?
This section explains the working of the tasks or instructions that you have just completed.
You will also find some other learning aids in the book.
[ xiii ]
Preface
Pop quiz – heading
These are short multiple-choice questions intended to help you test your own understanding.
Have a go hero – heading
These are practical challenges that give you ideas to experiment with what you have learned.
Conventions
In this book, you will find a number of styles of text that distinguish between different
kinds of information. Here are some examples of these styles, and an explanation of
their meaning.
Code words in text are shown as follows: "Notice that numpysum() does not need a
for loop."
A block of code is set as follows:
def numpysum(n):
a = numpy.arange(n) ** 2
b = numpy.arange(n) ** 3
c = a + b
return c
When we wish to draw your attention to a particular part of a code block, the relevant lines
or items are set in bold:
reals = np.isreal(xpoints)
print "Real number?", reals
Real number? [ True True True
True False False False False]
Any command-line input or output is written as follows:
>>>fromnumpy.testing import rundocs
>>>rundocs('docstringtest.py')
New terms and important words are shown in bold. Words that you see on the screen, in
menus or dialog boxes for example, appear in the text like this: "Clicking on the Next button
moves you to the next screen."
[ xiv ]
Preface
Warnings or important notes appear in a box like this.
Tips and tricks appear like this.
Reader feedback
Feedback from our readers is always welcome. Let us know what you think about this
book—what you liked or disliked. Reader feedback is important for us as it helps us develop
titles that you will really get the most out of.
To send us general feedback, simply e-mail feedback@packtpub.com, and mention the
book's title in the subject of your message.
If there is a topic that you have expertise in and you are interested in either writing or
contributing to a book, see our author guide at www.packtpub.com/authors.
Customer support
Now that you are the proud owner of a Packt book, we have a number of things to help you
to get the most from your purchase.
Downloading the example code
You can download the example code files from your account at http://www.packtpub.
com for all the Packt Publishing books you have purchased. If you purchased this book
elsewhere, you can visit http://www.packtpub.com/support and register to have the
files e-mailed directly to you.
Downloading the color images of this book
We also provide you with a PDF file that has color images of the screenshots/diagrams used
in this book. The color images will help you better understand the changes in the output.
You can download this file from https://www.packtpub.com/sites/default/files/
downloads/NumpyBeginner'sGuide_Third_Edition_ColorImages.pdf.
[ xv ]
Preface
Errata
Although we have taken every care to ensure the accuracy of our content, mistakes do happen.
If you find a mistake in one of our books—maybe a mistake in the text or the code—we
would be grateful if you could report this to us. By doing so, you can save other readers from
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[ xvi ]
1
NumPy Quick Start
Let's get started. We will install NumPy and related software on different
operating systems and have a look at some simple code that uses NumPy. This
chapter briefly introduces the IPython interactive shell. SciPy is closely related
to NumPy, so you will see the SciPy name appearing here and there. At the end
of this chapter, you will find pointers on how to find additional information
online if you get stuck or are uncertain about the best way to solve problems.
In this chapter, you will cover the following topics:
Install Python, SciPy, matplotlib, IPython, and NumPy on Windows, Linux,
and Macintosh
Do a short refresher of Python
Write simple NumPy code
Get to know IPython
Browse online documentation and resources
Python
NumPy is based on Python, so you need to have Python installed. On some operating
systems, Python is already installed. However, you need to check whether the Python version
corresponds with the NumPy version you want to install. There are many implementations of
Python, including commercial implementations and distributions. In this book, we focus on
the standard CPython implementation, which is guaranteed to be compatible with NumPy.
[1]
NumPy Quick Start
Time for action – installing Python on different operating
systems
NumPy has binary installers for Windows, various Linux distributions, and Mac OS X
at http://sourceforge.net/projects/numpy/files/. There is also a source
distribution, if you prefer that. You need to have Python 2.4.x or above installed on your
system. We will go through the various steps required to install Python on the following
operating systems:
Debian and Ubuntu: Python might already be installed on Debian and Ubuntu,
but the development headers are usually not. On Debian and Ubuntu, install the
python and python-dev packages with the following commands:
$ [sudo] apt-get install python
$ [sudo] apt-get install python-dev
Windows: The Windows Python installer is available at https://www.python.
org/downloads/. On this website, we can also find installers for Mac OS X and
source archives for Linux, UNIX, and Mac OS X.
Mac: Python comes preinstalled on Mac OS X. We can also get Python through
MacPorts, Fink, Homebrew, or similar projects.
Install, for instance, the Python 2.7 port by running the following command:
$ [sudo] port install python27
Linear Algebra PACKage (LAPACK) does not need to be present but, if it is,
NumPy will detect it and use it during the installation phase. It is recommended
that you install LAPACK for serious numerical analysis as it has useful numerical
linear algebra functionality.
What just happened?
We installed Python on Debian, Ubuntu, Windows, and the Mac OS X.
You can download the example code files for all the Packt books you have
purchased from your account at https://www.packtpub.com/. If you
purchased this book elsewhere, you can visit https://www.packtpub.
com/books/content/support and register to have the files e-mailed
directly to you.
[2]
Chapter 1
The Python help system
Before we start the NumPy introduction, let's take a brief tour of the Python help system,
in case you have forgotten how it works or are not very familiar with it. The Python help
system allows you to look up documentation from the interactive Python shell. A shell is
an interactive program, which accepts commands and executes them for you.
Time for action – using the Python help system
Depending on your operating system, you can access the Python shell with special
applications, usually a terminal of some sort.
1.
In such a terminal, type the following command to start a Python shell:
$ python
2.
You will get a short message with the Python version and other information and the
following prompt:
>>>
Type the following in the prompt:
>>> help()
Another message appears and the prompt changes as follows:
help>
3.
If you type, for instance, keywords as the message says, you get a list of keywords.
The topics command gives a list of topics. If you type any of the topic names (such
as LISTS) in the prompt, you get additional information about the topic. Typing q
quits the information screen. Pressing Ctrl + D together returns you to the normal
Python prompt:
>>>
Pressing Ctrl + D together again ends the Python shell session.
What just happened?
We learned about the Python interactive shell and the Python help system.
[3]
NumPy Quick Start
Basic arithmetic and variable assignment
In the Time for action – using the Python help system section, we used the Python shell to
look up documentation. We can also use Python as a calculator. By the way, this is just a
refresher, so if you are completely new to Python, I recommend taking some time to learn
the basics. If you put your mind to it, learning basic Python should not take you more than a
couple of weeks.
Time for action – using Python as a calculator
We can use Python as a calculator as follows:
1.
In a Python shell, add 2 and 2 as follows:
>>> 2 + 2
4
2.
Multiply 2 and 2 as follows:
>>> 2 * 2
4
3.
Divide 2 and 2 as follows:
>>> 2/2
1
4.
If you have programmed before, you probably know that dividing is a bit tricky since
there are different types of dividing. For a calculator, the result is usually adequate,
but the following division may not be what you were expecting:
>>> 3/2
1
We will discuss what this result is about in several later chapters of this book. Take
the cube of 2 as follows:
>>> 2 ** 3
8
What just happened?
We used the Python shell as a calculator and performed addition, multiplication, division,
and exponentiation.
[4]
Chapter 1
Time for action – assigning values to variables
Assigning values to variables in Python works in a similar way to most programming
languages.
1.
For instance, assign the value of 2 to a variable named var as follows:
>>> var = 2
>>> var
2
2.
We defined the variable and assigned it a value. In this Python code, the type of the
variable is not fixed. We can make the variable in to a list, which is a built-in Python
type corresponding to an ordered sequence of values. Assign a list to var as follows:
>>> var = [2, 'spam', 'eggs']
>>> var
[2, 'spam', 'eggs']
We can assign a new value to a list item using its index number (counting starts from
0). Assign a new value to the first list element:
>>> var
['ham', 'spam', 'eggs']
3.
We can also swap values easily. Define two variables and swap their values:
>>> a = 1
>>> b = 2
>>> a, b = b, a
>>> a
2
>>> b
1
What just happened?
We assigned values to variables and Python list items. This section is by no means
exhaustive; therefore, if you are struggling, please read Appendix B, Additional Online
Resources, to find recommended Python tutorials.
[5]
NumPy Quick Start
The print() function
If you haven't programmed in Python for a while or are a Python novice, you may be
confused about the Python 2 versus Python 3 discussions. In a nutshell, the latest version
Python 3 is not backward compatible with the older Python 2 because the Python
development team felt that some issues were fundamental and therefore warranted a
radical change. The Python team has committed to maintain Python 2 until 2020. This may
be problematic for the people who still depend on Python 2 in some way. The consequence
for the print() function is that we have two types of syntax.
Time for action – printing with the print() function
We can print using the print() function as follows:
1.
The old syntax is as follows:
>>> print 'Hello'
Hello
2.
The new Python 3 syntax is as follows:
>>> print('Hello')
Hello
The parentheses are now mandatory in Python 3. In this book, I try to use the
new syntax as much as possible; however, I use Python 2 to be on the safe side. To
enforce the syntax, each Python 2 script with print() calls in this book starts with:
>>> from __future__ import print_function
3.
Try to use the old syntax to get the following error message:
>>> print 'Hello'
File "", line 1
print 'Hello'
^
SyntaxError: invalid syntax
4.
To print a newline, use the following syntax:
>>> print()
5.
To print multiple items, separate them with commas:
>>> print(2, 'ham', 'egg')
2 ham egg
[6]
Chapter 1
6.
By default, Python separates the printed values with spaces and prints output to the
screen. You can customize these settings. Read more about this function by typing
the following command:
>>> help(print)
You can exit again by typing q.
What just happened?
We learned about the print() function and its relation to Python 2 and Python 3.
Code comments
Commenting code is a best practice with the goal of making code clearer for yourself and
other coders (see https://google-styleguide.googlecode.com/svn/trunk/
pyguide.html?showone=Comments#Comments). Usually, companies and other
organizations have policies regarding code comment such as comment templates. In this
book, I did not comment the code in such a fashion for brevity and because the text in the
book should clarify the code.
Time for action – commenting code
The most basic comment starts with a hash sign and continues until the end of the line:
1.
Comment code with this type of comment as follows:
>>> # Comment from hash to end of line
2.
However, if the hash sign is between single or double quotes, then we have a string,
which is an ordered sequence of characters:
>>> astring = '# This is not a comment'
>>> astring
'# This is not a comment'
3.
We can also comment multiple lines as a block. This is useful if you want to write a
more detailed description of the code. Comment multiple lines as follows:
"""
Chapter 1 of NumPy Beginners Guide.
Another line of comment.
"""
We refer to this type of comment as triple-quoted for obvious reasons.
It also is used to test code. You can read about testing in Chapter 8, Assuring
Quality with Testing.
[7]
NumPy Quick Start
The if statement
The if statement in Python has a bit different syntax to other languages, such as C++ and
Java. The most important difference is that indentation matters, which I hope you are
aware of.
Time for action – deciding with the if statement
We can use the if statement in the following ways:
1.
Check whether a number is negative as follows:
>>> if 42 < 0:
...
print('Negative')
... else:
...
print('Not negative')
...
Not negative
In the preceding example, Python decided that 42 is not negative. The else clause
is optional. The comparison operators are equivalent to the ones in C++, Java, and
similar languages.
2.
Python also has a chained branching logic compound statement for multiple tests
similar to the switch statement in C++, Java, and other programming languages.
Decide whether a number is negative, 0, or positive as follows:
>>> a = -42
>>> if a < 0:
...
print('Negative')
... elif a == 0:
...
print('Zero')
... else:
...
print('Positive')
...
Negative
This time, Python decided that 42 is negative.
What just happened?
We learned how to do branching logic in Python.
[8]
Chapter 1
The for loop
Python has a for statement with the same purpose as the equivalent construct in C++,
Pascal, Java, and other languages. However, the mechanism of looping is a bit different.
Time for action – repeating instructions with loops
We can use the for loop in the following ways:
1.
Loop over an ordered sequence, such as a list, and print each item as follows:
>>> food = ['ham', 'egg', 'spam']
>>> for snack in food:
...
print(snack)
...
ham
egg
spam
2.
And remember that, as always, indentation matters in Python. We loop over a range
of values with the built-in range() or xrange() functions. The latter function is
slightly more efficient in certain cases. Loop over the numbers 1-9 with a step of 2
as follows:
>>> for i in range(1, 9, 2):
...
print(i)
...
1
3
5
7
3.
The start and step parameter of the range() function are optional with default
values of 1. We can also prematurely end a loop. Loop over the numbers 0-9 and
break out of the loop when you reach 3:
>>> for i in range(9):
...
print(i)
...
if i == 3:
...
print('Three')
...
break
[9]
NumPy Quick Start
...
0
1
2
3
Three
4.
The loop stopped at 3 and we did not print the higher numbers. Instead of leaving
the loop, we can also get out of the current iteration. Print the numbers 0-4,
skipping 3 as follows:
>>> for i in range(5):
...
if i == 3:
...
print('Three')
...
continue
...
print(i)
...
0
1
2
Three
4
5.
The last line in the loop was not executed when we reached 3 because of the
continue statement. In Python, the for loop can have an else statement
attached to it. Add an else clause as follows:
>>> for i in range(5):
...
print(i)
... else:
...
print(i, 'in else clause')
...
0
1
2
3
4
(4, 'in else clause')
6.
Python executes the code in the else clause last. Python also has a while loop. I
do not use it that much because the for loop is more useful in my opinion.
[ 10 ]
Chapter 1
What just happened?
We learned how to repeat instructions in Python with loops. This section included the break
and continue statements, which exit and continue looping.
Python functions
Functions are callable blocks of code. We call functions by the name we give them.
Time for action – defining functions
Let's define the following simple function:
1.
Print Hello and a given name in the following way:
>>> def print_hello(name):
...
print('Hello ' + name)
...
Call the function as follows:
>>> print_hello('Ivan')
Hello Ivan
2.
Some functions do not have arguments, or the arguments have default values. Give
the function a default argument value as follows:
>>> def print_hello(name='Ivan'):
...
print('Hello ' + name)
...
>>> print_hello()
Hello Ivan
3.
Usually, we want to return a value. Define a function, which doubles input values
as follows:
>>> def double(number):
...
return 2 * number
...
>>> double(3)
6
[ 11 ]
NumPy Quick Start
What just happened?
We learned how to define functions. Functions can have default argument values and
return values.
Python modules
A file containing Python code is called a module. A module can import other modules,
functions in other modules, and other parts of modules. The filenames of Python modules
end with .py. The name of the module is the same as the filename minus the .py suffix.
Time for action – importing modules
Importing modules can be done in the following manner:
1.
If the filename is, for instance, mymodule.py, import it as follows:
>>> import mymodule
2.
The standard Python distribution has a math module. After importing it, list the
functions and attributes in the module as follows:
>>> import math
>>> dir(math)
['__doc__', '__file__', '__name__', '__package__', 'acos',
'acosh', 'asin', 'asinh', 'atan', 'atan2', 'atanh', 'ceil',
'copysign', 'cos', 'cosh', 'degrees', 'e', 'erf', 'erfc', 'exp',
'expm1', 'fabs', 'factorial', 'floor', 'fmod', 'frexp', 'fsum',
'gamma', 'hypot', 'isinf', 'isnan', 'ldexp', 'lgamma', 'log',
'log10', 'log1p', 'modf', 'pi', 'pow', 'radians', 'sin', 'sinh',
'sqrt', 'tan', 'tanh', 'trunc']
3.
Call the pow() function in the math module:
>>> math.pow(2, 3)
8.0
Notice the dot in the syntax. We can also import a function directly and call it by its
short name. Import and call the pow() function as follows:
>>> from math import pow
>>> pow(2, 3)
8.0
[ 12 ]
Chapter 1
4.
Python lets us define aliases for imported modules and functions. This is a good time
to introduce the import conventions we are going to use for NumPy and a plotting
library we will use a lot:
import numpy as np
import matplotlib.pyplot as plt
What just happened?
We learned about modules, importing modules, importing functions, calling functions in
modules, and the import conventions of this book. This concludes the Python refresher.
NumPy on Windows
Installing NumPy on Windows is straightforward. You only need to download an installer,
and a wizard will guide you through the installation steps.
Time for action – installing NumPy, matplotlib, SciPy, and
IPython on Windows
Installing NumPy on Windows is necessary but this is, fortunately, a straightforward task that
we will cover in detail. It is recommended that you install matplotlib, SciPy, and IPython.
However, this is not required to enjoy this book. The actions we will take are as follows:
1.
Download a NumPy installer for Windows from the SourceForge website
http://sourceforge.net/projects/numpy/files/.
Choose the appropriate NumPy version according to your Python version. In
the preceding screen shot, we chose numpy-1.9.2-win32-superpackpython2.7.exe.
[ 13 ]
NumPy Quick Start
2.
Open the EXE installer by double-clicking on it as shown in the following screen shot:
3.
4.
Now, we can see a description of NumPy and its features. Click on Next.
5.
In this example, Python 2.7 was found. Click on Next if Python is found; otherwise,
click on Cancel and install Python (NumPy cannot be installed without Python).
Click on Next. This is the point of no return. Well, kind of, but it is best to make sure
that you are installing to the proper directory and so on and so forth. Now the real
installation starts. This may take a while.
If you have Python installed, it should automatically be detected. If it is not
detected, your path settings might be wrong. At the end of this chapter, we have
listed resources in case you have problems with installing NumPy.
Install SciPy and matplotlib with the Enthought Canopy distribution (https://
www.enthought.com/products/canopy/). It might be necessary to put the
msvcp71.dll file in your C:\Windows\system32 directory. You can get it from
http://www.dll-files.com/dllindex/dll-files.shtml?msvcp71
A Windows IPython installer is available on the IPython website (see http://
ipython.org/).
What just happened?
We installed NumPy, SciPy, matplotlib, and IPython on Windows.
[ 14 ]
Chapter 1
NumPy on Linux
Installing NumPy and its related recommended software on Linux depends on the
distribution you have. We will discuss how you will install NumPy from the command line,
although you can probably use graphical installers; it depends on your distribution (distro).
The commands to install matplotlib, SciPy, and IPython are the same—only the package
names are different. Installing matplotlib, SciPy, and IPython is recommended, but optional.
Time for action – installing NumPy, matplotlib, SciPy, and
IPython on Linux
Most Linux distributions have NumPy packages. We will go through the necessary commands
for some of the most popular Linux distros:
Installing NumPy on Red Hat: Run the following instructions from the
command line:
$ yum install python-numpy
Installing NumPy on Mandriva: To install NumPy on Mandriva, run the following
command line instruction:
$ urpmi python-numpy
Installing NumPy on Gentoo: To install NumPy on Gentoo, run the following
command line instruction:
$ [sudo] emerge numpy
Installing NumPy on Debian and Ubuntu: On Debian or Ubuntu, type the following
on the command line:
$ [sudo] apt-get install python-numpy
The following table gives an overview of the Linux distributions and the
corresponding package names for NumPy, SciPy, matplotlib, and IPython:
Linux distribution
Arch Linux
NumPy
python-numpy
Debian
python-numpy
Fedora
numpy
Gentoo
dev-python/
numpy
SciPy
pythonscipy
pythonscipy
pythonscipy
scipy
[ 15 ]
matplotlib
pythonmatplotlib
pythonmatplotlib
pythonmatplotlib
matplotlib
IPython
ipython
ipython
ipython
ipython
NumPy Quick Start
Linux distribution
OpenSUSE
Slackware
NumPy
python-numpy,
python-numpydevel
numpy
SciPy
pythonscipy
matplotlib
pythonmatplotlib
IPython
ipython
scipy
matplotlib
ipython
NumPy on Mac OS X
You can install NumPy, matplotlib, and SciPy on the Mac OS X with a GUI installer (not
possible for all versions) or from the command line with a port manager such as MacPorts,
Homebrew, or Fink, depending on your preference. You can also install using a script from
https://github.com/fonnesbeck/ScipySuperpack.
Time for action – installing NumPy, SciPy, matplotlib, and
IPython with MacPorts or Fink
Alternatively, we can install NumPy, SciPy, matplotlib, and IPython through the MacPorts
route or with Fink. The following installation steps show how to install all these packages:
Installing with MacPorts: Type the following command:
$ [sudo] port install py-numpy py-scipy py-matplotlib py-ipython
Installing with Fink: Fink also has packages for NumPy—scipy-core-py24, scipycore-py25, and scipy-core-py26. The SciPy packages are scipy-py24, scipypy25 and scipy-py26. We can install NumPy and the additional recommended
packages, referring to this book on Python 2.7, using the following command:
$ fink install scipy-core-py27 scipy-py27 matplotlib-py27
What just happened?
We installed NumPy and the additional recommended software on Mac OS X with
MacPorts and Fink.
Building from source
We can retrieve the source code for NumPy with git as follows:
$ git clone git://github.com/numpy/numpy.git numpy
Alternatively, download the source from http://sourceforge.net/projects/numpy/
files/.
[ 16 ]
Chapter 1
Install in /usr/local with the following command:
$ python setup.py build
$ [sudo] python setup.py install --prefix=/usr/local
To build, we need a C compiler such as GCC and the Python header files in the python-dev
or python-devel packages.
Arrays
After going through the installation of NumPy, it's time to have a look at NumPy arrays.
NumPy arrays are more efficient than Python lists when it comes to numerical operations.
NumPy code requires less explicit loops than the equivalent Python code.
Time for action – adding vectors
Imagine that we want to add two vectors called a and b (see https://www.khanacademy.
org/science/physics/one-dimensional-motion/displacement-velocitytime/v/introduction-to-vectors-and-scalars). Vector is used here in the
mathematical sense meaning a one-dimensional array. We will learn in Chapter 5, Working
with Matrices and ufuncs, about specialized NumPy arrays, which represent matrices. Vector
a holds the squares of integers 0 to n, for instance, if n is equal to 3, then a is equal to (0,1,
4). Vector b holds the cubes of integers 0 to n, so if n is equal to 3, then b is equal to (0,1,
8). How will you do that using plain Python? After we come up with a solution, we will
compare it to the NumPy equivalent.
1.
Adding vectors using pure Python: The following function solves the vector addition
problem using pure Python without NumPy:
def pythonsum(n):
a = range(n)
b = range(n)
c = []
for i in range(len(a)):
a[i] = i ** 2
b[i] = i ** 3
c.append(a[i] + b[i])
return c
[ 17 ]
NumPy Quick Start
Downloading the example code files
You can download the example code files from your account at
http://www.packtpub.com for all the Packt Publishing books you
have purchased. If you purchased this book elsewhere, you can visit
http://www.packtpub.com/support and register to have the
files e-mailed directly to you.
2.
Adding vectors using NumPy: Following is a function that achieves the same result
with NumPy:
def
a
b
c
numpysum(n):
= np.arange(n) ** 2
= np.arange(n) ** 3
= a + b
return c
Notice that numpysum() does not need a for loop. Also, we used the arange() function
from NumPy that creates a NumPy array for us with integers 0 to n. The arange() function
was imported; that is why it is prefixed with numpy (actually, it is customary to abbreviate it
via an alias to np).
Now comes the fun part. The preface mentions that NumPy is faster when it comes to
array operations. How much faster is NumPy, though? The following program will show us
by measuring the elapsed time, in microseconds, for the numpysum() and pythonsum()
functions. It also prints the last two elements of the vector sum. Let's check that we get the
same answers by using Python and NumPy:
#!/usr/bin/env/python
from __future__ import print_function
import sys
from datetime import datetime
import numpy as np
"""
Chapter 1 of NumPy Beginners Guide.
This program demonstrates vector addition the Python way.
Run from the command line as follows
python vectorsum.py n
where n is an integer that specifies the size of the vectors.
[ 18 ]
Chapter 1
The first vector to be added contains the squares of 0 up to n.
The second vector contains the cubes of 0 up to n.
The program prints the last 2 elements of the sum and the elapsed
time.
"""
def numpysum(n):
a = np.arange(n) ** 2
b = np.arange(n) ** 3
c = a + b
return c
def pythonsum(n):
a = range(n)
b = range(n)
c = []
for i in range(len(a)):
a[i] = i ** 2
b[i] = i ** 3
c.append(a[i] + b[i])
return c
size = int(sys.argv[1])
start = datetime.now()
c = pythonsum(size)
delta = datetime.now() - start
print("The last 2 elements of the sum", c[-2:])
print("PythonSum elapsed time in microseconds", delta.microseconds)
start = datetime.now()
c = numpysum(size)
delta = datetime.now() - start
print("The last 2 elements of the sum", c[-2:])
print("NumPySum elapsed time in microseconds", delta.microseconds)
[ 19 ]
NumPy Quick Start
The output of the program for 1000, 2000, and 3000 vector elements is as follows:
$ python vectorsum.py 1000
The last 2 elements of the sum [995007996, 998001000]
PythonSum elapsed time in microseconds 707
The last 2 elements of the sum [995007996 998001000]
NumPySum elapsed time in microseconds 171
$ python vectorsum.py 2000
The last 2 elements of the sum [7980015996, 7992002000]
PythonSum elapsed time in microseconds 1420
The last 2 elements of the sum [7980015996 7992002000]
NumPySum elapsed time in microseconds 168
$ python vectorsum.py 4000
The last 2 elements of the sum [63920031996, 63968004000]
PythonSum elapsed time in microseconds 2829
The last 2 elements of the sum [63920031996 63968004000]
NumPySum elapsed time in microseconds 274
What just happened?
Clearly, NumPy is much faster than the equivalent normal Python code. One thing is certain,
we get the same results whether we use NumPy or not. However, the result printed differs
in representation. Notice that the result from the numpysum() function does not have any
commas. How come? Obviously, we are not dealing with a Python list but with a NumPy
array. It was mentioned in the Preface that NumPy arrays are specialized data structures for
numerical data. We will learn more about NumPy arrays in the next chapter.
Pop quiz – Functioning of the arange() function
Q1. What does arange(5) do?
1. Creates a Python list of 5 elements with the values 1-5.
2. Creates a Python list of 5 elements with the values 0-4.
3. Creates a NumPy array with the values 1-5.
4. Creates a NumPy array with the values 0-4.
5. None of the above.
[ 20 ]
Chapter 1
Have a go hero – continue the analysis
The program we used to compare the speed of NumPy and regular Python is not very
scientific. We should at least repeat each measurement a couple of times. It will be nice to
be able to calculate some statistics such as average times. Also, you might want to show
plots of the measurements to friends and colleagues.
Hints to help can be found in the online documentation and the resources listed
at the end of this chapter. NumPy has statistical functions that can calculate
averages for you. I recommend using matplotlib to produce plots. Chapter 9,
Plotting with matplotlib, gives a quick overview of matplotlib.
IPython – an interactive shell
Scientists and engineers are used to experiment. Scientists created IPython with
experimentation in mind. Many view the interactive environment that IPython provides
as a direct answer to MATLAB, Mathematica, and Maple. You can find more information,
including installation instructions, at http://ipython.org/.
IPython is free, open source, and available for Linux, UNIX, Mac OS X, and Windows. The
IPython authors only request that you cite IPython in any scientific work that uses IPython.
The following is a list of the basic IPython features:
Tab completion
History mechanism
Inline editing
Ability to call external Python scripts with %run
Access to system commands
Pylab switch
Access to Python debugger and profiler
The Pylab switch imports all the SciPy, NumPy, and matplotlib packages. Without this switch,
we will have to import every package we need ourselves.
[ 21 ]
NumPy Quick Start
All we need to do is enter the following instruction on the command line:
$ ipython --pylab
IPython 2.4.1 -- An enhanced Interactive Python.
?
-> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help
-> Python's own help system.
object?
-> Details about 'object', use 'object??' for extra details.
Using matplotlib backend: MacOSX
In [1]: quit()
The quit()command or Ctrl + D quits the IPython shell. We may want to be able to go back
to our experiments. In IPython, it is easy to save a session for later:
In [1]: %logstart
Activating auto-logging. Current session state plus future input saved.
Filename
: ipython_log.py
Mode
: rotate
Output logging : False
Raw input log
: False
Timestamping
: False
State
: active
Let's say we have the vector addition program that we made in the current directory. Run
the script as follows:
In [1]: ls
README
vectorsum.py
In [2]: %run -i vectorsum.py 1000
As you probably remember, 1000 specifies the number of elements in a vector. The -d
switch of %run starts an ipdb debugger with c the script is started. n steps through the
code. Typing quit at the ipdb prompt exits the debugger:
In [2]: %run -d vectorsum.py 1000
*** Blank or comment
*** Blank or comment
Breakpoint 1 at: /Users/…/vectorsum.py:3
[ 22 ]
Chapter 1
Enter c at the ipdb> prompt to start your script.
>(1)()
ipdb> c
> /Users/…/vectorsum.py(3)()
2
1---> 3 import sys
4 from datetime import datetime
ipdb> n
>
/Users/…/vectorsum.py(4)()
1
3 import sys
----> 4 from datetime import datetime
5 import numpy
ipdb> n
> /Users/…/vectorsum.py(5)()
4 from datetime import datetime
----> 5 import numpy
6
ipdb> quit
We can also profile our script by passing the -p option to %run:
In [4]: %run -p vectorsum.py 1000
1058 function calls (1054 primitive calls) in 0.002 CPU seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
1 0.001
0.001
0.001
0.001 vectorsum.py:28(pythonsum)
1 0.001
0.001
0.002
0.002 {execfile}
1000 0.000
0.0000.0000.000 {method 'append' of 'list' objects}
1 0.000
0.000
0.002
0.002 vectorsum.py:3()
1 0.000
0.0000.0000.000 vectorsum.py:21(numpysum)
3
0.000
0.0000.0000.000 {range}
1
0.000
0.0000.0000.000 arrayprint.py:175(_array2string)
3/1
0.000
0.0000.0000.000 arrayprint.py:246(array2string)
[ 23 ]
NumPy Quick Start
2
0.000
0.0000.0000.000 {method 'reduce' of 'numpy.ufunc' objects}
4
0.000
0.0000.0000.000 {built-in method now}
2
0.000
0.0000.0000.000 arrayprint.py:486(_formatInteger)
2
0.000
0.0000.0000.000 {numpy.core.multiarray.arange}
1
0.000
0.0000.0000.000 arrayprint.py:320(_formatArray)
3/1
0.000
0.0000.0000.000 numeric.py:1390(array_str)
1
0.000
0.0000.0000.000 numeric.py:216(asarray)
2
0.000
0.0000.0000.000 arrayprint.py:312(_extendLine)
1
0.000
0.0000.0000.000 fromnumeric.py:1043(ravel)
2
0.000
0.0000.0000.000 arrayprint.py:208()
1
0.000
0.000
11
0.000
0.002
0.002:1()
0.0000.0000.000 {len}
2
0.000
0.0000.0000.000 {isinstance}
1
0.000
0.0000.0000.000 {reduce}
1
0.000
0.0000.0000.000 {method 'ravel' of 'numpy.ndarray' objects}
4
0.000
0.0000.0000.000 {method 'rstrip' of 'str' objects}
3
0.000
0.0000.0000.000 {issubclass}
2
0.000
0.0000.0000.000 {method 'item' of 'numpy.ndarray' objects}
1
0.000
0.0000.0000.000 {max}
1
0.000
objects}
0.0000.0000.000 {method 'disable' of '_lsprof.Profiler'
This gives us a bit more insight in to the workings of our program. In addition, we can now
identify performance bottlenecks. The %hist command shows the commands history:
In [2]: a=2+2
In [3]: a
Out[3]: 4
In [4]: %hist
1: _ip.magic("hist ")
2: a=2+2
3: a
I hope you agree that IPython is a really useful tool!
[ 24 ]
Chapter 1
Online resources and help
When we are in IPython's pylab mode, we can open manual pages for NumPy functions
with the help command. It is not necessary to know the name of a function. We can type
a few characters and then let tab completion do its work. Let's, for instance, browse the
available information for the arange() function:
In [2]: help ar
In [2]: help arange
Another option is to put a question mark behind the function name:
In [3]: arange?
The main documentation website for NumPy and SciPy is at http://docs.scipy.org/
doc/. Through this web page, we can browse the NumPy reference at http://docs.
scipy.org/doc/numpy/reference/, the user guide, and several tutorials.
The popular Stack Overflow software development forum has hundreds of questions tagged
numpy. To view them, go to http://stackoverflow.com/questions/tagged/numpy.
If you are really stuck with a problem or you want to be kept informed of NumPy
development, you can subscribe to the NumPy discussion mailing list. The e-mail address
is numpy-discussion@scipy.org. The number of e-mails per day is not too high with
almost no spam to speak of. Most importantly, the developers actively involved with NumPy
also answer questions asked on the discussion group. The complete list can be found at
http://www.scipy.org/scipylib/mailing-lists.html.
For IRC users, there is an IRC channel on irc://irc.freenode.net. The channel is called
#scipy, but you can also ask NumPy questions since SciPy users also have knowledge of
NumPy, as SciPy is based on NumPy. There are at least 50 members on the SciPy channel at
all times.
[ 25 ]
NumPy Quick Start
Summary
In this chapter, we installed NumPy and other recommended software that we will be using
in some sections of this book. We got a vector addition program working and convinced
ourselves that NumPy has superior performance. You were introduced to the IPython
interactive shell. In addition, you explored the available NumPy documentation and
online resources.
In the next chapter, you will take a look under the hood and explore some fundamental
concepts including arrays and data types.
[ 26 ]
2
Beginning with NumPy Fundamentals
After installing NumPy and getting some code to work, it's time to cover
NumPy basics.
The topics we shall cover in this chapter are as follows:
Data types
Array types
Type conversions
Array creation
Indexing
Slicing
Shape manipulation
Before we start, let me make a few remarks about the code examples in this chapter. The
code snippets in this chapter show input and output from several IPython sessions. Recall
that IPython was introduced in Chapter 1, NumPy Quick Start, as the interactive Python shell
of choice for scientific computing. The advantages of IPython are the --pylab switch that
imports many scientific computing Python packages, including NumPy, and the fact that
it is not necessary to explicitly call the print() function to display variable values. Other
features include easy parallel computation and the notebook interface in the form of
a persistent worksheet in a web browser.
However, the source code delivered alongside the book is a regular Python code that uses
import and print statements.
[ 27 ]
Beginning with NumPy Fundamentals
NumPy array object
NumPy has a multidimensional array object called ndarray. It consists of two parts:
The actual data
Some metadata describing the data
The majority of array operations leave the raw data untouched. The only aspect that changes
is the metadata.
In the previous chapter, we have already learned how to create an array using the arange()
function. Actually, we created a one-dimensional array that contained a set of numbers.
The ndarray object can have more than one dimension.
The NumPy array is in general homogeneous (there is a special array type that is
heterogeneous as described in the Time for action – creating a record data type section)—the
items in the array have to be of the same type. The advantage is that, if we know that the items
in the array are of the same type, it is easy to determine the storage size required for the array.
NumPy arrays are indexed starting from 0, just like in Python. Data types are represented by
special objects. We will discuss these objects comprehensively in this chapter.
Let's create an array with the arange() function again. Get the data type of an array using
the following code:
In: a = arange(5)
In: a.dtype
Out: dtype('int64')
The data type of array a is int64 (at least on my machine), but you may get int32 as
output if you are using 32-bit Python. In both the cases, we are dealing with integers
(64-bit or 32-bit). Besides the data type of an array, it is important to know its shape.
In Chapter 1, NumPy Quick Start, we demonstrated how to create a vector (actually,
a one-dimensional NumPy array). A vector is commonly used in mathematics, but most
of the time, we need higher dimensional objects. Determine the shape of the vector we
created a few minutes ago. The following code is an example of creating a vector:
In [4]: a
Out[4]: array([0, 1, 2, 3, 4])
In: a.shape
Out: (5,)
As you can see, the vector has five elements with values ranging from 0 to 4. The shape
attribute of the array is a tuple, in this case a tuple of 1 element, which contains the length
in each dimension.
[ 28 ]
Chapter 2
A tuple in Python is an immutable (it can't change) sequence of values. Once
tuples are created, we are not allowed to change the values of tuple elements
or append new elements. This makes tuples safer than lists because you can't
mutate them by accident. A common use case for tuples is as return value of
functions. For more examples, have a look at the Introducing Tuples section of
Chapter 3, Dive into Python, available at http://www.diveintopython.
net/native_data_types/tuples.html.
Time for action – creating a multidimensional array
Now that we know how to create a vector, we are ready to create a multidimensional NumPy
array. After we create the array, we will again want to display its shape:
1.
Create a two-by-two array:
In: m = array([arange(2), arange(2)])
In: m
Out:
array([[0, 1],
[0, 1]])
2.
Show the array shape:
In: m.shape
Out: (2, 2)
What just happened?
We created a two-by-two array with the arange() and array() functions we have come to
trust and love. Without any warning, the array() function appeared on the stage.
The array() function creates an array from an object that you give to it. The object needs
to be array-like, for instance, a Python list. In the preceding example, we passed in a list of
arrays. The object is the only required argument of the array() function. NumPy functions
tend to have a lot of optional arguments with predefined defaults. View the documentation
for this function from the IPython shell with the help() function given here:
In [1]: help(array)
Or use the following shorthand:
In [2]: array?
Of course, you can substitute array in this example with another NumPy function you are
interested in.
[ 29 ]
Beginning with NumPy Fundamentals
Pop quiz – the shape of ndarray
Q1. How is the shape of an ndarray stored?
1. It is stored in a comma-separated string.
2. It is stored in a list.
3. It is stored in a tuple.
Have a go hero – create a three-by-three array
It shouldn't be too hard now to create a three-by-three array. Give it a go and check whether
the array shape is as expected.
Selecting elements
From time to time, we will want to select a particular element of an array. We will take a look
at how to do this, but, first, create a two-by-two array again:
In: a = array([[1,2],[3,4]])
In: a
Out:
array([[1, 2],
[3, 4]])
The array was created this time by passing a list of lists to the array() function. We will
now select one by one each item of the matrix. Remember, the indices are numbered
starting from 0:
In: a[0,0]
Out: 1
In: a[0,1]
Out: 2
In: a[1,0]
Out: 3
In: a[1,1]
Out: 4
As you can see, selecting elements of the array is pretty simple. For the array a, we just use
the notation a[m,n], where m and n are the indices of the item in the array (the array can
have even more dimensions than in this example). This screenshot shows a simple example
of an array:
[ 30 ]
Chapter 2
NumPy numerical types
Python has an integer type, a float type, and a complex type; however, this is not enough
for scientific computing and, for this reason, NumPy has a lot more data types with varying
precision, dependent on memory requirements.
Integers represent whole numbers, such as -1, 0, and 1. Floating-point
numbers correspond to real numbers as used in mathematics, for example,
fractions or irrational numbers such as pi. Because of the way computers
work, we are able to represent integers exactly, but floating-point numbers
are approximated. Complex numbers can have an imaginary component
usually denoted with i or j. By definition, i is the square root of -1. For
instance, 2.5 + 3.7i is a complex number (for more information, refer
to https://www.khanacademy.org/math/precalculus/
imaginary_complex_precalc).
In practice, we need even more types with varying precision and, therefore, different
memory size of the type. The majority of the NumPy numerical types end with a number.
This number indicates the number of bits associated with the type. The following table
(adapted from the NumPy user guide) gives an overview of NumPy numerical types:
Type
bool
Description
inti
Platform integer (normally either int32 or int64)
int8
Byte (-128 to 127)
int16
Integer (-32768 to 32767)
int32
Integer (-2 ** 31 to 2 ** 31 -1)
int64
Integer (-2 ** 63 to 2 ** 63 -1)
uint8
Unsigned integer (0 to 255)
uint16
Unsigned integer (0 to 65535)
uint32
Unsigned integer (0 to 2 ** 32 - 1)
uint64
Unsigned integer (0 to 2 ** 64 - 1)
float16
Half precision float: sign bit, 5 bits exponent, 10 bits mantissa
float32
Single precision float: sign bit, 8 bits exponent, 23 bits mantissa
float64 or float
Double precision float: sign bit, 11 bits exponent, 52 bits mantissa
Boolean (True or False) stored as a bit
[ 31 ]
Beginning with NumPy Fundamentals
Type
complex64
Description
complex128 or
complex
Complex number, represented by two 64-bit floats (real and
imaginary components)
Complex number, represented by two 32-bit floats (real and
imaginary components)
For floating-point types, we can request information with the finfo() function given here:
In: finfo(float16)
Out: finfo(resolution=0.0010004, min=-6.55040e+04, max=6.55040e+04,
dtype=float16)
For each data type, there exists a corresponding conversion function:
In: float64(42)
Out: 42.0
In: int8(42.0)
Out: 42
In: bool(42)
Out: True
In: bool(0)
Out: False
In: bool(42.0)
Out: True
In: float(True)
Out: 1.0
In: float(False)
Out: 0.0
Many functions have a data type argument, which is often optional:
In: arange(7, dtype=uint16)
Out: array([0, 1, 2, 3, 4, 5, 6], dtype=uint16)
It is important to know that you are not allowed to convert a complex number into an integer
or float. Trying to do that triggers a TypeError, as shown in the following screenshot:
The same goes for conversion of a complex number into a float.
[ 32 ]
Chapter 2
An exception in Python is an abnormal condition, which we usually try
to avoid. A TypeError is a Python built-in exception, occurring when
we specify the wrong type for an argument.
The j part is the imaginary coefficient of the complex number. However, you can convert a
float in to a complex number, for instance, complex(1.0).
Data type objects
Data type objects are instances of the numpy.dtype class. Once again, arrays have a data
type. To be precise, every element in a NumPy array has the same data type. The data type
object can tell you the size of the data in bytes. The size in bytes is given by the itemsize
attribute of the dtype class:
In: a.dtype.itemsize
Out: 8
Character codes
Character codes are included for backward compatibility with Numeric. Numeric is
the predecessor of NumPy. Their use is not recommended, but the codes are provided
here because they pop up in several places. We should instead use the dtype objects.
The table shows the character codes:
Type
Integer
Character code
i
Unsigned integer
u
Single precision float
f
Double precision float
d
Boolean
b
Complex
D
String
S
Unicode
U
Void
V
Look at the following code to create an array of single precision floats:
In: arange(7, dtype='f')
Out: array([ 0., 1., 2.,
3.,
4.,
5.,
Likewise this creates an array of complex numbers.
[ 33 ]
6.], dtype=float32)
Beginning with NumPy Fundamentals
In: arange(7, dtype='D')
Out: array([ 0.+0.j, 1.+0.j,
6.+0.j])
2.+0.j,
3.+0.j,
4.+0.j,
5.+0.j,
The dtype constructors
Python classes have functions, which are called methods, if they belong to a class. Some of
these methods are special and used to create new objects. These specialized methods are
called constructors.
You can read more about Python classes at https://docs.python.
org/2/tutorial/classes.html.
We have a variety of ways to create data types. Take the case of floating point data:
Use the general Python float:
In: dtype(float)
Out: dtype('float64')
Specify a single precision float with a character code:
In: dtype('f')
Out: dtype('float32')
Use a double precision float character code:
In: dtype('d')
Out: dtype('float64')
We can give the data type constructor a two-character code. The first character
signifies the type and the second character is a number specifying the number of
bytes in the type (the numbers 2, 4, and 8 correspond to 16, 32, and 64-bit floats):
In: dtype('f8')
Out: dtype('float64')
A listing of all full data type names can be found with the sctypeDict.keys() function:
In: sctypeDict.keys()
Out: [0, …
'i2',
'int0']
[ 34 ]
Chapter 2
The dtype attributes
The dtype class has a number of useful attributes. For example, get information about the
character code of a data type through the attributes of dtype:
In: t = dtype('Float64')
In: t.char
Out: 'd'
The type attribute corresponds to the type of object of the array elements:
In: t.type
Out:
The str attribute of the dtype class gives a string representation of the data type. It starts
with a character representing endianness, if appropriate, then a character code, followed by
a number corresponding to the number of bytes that each array item requires. Endianness,
here, refers to the way bytes are ordered within a 32- or 64-bit word. In big-endian order, the
most significant byte is stored first, indicated by >. In little-endian order, the least significant
byte is stored first, indicated by <:
In: t.str
Out: '
In: for item in f: print item
.....:
0
1
2
3
It is possible to get an element directly with the flatiter object:
In: b.flat[2]
Out: 2
And, it is also possible to directly get multiple elements:
In: b.flat[[1,3]]
Out: array([1, 3])
The flat attribute is settable. Setting the value of the flat attribute leads to
overwriting the values of the whole array:
In: b.flat = 7
In: b
Out:
array([[7, 7],
[7, 7]])
Or, it can also lead to overwriting the values of selected elements:
In: b.flat[[1,3]] = 1
In: b
Out:
array([[7, 1],
[7, 1]])
The following diagram shows the different types of attributes of the ndarray class:
[ 50 ]
Chapter 2
Time for action – converting arrays
Convert a NumPy array to a Python list with the tolist() function:
1.
Convert to a list:
In: b
Out: array([ 1.+1.j, 3.+2.j])
In: b.tolist()
Out: [(1+1j), (3+2j)]
2.
The astype() function converts the array to an array of the specified type:
In: b
Out: array([ 1.+1.j, 3.+2.j])
In: b.astype(int)
/usr/local/bin/ipython:1: ComplexWarning: Casting complex values
to real discards the imaginary part
#!/usr/bin/python
Out: array([1, 3])
We are losing the imaginary part when casting from the NumPy complex
type (not the plain vanilla Python one) to int. The astype() function
also accepts the name of a type as a string.
In: b.astype('complex')
Out: array([ 1.+1.j, 3.+2.j])
It won't show any warning this time because we used the proper data type.
What just happened?
We converted NumPy arrays to a list and to arrays of different data types. The code for this
example is in the arrayconversion.py file in this book's code bundle.
Summary
In this chapter, you learned a lot about NumPy fundamentals: data types and arrays. Arrays
have several attributes describing them. You learned that one of these attributes is the data
type, which, in NumPy, is represented by a fully-fledged object.
NumPy arrays can be sliced and indexed in an efficient manner, just like Python lists. NumPy
arrays have the added ability of working with multiple dimensions.
[ 51 ]
Beginning with NumPy Fundamentals
The shape of an array can be manipulated in many ways—stacking, resizing, reshaping,
and splitting. A great number of convenience functions for shape manipulation were
demonstrated in this chapter.
Having learned about the basics, it's time to move on to the study of commonly used
functions in Chapter 3, Getting Familiar with Commonly Used Functions, which includes
basic statistical and mathematical functions.
[ 52 ]
3
Getting Familiar with Commonly
Used Functions
In this chapter, we will have a look at common NumPy functions. In particular,
we will learn how to load data from files by using an example involving
historical stock prices. Also, we will get to see the basic NumPy mathematical
and statistical functions.
We will learn how to read from and write to files. Also, we will get a taste of the
functional programming and linear algebra possibilities in NumPy.
In this chapter, we shall cover the following topics:
Functions working on arrays
Loading arrays from files
Writing arrays to files
Simple mathematical and statistical functions
File I/O
First, we will learn about file I/O with NumPy. Data is usually stored in files. You would not
get far if you were not able to read from and write to files.
[ 53 ]
Getting Familiar with Commonly Used Functions
Time for action – reading and writing files
As an example of file I/O, we will create an identity matrix and store its contents in a file.
In this and other chapters, we will use the following line by convention
to import NumPy:
import numpy as np
Perform the following steps to do so:
1.
The identity matrix is a square matrix with ones on the main diagonal and zeros for
the rest (see https://www.khanacademy.org/math/precalculus/precalcmatrices/zero-identity-matrix-tutorial/v/identity-matrix).
The identity matrix can be created with the eye() function. The only argument that
we need to give the eye() function is the number of ones. So, for instance, for a
two-by-two matrix, write the following code:
i2 = np.eye(2)
print(i2)
The output is:
[[ 1. 0.]
[ 0. 1.]]
2.
Save the data in a plain text file with the savetxt() function. Specify the name of
the file that we want to save the data in and the array containing the data itself:
np.savetxt("eye.txt", i2)
A file called eye.txt should have been created in the same directory as the Python script.
What just happened?
Reading and writing files is a necessary skill for data analysis. We wrote to a file with
savetxt(). We made an identity matrix with the eye() function.
Instead of a filename, we can also provide a file handle. A file handle is a term
in many programming languages, which means a variable pointing to a file, like
a postal address. For more information on how to get a file handle in Python,
please refer to http://www.diveintopython3.net/files.html.
[ 54 ]
Chapter 3
You can check for yourself whether the contents are as expected. The code for this example
can be downloaded from the book support website: https://www.packtpub.com/
books/content/support (see save.py)
import numpy as np
i2 = np.eye(2)
print(i2)
np.savetxt("eye.txt", i2))
Comma-seperated value files
Files in the Comma-seperated value (CSV) format are encountered quite frequently. Often,
the CSV file is just a dump from a database. Usually, each field in the CSV file corresponds to
a database table column. As we all know, spreadsheet programs, such as Excel, can produce
CSV files, as well.
Time for action – loading from CSV files
How do we deal with CSV files? Luckily, the loadtxt() function can conveniently read CSV
files, split up the fields, and load the data into NumPy arrays. In the following example, we
will load historical stock price data for Apple (the company, not the fruit). The data is in CSV
format and is part of the code bundle for this book. The first column contains a symbol that
identifies the stock. In our case, it is AAPL. Second is the date in dd-mm-yyyy format. The
third column is empty. Then, in order, we have the open, high, low, and close price. Last,
but not least, is the trading volume of the day. This is what a line looks like:
AAPL,28-01-2011, ,344.17,344.4,333.53,336.1,21144800
For now, we are only interested in the close price and volume. In the preceding sample, that
will be 336.1 and 21144800. Store the close price and volume in two arrays as follows:
c,v=np.loadtxt('data.csv', delimiter=',', usecols=(6,7), unpack=True)
As you can see, data is stored in the data.csv file. We have set the delimiter to, (comma),
since we are dealing with a CSV file. The usecols parameter is set through a tuple to get
the seventh and eighth fields, which correspond to the close price and volume. The unpack
argument is set to True, which means that data will be unpacked and assigned to the c and
v variables that will hold the close price and volume, respectively.
[ 55 ]
Getting Familiar with Commonly Used Functions
Volume Weighted Average Price
Volume Weighted Average Price (VWAP) is a very important quantity in finance. It represents
an average price for a financial asset (see https://www.khanacademy.org/math/
probability/descriptive-statistics/old-stats-videos/v/statistics-theaverage). The higher the volume, the more significant a price move typically is. VWAP is
often used in algorithmic trading and is calculated using volume values as weights.
Time for action – calculating Volume Weighted Average Price
The following are the actions that we will take:
1.
2.
Read the data into arrays.
Calculate VWAP:
from __future__ import print_function
import numpy as np
c,v=np.loadtxt('data.csv', delimiter=',', usecols=(6,7),
unpack=True)
vwap = np.average(c, weights=v)
print("VWAP =", vwap)
The output is as follows:
VWAP = 350.589549353
What just happened?
That wasn't very hard, was it? We just called the average() function and set its weights
parameter to use the v array for weights. By the way, NumPy also has a function to calculate
the arithmetic mean. This is an unweighted average with all the weights equal to 1.
The mean() function
The mean() function is quite friendly and not so mean. This function calculates the
arithmetic mean of an array.
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Chapter 3
The arithmetic mean is given by the following formula:
1 n
∑ ai
n i =1
It sums the values in an array a and divides the sum by the number
of elements n (see https://www.khanacademy.org/math/
probability/descriptive-statistics/central_
tendency/e/mean_median_and_mode).
Let's see it in action:
print("mean =", np.mean(c))
As a result, we get the following printout:
mean = 351.037666667
Time-weighted average price
In finance, time-weighted average price (TWAP) is another average price measure. Now that
we are at it, let's compute the TWAP too. It is just a variation on a theme really. The idea is
that recent price quotes are more important, so we should give recent prices higher weights.
The easiest way is to create an array with the arange() function of increasing values from
zero to the number of elements in the close price array. This is not necessarily the correct
way. In fact, most of the examples concerning stock price analysis in this book are only
illustrative. The following is the TWAP code:
t = np.arange(len(c))
print("twap =", np.average(c, weights=t))
It produces the following output:
twap = 352.428321839
The TWAP is even higher than the mean.
Pop quiz – computing the weighted average
Q1. Which function returns the weighted average of an array?
1.
2.
3.
4.
weighted average
waverage
average
avg
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Getting Familiar with Commonly Used Functions
Have a go hero – calculating other averages
Try doing the same calculation using the open price. Calculate the mean for the volume and
the other prices.
Value range
Usually, we don't only want to know the average or arithmetic mean of a set of values,
which are in the middle, to know we also want the extremes, the full range—the highest and
lowest values. The sample data that we are using here already has those values per day—the
high and low price. However, we need to know the highest value of the high price and the
lowest price value of the low price.
Time for action – finding highest and lowest values
The min() and max() functions are the answer for our requirement. Perform the following
steps to find the highest and lowest values:
1.
First, read our file again and store the values for the high and low prices into arrays:
h,l=np.loadtxt('data.csv', delimiter=',', usecols=(4,5),
unpack=True)
The only thing that changed is the usecols parameter, since the high and low
prices are situated in different columns.
2.
The following code gets the price range:
print("highest =", np.max(h))
print("lowest =", np.min(l))
These are the values returned:
highest = 364.9
lowest = 333.53
Now, it's easy to get a midpoint, so it is left as an exercise for you to attempt.
3.
NumPy allows us to compute the spread of an array with a function called ptp().
The ptp() function returns the difference between the maximum and minimum
values of an array. In other words, it is equal to max(array)—min(array). Call
the ptp() function:
print("Spread high price", np.ptp(h))
print("Spread low price", np.ptp(l))
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Chapter 3
You will see this text printed:
Spread high price 24.86
Spread low price 26.97
What just happened?
We defined a range of highest to lowest values for the price. The highest value was given by
applying the max() function to the high price array. Similarly, the lowest value was found
by calling the min() function to the low price array. We also calculated the peak-to-peak
distance with the ptp() function:
from __future__ import print_function
import numpy as np
h,l=np.loadtxt('data.csv', delimiter=',', usecols=(4,5), unpack=True)
print("highest =", np.max(h))
print("lowest =", np.min(l))
print((np.max(h) + np.min(l)) /2)
print("Spread high price", np.ptp(h))
print("Spread low price", np.ptp(l))
Statistics
Stock traders are interested in the most probable close price. Common sense says that
this should be close to some kind of an average as the price dances around a mean, due to
random fluctuations. The arithmetic mean and weighted average are ways to find the center
of a distribution of values. However, neither are robust and both are sensitive to outliers.
Outliers are extreme values that are much bigger or smaller than the typical values in a
dataset. Usually, outliers are caused by a rare phenomenon or a measurement error. For
instance, if we have a close price value of a million dollars, this will influence the outcome
of our calculations.
Time for action – performing simple statistics
We can use some kind of threshold to weed out outliers, but there is a better way. It is called
the median, and it basically picks the middle value of a sorted set of values (see https://
www.khanacademy.org/math/probability/descriptive-statistics/central_
tendency/e/mean_median_and_mode). One half of the data is below the median and the
other half is above it. For example, if we have the values of 1, 2, 3, 4, and 5, then the median
will be 3, since it is in the middle.
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Getting Familiar with Commonly Used Functions
These are the steps to calculate the median:
1.
Create a new Python script and call it simplestats.py. You already know how to
load the data from a CSV file into an array. So, copy that line of code and make sure
that it only gets the close price. The code should appear like this:
c=np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
2.
The function that will do the magic for us is called median(). We will call it and
print the result immediately. Add the following line of code:
print("median =", np.median(c))
The program prints the following output:
median = 352.055
3.
Since it is our first time using the median() function, we would like to check
whether this is correct. Obviously, we can do it by just going through the file and
finding the correct value, but that is no fun. Instead, we will just mimic the median
algorithm by sorting the close price array and printing the middle value of the sorted
array. The msort() function does the first part for us. Call the function, store the
sorted array, and then print it:
sorted_close = np.msort(c)
print("sorted =", sorted_close)
This prints the following output:
Yup, it works! Let's now get the middle value of the sorted array:
N = len(c)
print "middle =", sorted[(N - 1)/2]
The preceding snippet gives us the following output:
middle = 351.99
4.
Hey, that's a different value than the one the median() function gave us. How
come? Upon further investigation, we find that the median() function return value
doesn't even appear in our file. That's even stranger! Before filing bugs with the
NumPy team, let's have a look at the documentation:
$ python
>>> import numpy as np
>>> help(np.median)
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Chapter 3
This mystery is easy to solve. It turns out that our naive algorithm only works for
arrays with odd lengths. For even-length arrays, the median is calculated from the
average of the two array values in the middle. Therefore, type the following code:
print("average middle =", (sorted[N /2] + sorted[(N - 1) / 2]) /
2)
This prints the following output:
average middle = 352.055
5.
Another statistical measure that we are concerned with is variance. Variance tells
us how much a variable varies (see https://www.khanacademy.org/math/
probability/descriptive-statistics/variance_std_deviation/e/
variance). In our case, it also tells us how risky an investment is, since a stock
price that varies too wildly is bound to get us into trouble.
Calculate the variance of the close price (with NumPy, this is just a one-liner):
print("variance =", np.var(c))
This gives us the following output:
variance = 50.1265178889
6.
Not that we don't trust NumPy or anything, but let's double-check using the
definition of variance, as found in the documentation. Mind you, this definition
might be different than the one in your statistics book, but that is quite common
in the field of statistics.
The population variance is defined as the mean
of the square of deviations from the mean, divided by the
number of elements in the array:
1 n
2
a
−
mean
(
)
∑
i
n i =1
Some books tell us to divide by the number of elements in the array minus one (this
is called a sample variance):
print("variance from definition =", np.mean((c - c.mean())**2))
The output is as follows:
variance from definition = 50.1265178889
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Getting Familiar with Commonly Used Functions
What just happened?
Maybe you noticed something new. We suddenly called the mean() function on the c
array. Yes, this is legal, because the ndarray class has a mean() method. This is for your
convenience. For now, just keep in mind that this is possible. The code for this example
can be found in simplestats.py:
from __future__ import print_function
import numpy as np
c=np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
print("median =", np.median(c))
sorted = np.msort(c)
print("sorted =", sorted)
N = len(c)
print("middle =", sorted[(N - 1)/2])
print("average middle =", (sorted[N /2] + sorted[(N - 1) / 2]) / 2)
print("variance =", np.var(c))
print("variance from definition =", np.mean((c - c.mean())**2))
Stock returns
In academic literature, it is more common to base analysis on stock returns and log returns
of the close price. Simple returns are just the rate of change from one value to the next.
Logarithmic returns, or log returns, are determined by taking the log of all the prices and
calculating the differences between them. In high school, we learned that:
a
log ( a ) − log ( b ) = log
b
Log returns, therefore, also measure the rate of change. Returns are dimensionless, since,
in the act of dividing, we divide dollar by dollar (or some other currency). Anyway, investors
are most likely to be interested in the variance or standard deviation of the returns, as this
represents risk.
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Chapter 3
Time for action – analyzing stock returns
Perform the following steps to analyze stock returns:
1.
First, let's calculate simple returns. NumPy has the diff() function that returns an
array that is built up of the difference between two consecutive array elements. This
is sort of like differentiation in calculus (the derivative of price with respect to time).
To get the returns, we also have to divide by the value of the previous day. We must
be careful though. The array returned by diff() is one element shorter than the
close prices array. After careful deliberation, we get the following code:
returns = np.diff( arr ) / arr[ : -1]
Notice that we don't use the last value in the divisor. The standard deviation is
equal to the square root of variance. Compute the standard deviation using the
std() function:
print("Standard deviation =", np.std(returns))
This results in the following output:
Standard deviation = 0.0129221344368
2.
The log return or logarithmic return is even easier to calculate. Use the log()
function to get the natural logarithm of the close price and then unleash the
diff() function on the result:
logreturns = np.diff(np.log(c))
Normally, we have to check that the input array doesn't have zeros or negative
numbers. If it does, we will get an error. Stock prices are, however, always positive,
so we didn't have to check.
3.
Quite likely, we will be interested in days when the return is positive. In the current
setup, we can get the next best thing with the where() function, which returns the
indices of an array that satisfies a condition. Just type the following code:
posretindices = np.where(returns > 0)
print("Indices with positive returns", posretindices)
This gives us a number of indices for the array elements that are positive as a tuple,
recognizable by the round brackets on both sides of the printout:
Indices with positive returns (array([ 0, 1, 4, 5,
10, 11, 12, 16, 17, 18, 19, 21, 22, 23, 25, 28]),)
[ 63 ]
6,
7,
9,
Getting Familiar with Commonly Used Functions
4.
In investing, volatility measures price variation of a financial security. Historical
volatility is calculated from historical price data. The logarithmic returns are
interesting if you want to know the historical volatility—for instance, the annualized
or monthly volatility. The annualized volatility is equal to the standard deviation of
the log returns as a ratio of its mean, divided by one over the square root of the
number of business days in a year, usually one assumes 252. Calculate it with the
std() and mean() functions, as in the following code:
annual_volatility = np.std(logreturns)/np.mean(logreturns)
annual_volatility = annual_volatility / np.sqrt(1./252.)
print(annual_volatility)
Take notice of the division within the sqrt() function. Since, in Python, integer
division works differently than float division, we needed to use floats to make
sure that we get the proper results. The monthly volatility is similarly given by
the following code:
print("Monthly volatility", annual_volatility * np.sqrt(1./12.))
What just happened?
We calculated the simple stock returns with the diff() function, which calculates
differences between sequential elements. The log() function computes the natural
logarithms of array elements. We used it to calculate the logarithmic returns. At the
end of this section, we calculated the annual and monthly volatility (see returns.py):
from __future__ import print_function
import numpy as np
c=np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
returns = np.diff( c ) / c[ : -1]
print("Standard deviation =", np.std(returns))
logreturns = np.diff( np.log(c) )
posretindices = np.where(returns > 0)
print("Indices with positive returns", posretindices)
annual_volatility = np.std(logreturns)/np.mean(logreturns)
annual_volatility = annual_volatility / np.sqrt(1./252.)
print("Annual volatility", annual_volatility)
print("Monthly volatility", annual_volatility * np.sqrt(1./12.))
[ 64 ]
Chapter 3
Dates
Do you sometimes have the Monday blues or Friday fever? Ever wondered whether
the stock market suffers from these phenomena? Well, I think this certainly warrants
extensive research.
Time for action – dealing with dates
First, we will read the close price data. Second, we will split the prices according to the day
of the week. Third, for each weekday, we will calculate the average price. Finally, we will
find out which day of the week has the highest average and which has the lowest average.
A word of warning before we commence: you might be tempted to use the result to buy
stock on one day and sell on the other. However, we don't have enough data to make this
kind of decisions.
Coders hate dates because they are so complicated! NumPy is very much oriented toward
floating point operations. For this reason, we need to take extra effort to process dates. Try it
out yourself; put the following code in a script or use the one that comes with this book:
dates, close=np.loadtxt('data.csv', delimiter=',',
usecols=(1,6), unpack=True)
Execute the script and the following error will appear:
ValueError: invalid literal for float(): 28-01-2011
Now, perform the following steps to deal with dates:
1.
Obviously, NumPy tried to convert the dates into floats. What we have to do is tell
NumPy explicitly how to convert the dates. The loadtxt() function has a special
parameter for this purpose. The parameter is called converters and is a dictionary
that links columns with the so-called converter functions. It is our responsibility to
write the converter function. Write the function down:
# Monday 0
# Tuesday 1
# Wednesday 2
# Thursday 3
# Friday 4
# Saturday 5
# Sunday 6
def datestr2num(s):
return datetime.datetime.strptime(s, "%d-%m-%Y").date().
weekday()
[ 65 ]
Getting Familiar with Commonly Used Functions
We give the datestr2num() function dates as a string, such as 28-01-2011. The
string is first turned into a datetime object, using a specified format %d-%m-%Y. By
the way, this is standard Python and is not related to NumPy itself (see https://
docs.python.org/2/library/datetime.html#strftime-and-strptimebehavior). Second, the datetime object is turned into a day. Finally, the weekday
method is called on the date to return a number. As you can read in the comments,
the number is between 0 and 6. 0 is, for instance, Monday, and 6 is Sunday. The actual
number, of course, is not important for our algorithm; it is only used as identification.
2.
Now, hook up our date converter function:
dates, close=np.loadtxt('data.csv', delimiter=',', usecols=(1,6),
converters={1: datestr2num}, unpack=True)
print "Dates =", dates
This prints the following output:
Dates = [ 4. 0. 1. 2. 3. 4. 0. 1. 2. 3. 4. 0. 1.
3. 4. 1. 2. 4. 0. 1. 2. 3. 4. 0. 1. 2. 3. 4.]
2.
No Saturdays and Sundays, as you can see. Exchanges are closed over the weekend.
3.
We will now make an array that has five elements for each day of the week. Initialize
the values of the array to 0:
averages = np.zeros(5)
This array will hold the averages for each weekday.
4.
We already learned about the where() function that returns indices of the array
for elements that conform to a specified condition. The take() function can use
these indices and takes the values of the corresponding array items. We will use the
take() function to get the close prices for each weekday. In the following loop,
we go through the date values 0 to 4, better known as Monday to Friday. We get
the indices with the where() function for each day and store it in the indices
array. Then, we retrieve the values corresponding to the indices, using the take()
function. Finally, compute an average for each weekday and store it in the averages
array, like this:
for i in range(5):
indices = np.where(dates == i)
prices = np.take(close, indices)
avg = np.mean(prices)
print("Day", i, "prices", prices, "Average", avg)
averages[i] = avg
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Chapter 3
The loop prints the following output:
Day 0 prices [[ 339.32
351.79
351.88
359.18
353.21
355.36]] Average
Day 1 prices [[ 345.03
Average 350.635
355.2
359.9
338.61
349.31
355.76]]
Day 2 prices [[ 344.32
Average 352.136666667
358.16
363.13
342.62
352.12
352.47]]
Day 3 prices [[ 343.44
Average 350.898333333
354.54
358.3
342.88
359.56
346.67]]
356.85
350.56
348.16
360.
Day 4 prices [[ 336.1
346.5
351.99]] Average 350.022857143
5.
If you want, you can go ahead and find out which day has the highest average, and
which the lowest. However, it is just as easy to find this out with the max() and
min() functions, as shown here:
top = np.max(averages)
print("Highest average", top)
print("Top day of the week", np.argmax(averages))
bottom = np.min(averages)
print("Lowest average", bottom)
print("Bottom day of the week", np.argmin(averages))
The output is as follows:
Highest average 352.136666667
Top day of the week 2
Lowest average 350.022857143
Bottom day of the week 4
What just happened?
The argmin() function returned the index of the lowest value in the averages array.
The index returned was 4, which corresponds to Friday. The argmax() function returned
the index of the highest value in the averages array. The index returned was 2, which
corresponds to Wednesday (see weekdays.py):
from __future__ import print_function
import numpy as np
from datetime import datetime
# Monday 0
# Tuesday 1
# Wednesday 2
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Getting Familiar with Commonly Used Functions
# Thursday 3
# Friday 4
# Saturday 5
# Sunday 6
def datestr2num(s):
return datetime.strptime(s, "%d-%m-%Y").date().weekday()
dates, close=np.loadtxt('data.csv', delimiter=',', usecols=(1,6),
converters={1: datestr2num}, unpack=True)
print("Dates =", dates)
averages = np.zeros(5)
for i in range(5):
indices = np.where(dates == i)
prices = np.take(close, indices)
avg = np.mean(prices)
print("Day", i, "prices", prices, "Average", avg)
averages[i] = avg
top = np.max(averages)
print("Highest average", top)
print("Top day of the week", np.argmax(averages))
bottom = np.min(averages)
print("Lowest average", bottom)
print("Bottom day of the week", np.argmin(averages))
Have a go hero – looking at VWAP and TWAP
Hey, that was fun! For the sample data, it appears that Friday is the cheapest day and
Wednesday is the day when your Apple stock will be worth the most. Ignoring the fact that
we have very little data, is there a better method to compute the averages? Shouldn't we
involve volume data as well? Maybe it makes more sense to you to do a time-weighted
average. Give it a go! Calculate the VWAP and TWAP. You can find some hints on how to go
about doing this at the beginning of this chapter.
[ 68 ]
Chapter 3
Time for action – using the datetime64 data type
The datetime64 data type was introduced in NumPy 1.7.0 (see http://docs.scipy.
org/doc/numpy/reference/arrays.datetime.html).
1.
To learn about the datetime64 data type, start a Python shell and import NumPy
as follows:
$ python
>>> import numpy as np
Create a datetime64 from a string (you can use another date if you like):
>>> np.datetime64('2015-04-22')
numpy.datetime64('2015-04-22')
In the preceding code, we created a datetime64 for April 22, 2015, which happens
to be Earth Day. We used the YYYY-MM-DD format, where Y corresponds to the year,
M corresponds to the month, and D corresponds to the day of the month. NumPy
uses the ISO 8601 standard (see http://en.wikipedia.org/wiki/ISO_8601).
This is an international standard to represent dates and times. ISO 8601 allows the
YYYY-MM-DD, YYYY-MM, and YYYYMMDD formats. Check for yourself, as follows:
>>> np.datetime64('2015-04-22')
numpy.datetime64('2015-04-22')
>>> np.datetime64('2015-04')
numpy.datetime64('2015-04')
2.
By default, ISO 8601 uses the local time zone. Times can be specified using the
format T[hh:mm:ss]. For example, define January 1, 1677 at 8:19 p.m. as follows:
>>> local = np.datetime64('1677-01-01T20:19')
>>> local
numpy.datetime64('1677-01-01T20:19Z')
Additionally, a string in the format [hh:mm] specifies an offset that is relative to the
UTC time zone. Create a datetime64 with 9 hours offset, as follows:
>>> with_offset = np.datetime64('1677-01-01T20:19-0900')
>>> with_offset
numpy.datetime64('1677-01-02T05:19Z')
The Z at the end stands for Zulu time, which is how UTC is sometimes referred to.
Subtract the two datetime64 objects from each other:
>>> local - with_offset
numpy.timedelta64(-540,'m')
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Getting Familiar with Commonly Used Functions
The subtraction creates a NumPy timedelta64 object, which in this case, indicates
a 540 minute difference. We can also add or subtract a number of days to a
datetime64 object. For instance, April 22, 2015 happens to be a Wednesday. With
the arange() function, create an array holding all the Wednesdays from April 22,
2015 until May 22, 2015 as follows:
>>> np.arange('2015-04-22', '2015-05-22', 7, dtype='datetime64')
array(['2015-04-22', '2015-04-29', '2015-05-06', '2015-05-13',
'2015-05-20'], dtype='datetime64[D]')
Note that in this case, it is mandatory to specify the dtype argument, otherwise
NumPy thinks that we are dealing with strings.
What just happened?
We learned about the NumPy datetime64 type. This data type allows us to manipulate
dates and times with ease. Its features include simple arithmetic and creation of arrays
using the normal NumPy capabilities.
Weekly summary
The data that we used in the previous Time for action section is end-of-day data. In essence,
it is summarized data compiled from the trade data for a certain day. If you are interested in
the market and have decades of data, you might want to summarize and compress the data
even further. Let's summarize the data of Apple stocks to give us weekly summaries.
Time for action – summarizing data
The data we will summarize will be for a whole business week, running from Monday
to Friday. During the period covered by the data, there was one holiday on February 21,
President's Day. This happened to be a Monday and the US stock exchanges were closed on
this day. As a consequence, there is no entry for this day, in the sample. The first day in the
sample is a Friday, which is inconvenient. Use the following instructions to summarize data:
1.
To simplify, just have a look at the first three weeks in the sample— later, you can
have a go at improving this:
close = close[:16]
dates = dates[:16]
We will be building on the code from the previous Time for action section.
[ 70 ]
Chapter 3
2.
Commencing, we will find the first Monday in our sample data. Recall that Mondays
have the code 0 in Python. This is what we will put in the condition of the where()
function. Then, we will need to extract the first element that has index 0. The result
will be a multidimensional array. Flatten this with the ravel() function:
# get first Monday
first_monday = np.ravel(np.where(dates == 0))[0]
print("The first Monday index is", first_monday)
This will print the following output:
The first Monday index is 1
3.
The next logical step is to find the Friday before last Friday in the sample. The
logic is similar to the one for finding the first Monday, and the code for Friday is 4.
Additionally, we are looking for the second to last element with index 2:
# get last Friday
last_friday = np.ravel(np.where(dates == 4))[-2]
print("The last Friday index is", last_friday)
This will give us the following output:
The last Friday index is 15
4.
Next, create an array with the indices of all the days in the three weeks:
weeks_indices = np.arange(first_monday, last_friday + 1)
print("Weeks indices initial", weeks_indices)
5.
Split the array in pieces of size 5 with the split() function:
weeks_indices = np.split(weeks_indices, 3)
print("Weeks indices after split", weeks_indices)
This splits the array as follows:
Weeks indices after split [array([1, 2, 3, 4, 5]), array([ 6,
8, 9, 10]), array([11, 12, 13, 14, 15])]
6.
7,
In NumPy, array dimensions are called axes. Now, we will get fancy with the apply_
along_axis() function. This function calls another function, which we will provide,
to operate on each of the elements of an array. Currently, we have an array with three
elements. Each array item corresponds to one week in our sample and contains indices
of the corresponding items. Call the apply_along_axis() function by supplying
the name of our function, called summarize(), which we will define shortly.
Furthermore, specify the axis or dimension number (such as 1), the array to operate
on, and a variable number of arguments for the summarize() function, if any:
weeksummary = np.apply_along_axis(summarize, 1,
weeks_indices, open, high, low, close)
print("Week summary", weeksummary)
[ 71 ]
Getting Familiar with Commonly Used Functions
7.
For each week, the summarize() function returns a tuple that holds the open,
high, low, and close price for the week, similar to end-of-day data:
def summarize(a, o, h, l, c):
monday_open = o[a[0]]
week_high = np.max( np.take(h, a) )
week_low = np.min( np.take(l, a) )
friday_close = c[a[-1]]
return("APPL", monday_open, week_high,
week_low, friday_close)
Notice that we used the take() function to get the actual values from indices.
Calculating the high and low values for the week was easily done with the max()
and min() functions. The open for the week is the open for the first day in the
week—Monday. Likewise, the close is the close for the last day of the week—Friday:
Week summary [['APPL' '335.8' '346.7' '334.3' '346.5']
['APPL' '347.89' '360.0' '347.64' '356.85']
['APPL' '356.79' '364.9' '349.52' '350.56']]
8.
Store the data in a file with the NumPy savetxt() function:
np.savetxt("weeksummary.csv", weeksummary, delimiter=",",
fmt="%s")
As you can see, have specified a filename, the array we want to store, a delimiter
(in this case a comma), and the format we want to store floating point numbers in.
The format string starts with a percent sign. Second is an optional flag. The—flag
means left justify, 0 means left pad with zeros, and + means precede with + or -.
Third is an optional width. The width indicates the minimum number of characters.
Fourth, a dot is followed by a number linked to precision. Finally, there comes a
character specifier; in our example, the character specifier is a string. The character
codes are described as follows:
Character code
c
Description
d or i
signed decimal integer
e or E
scientific notation with e or E.
f
decimal floating point
g,G
use the shorter of e,E or f
o
signed octal
character
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Chapter 3
Character code
s
Description
u
unsigned decimal integer
x,X
unsigned hexadecimal integer
string of characters
View the generated file in your favorite editor or type at the command line:
$ cat weeksummary.csv
APPL,335.8,346.7,334.3,346.5
APPL,347.89,360.0,347.64,356.85
APPL,356.79,364.9,349.52,350.56
What just happened?
We did something that is not even possible in some programming languages. We defined a
function and passed it as an argument to the apply_along_axis() function.
The programming paradigm described here is called functional programming.
You can read more about functional programming in Python at
https://docs.python.org/2/howto/functional.html.
Arguments for the summarize() function were neatly passed by apply_along_axis()
(see weeksummary.py):
from __future__ import print_function
import numpy as np
from datetime import datetime
# Monday 0
# Tuesday 1
# Wednesday 2
# Thursday 3
# Friday 4
# Saturday 5
# Sunday 6
def datestr2num(s):
return datetime.strptime(s, "%d-%m-%Y").date().weekday()
dates, open, high, low, close=np.loadtxt('data.csv', delimiter=',',
usecols=(1, 3, 4, 5, 6), converters={1: datestr2num}, unpack=True)
close = close[:16]
dates = dates[:16]
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Getting Familiar with Commonly Used Functions
# get first Monday
first_monday = np.ravel(np.where(dates == 0))[0]
print("The first Monday index is", first_monday)
# get last Friday
last_friday = np.ravel(np.where(dates == 4))[-1]
print("The last Friday index is", last_friday)
weeks_indices = np.arange(first_monday, last_friday + 1)
print("Weeks indices initial", weeks_indices)
weeks_indices = np.split(weeks_indices, 3)
print("Weeks indices after split", weeks_indices)
def summarize(a, o, h, l, c):
monday_open = o[a[0]]
week_high = np.max( np.take(h, a) )
week_low = np.min( np.take(l, a) )
friday_close = c[a[-1]]
return("APPL", monday_open, week_high, week_low, friday_close)
weeksummary = np.apply_along_axis(summarize, 1, weeks_indices, open,
high, low, close)
print("Week summary", weeksummary)
np.savetxt("weeksummary.csv", weeksummary, delimiter=",", fmt="%s")
Have a go hero – improving the code
Change the code to deal with a holiday. Time the code to see how big the speedup due to
apply_along_axis() is.
Average True Range
The Average True Range (ATR) is a technical indicator that measures volatility of stock prices.
The ATR calculation is not important further but will serve as an example of several NumPy
functions, including the maximum() function.
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Chapter 3
Time for action – calculating the Average True Range
To calculate the ATR, perform the following steps:
1.
The ATR is based on the low and high price of N days, usually the last 20 days.
N = 5
h = h[-N:]
l = l[-N:]
2.
We also need to know the close price of the previous day:
previousclose = c[-N -1: -1]
For each day, we calculate the following:
The daily range—the difference between the high and low price:
h – l
The difference between the high and previous close:
h – previousclose
The difference between the previous close and the low price:
previousclose – l
3.
The max() function returns the maximum of an array. Based on those three values,
we calculate the so-called true range, which is the maximum of these values. We are
now interested in the element-wise maxima across arrays—meaning the maxima of
the first elements in the arrays, the second elements in the arrays, and so on. Use
the NumPy maximum() function instead of the max() function for this purpose:
truerange = np.maximum(h - l, h - previousclose, previousclose l)
4.
Create an atr array of size N and initialize its values to 0:
atr = np.zeros(N)
5.
The first value of the array is just the average of the truerange array:
atr[0] = np.mean(truerange)
Calculate the other values with the following formula:
( ( N − 1) PATR + TR )
N
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Getting Familiar with Commonly Used Functions
Here, PATR is the previous day's ATR; TR is the true range:
for i in range(1, N):
atr[i] = (N - 1) * atr[i - 1] + truerange[i]
atr[i] /= N
What just happened?
We formed three arrays, one for each of the three ranges—daily range, the gap between the
high of today and the close of yesterday, and the gap between the close of yesterday and the
low of today. This tells us how much the stock price moved and, therefore, how volatile it is.
The algorithm requires us to find the maximum value for each day. The max() function that
we used before can give us the maximum value within an array, but that is not what we want
here. We need the maximum value across arrays, so we want the maximum value of the first
elements in the three arrays, the second elements, and so on. In preceding Time for action
section, we saw that the maximum() function can do this. After this, we computed a moving
average of the true range values (see atr.py):
from __future__ import print_function
import numpy as np
h, l, c = np.loadtxt('data.csv', delimiter=',', usecols=(4, 5, 6),
unpack=True)
N = 5
h = h[-N:]
l = l[-N:]
print("len(h)", len(h), "len(l)", len(l))
print("Close", c)
previousclose = c[-N -1: -1]
print("len(previousclose)", len(previousclose))
print("Previous close", previousclose)
truerange = np.maximum(h - l, h - previousclose, previousclose - l)
print("True range", truerange)
atr = np.zeros(N)
atr[0] = np.mean(truerange)
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Chapter 3
for i in range(1, N):
atr[i] = (N - 1) * atr[i - 1] + truerange[i]
atr[i] /= N
print("ATR", atr)
In the following sections, we will learn better ways to calculate moving averages.
Have a go hero – taking the minimum() function for a spin
Besides the maximum() function, there is a minimum() function. You can probably guess
what it does. Make a small script or start an interactive session in IPython to test your
assumptions.
Simple Moving Average
The Simple Moving Average (SMA) is commonly used to analyze time-series data. To
calculate it, we define a moving window of N periods, N days in our case. We move this
window along the data and calculate the mean of the values inside the window.
Time for action – computing the Simple Moving Average
The moving average is easy enough to compute with a few loops and the mean() function,
but NumPy has a better alternative—the convolve() function. The SMA is, after all,
nothing more than a convolution with equal weights or, if you like, unweighted.
Convolution is a mathematical operation on two functions defined as the
integral of the product of the two functions after one of the functions is
reversed and shifted.
∞
∞
( f ∗ g )( t ) = ∫−∞ f (τ ) g ( t − τ ) dτ = ∫−∞ f ( t − τ ) g (τ ) dτ
Convolution is described on Wikipedia at https://en.wikipedia.org/
wiki/Convolution. Khan Academy also has a tutorial on convolution
at https://www.khanacademy.org/math/differentialequations/laplace-transform/convolution-integral/v/
introduction-to-the-convolution.
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Getting Familiar with Commonly Used Functions
Use the following steps to compute the SMA:
1.
Use the ones() function to create an array of size N and elements initialized to 1,
and then, divide the array by N to give us the weights:
N = 5
weights = np.ones(N) / N
print("Weights", weights)
For N = 5, this gives us the following output:
Weights [ 0.2
2.
0.2
0.2
0.2
0.2]
Now, call the convolve() function with these weights:
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,),
unpack=True)
sma = np.convolve(weights, c)[N-1:-N+1]
3.
From the array returned by convolve(), we extracted the data in the center of size
N. The following code makes an array of time values and plots with matplotlib
that we will cover in a later chapter:
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,),
unpack=True)
sma = np.convolve(weights, c)[N-1:-N+1]
t = np.arange(N - 1, len(c))
plt.plot(t, c[N-1:], lw=1.0, label="Data")
plt.plot(t, sma, '--', lw=2.0, label="Moving average")
plt.title("5 Day Moving Average")
plt.xlabel("Days")
plt.ylabel("Price ($)")
plt.grid()
plt.legend()
plt.show()
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Chapter 3
In the following chart, the smooth dashed line is the 5 day SMA and the jagged thin
line is the close price:
What just happened?
We computed the SMA for the close stock price. It turns out that the SMA is just a signal
processing technique—a convolution with weights 1/N, where N is the size of the moving
average window. We learned that the ones() function can create an array with ones and
the convolve() function calculates the convolution of a dataset with specified weights
(see sma.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
N = 5
weights = np.ones(N) / N
print("Weights", weights)
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
sma = np.convolve(weights, c)[N-1:-N+1]
t = np.arange(N - 1, len(c))
plt.plot(t, c[N-1:], lw=1.0, label="Data")
plt.plot(t, sma, '--', lw=2.0, label="Moving average")
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Getting Familiar with Commonly Used Functions
plt.title("5 Day Moving Average")
plt.xlabel("Days")
plt.ylabel("Price ($)")
plt.grid()
plt.legend()
plt.show()
Exponential Moving Average
The Exponential Moving Average (EMA) is a popular alternative to the SMA. This method
uses exponentially decreasing weights. The weights for points in the past decrease
exponentially but never reach zero. We will learn about the exp() and linspace()
functions while calculating the weights.
Time for action – calculating the Exponential Moving Average
Given an array, the exp() function calculates the exponential of each array element. For
example, look at the following code:
x = np.arange(5)
print("Exp", np.exp(x))
It gives the following output:
Exp [
1.
2.71828183
7.3890561
20.08553692
54.59815003]
The linspace() function takes as parameters a start value, a stop value, and optionally an
array size. It returns an array of evenly spaced numbers. This is an example:
print("Linspace", np.linspace(-1, 0, 5))
This will give us the following output:
Linspace [-1.
-0.75 -0.5
-0.25
0.
]
Calculate the EMA for our data:
1.
Now, back to the weights, calculate them with exp() and linspace():
N = 5
weights = np.exp(np.linspace(-1., 0., N))
2.
Normalize the weights with the ndarray sum() method:
weights /= weights.sum()
print("Weights", weights)
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Chapter 3
For N = 5, we get these weights:
Weights [ 0.11405072
0.31002201]
3.
0.14644403
0.18803785
0.24144538
After this, use the convolve() function that we learned about in the SMA section
and also plot the results:
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,),
unpack=True)
ema = np.convolve(weights, c)[N-1:-N+1]
t = np.arange(N - 1, len(c))
plt.plot(t, c[N-1:], lw=1.0, label='Data')
plt.plot(t, ema, '--', lw=2.0, label='Exponential Moving Average')
plt.title('5 Days Exponential Moving Average')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.legend()
plt.grid()
plt.show()
This gives us a nice chart where, again, the close price is the thin jagged line and the
EMA is the smooth dashed line:
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Getting Familiar with Commonly Used Functions
What just happened?
We calculated the EMA of the close price. First, we computed exponentially decreasing
weights with the exp() and linspace() functions. The linspace() function gave us
an array with evenly spaced elements, and, then, we calculated the exponential for these
numbers. We called the ndarray sum() method in order to normalize the weights. After
this, we applied the convolve() trick that we learned in the SMA section (see ema.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(5)
print("Exp", np.exp(x))
print("Linspace", np.linspace(-1, 0, 5))
# Calculate weights
N = 5
weights = np.exp(np.linspace(-1., 0., N))
# Normalize weights
weights /= weights.sum()
print("Weights", weights)
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
ema = np.convolve(weights, c)[N-1:-N+1]
t = np.arange(N - 1, len(c))
plt.plot(t, c[N-1:], lw=1.0, label='Data')
plt.plot(t, ema, '--', lw=2.0, label='Exponential Moving Average')
plt.title('5 Days Exponential Moving Average')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.legend()
plt.grid()
plt.show()
Bollinger Bands
Bollinger Bands are yet another technical indicator. Yes, there are thousands of them. This
one is named after its inventor and indicates a range for the price of a financial security. It
consists of three parts:
1. A Simple Moving Average.
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Chapter 3
2. An upper band of two standard deviations above this moving average—the
standard deviation is derived from the same data with which the moving average
is calculated.
3. A lower band of two standard deviations below the moving average.
Time for action – enveloping with Bollinger Bands
We already know how to calculate the SMA. So, if you need to refresh your memory, please
review the Time for action – computing the simple average section in this chapter. This
example will introduce the NumPy fill() function. The fill() function sets the value of
an array to a scalar value. The function should be faster than array.flat = scalar or
setting the values of the array one-by-one in a loop. Perform the following steps to envelope
with the Bollinger Bands:
1.
Starting with an array called sma that contains the moving average values, we
will loop through all the datasets corresponding to those values. After forming
the dataset, calculate the standard deviation. Note that at a certain point, it
will be necessary to calculate the difference between each data point and the
corresponding average value. If we do not have NumPy, we will loop through these
points and subtract each of the values one-by-one from the corresponding average.
However, the NumPy fill() function allows us to construct an array that has
elements set to the same value. This enables us to save on one loop and subtract
arrays in one go:
deviation = []
C = len(c)
for i in range(N - 1, C):
if i + N < C:
dev = c[i: i + N]
else:
dev = c[-N:]
averages = np.zeros(N)
averages.fill(sma[i - N - 1])
dev = dev - averages
dev = dev ** 2
dev = np.sqrt(np.mean(dev))
deviation.append(dev)
deviation = 2 * np.array(deviation)
print(len(deviation), len(sma))
upperBB = sma + deviation
lowerBB = sma - deviation
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Getting Familiar with Commonly Used Functions
2.
To plot, we will use the following code (don't worry about it now; we will see how
this works in Chapter 9, Plotting with matplotlib):
t = np.arange(N - 1, C)
plt.plot(t, c_slice, lw=1.0, label='Data')
plt.plot(t, sma, '--', lw=2.0, label='Moving Average')
plt.plot(t, upperBB, '-.', lw=3.0, label='Upper Band')
plt.plot(t, lowerBB, ':', lw=4.0, label='Lower Band')
plt.title('Bollinger Bands')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.grid()
plt.legend()
plt.show()
Following is a chart showing the Bollinger Bands for our data. The jagged thin line in
the middle represents the close price, and the dashed, smoother line crossing it is
the moving average:
[ 84 ]
Chapter 3
What just happened?
We worked out the Bollinger Bands that envelope the close price of our data. More
importantly, we got acquainted with the NumPy fill() function. This function fills
an array with a scalar value. This is the only parameter of the fill() function (see
bollingerbands.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
N = 5
weights = np.ones(N) / N
print("Weights", weights)
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
sma = np.convolve(weights, c)[N-1:-N+1]
deviation = []
C = len(c)
for i in range(N - 1, C):
if i + N < C:
dev = c[i: i + N]
else:
dev = c[-N:]
averages = np.zeros(N)
averages.fill(sma[i - N - 1])
dev = dev - averages
dev = dev ** 2
dev = np.sqrt(np.mean(dev))
deviation.append(dev)
deviation = 2 * np.array(deviation)
print(len(deviation), len(sma))
upperBB = sma + deviation
lowerBB = sma - deviation
c_slice = c[N-1:]
between_bands = np.where((c_slice < upperBB) & (c_slice > lowerBB))
print(lowerBB[between_bands])
print(c[between_bands])
print(upperBB[between_bands])
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Getting Familiar with Commonly Used Functions
between_bands = len(np.ravel(between_bands))
print("Ratio between bands", float(between_bands)/len(c_slice))
t = np.arange(N - 1, C)
plt.plot(t, c_slice, lw=1.0, label='Data')
plt.plot(t, sma, '--', lw=2.0, label='Moving Average')
plt.plot(t, upperBB, '-.', lw=3.0, label='Upper Band')
plt.plot(t, lowerBB, ':', lw=4.0, label='Lower Band')
plt.title('Bollinger Bands')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.grid()
plt.legend()
plt.show()
Have a go hero – switching to Exponential Moving Average
It is customary to choose the SMA to center the Bollinger Band on. The second most popular
choice is the EMA, so try that as an exercise. You can find a suitable example in this chapter,
if you need pointers.
Check whether the fill() function is faster or is as fast as array.flat = scalar, or
setting the value in a loop.
Linear model
Many phenomena in science have a related linear relationship model. The NumPy linalg
package deals with linear algebra computations. We will begin with the assumption that a
price value can be derived from N previous prices based on a linear relationship relation.
Time for action – predicting price with a linear model
Keeping an open mind, let's assume that we can express a stock price p as a linear
combination of previous values, that is, a sum of those values multiplied by certain
coefficients we need to determine:
pt = b + ∑ i =1 at −i pt −i
N
[ 86 ]
Chapter 3
In linear algebra terms, this boils down to finding a least-squares method (see https://
www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_
projections/v/linear-algebra-least-squares-approximation).
Independently of each other, the astronomers Legendre and
Gauss created the least squares method around 1805 (see
http://en.wikipedia.org/wiki/Least_squares).
The method was initially used to analyze the motion of celestial
bodies. The algorithm minimizes the sum of the squared residuals
(the difference between measured and predicted values):
n
∑ ( measured
i =1
i
− predictedi )
2
The recipe goes as follows:
1.
First, form a vector b containing N price values:
b = c[-N:]
b = b[::-1]
print("b", x)
The result is as follows:
b [ 351.99
2.
346.67
352.47
355.76
355.36]
Second, pre-initialize the matrix A to be N-by-N and contain zeros:
A = np.zeros((N, N), float)
Print("Zeros N by N", A)
The following should be printed on your screen:
Zeros N by N [[ 0.
3.
0.
[ 0.
0.
0.
0.
0.]
[ 0.
0.
0.
0.
0.]
[ 0.
0.
0.
0.
0.]
[ 0.
0.
0.
0.
0.]]
0.
0.
0.]
Third, fill the matrix A with N preceding price values for each value in b:
for i in range(N):
A[i, ] = c[-N - 1 - i: - 1 - i]
print("A", A)
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Getting Familiar with Commonly Used Functions
Now, A looks like this:
A [[ 360.
4.
355.36
355.76
352.47
346.67]
[ 359.56
360.
355.36
355.76
352.47]
[ 352.12
359.56
360.
355.36
355.76]
[ 349.31
352.12
359.56
360.
355.36]
[ 353.21
349.31
352.12
359.56
360.
]]
The objective is to determine the coefficients that satisfy our linear model by solving
the least squares problem. Employ the lstsq() function of the NumPy linalg
package to do this:
(x, residuals, rank, s) = np.linalg.lstsq(A, b)
print(x, residuals, rank, s)
The result is as follows:
[ 0.78111069 -1.44411737 1.63563225 -0.89905126 0.92009049]
[] 5 [ 1.77736601e+03
1.49622969e+01
8.75528492e+00
5.15099261e+00
1.75199608e+00]
The tuple returned contains the coefficient x that we were after, an array comprising
residuals, the rank of matrix A, and the singular values of A.
5.
Once we have the coefficients of our linear model, we can predict the next
price value. Compute the dot product (with the NumPy dot() function) of the
coefficients and the last known N prices:
print(np.dot(b, x))
The dot product (see https://www.khanacademy.org/math/linearalgebra/vectors_and_spaces/dot_cross_products/v/vector-dotproduct-and-vector-length) is the linear combination of the coefficients b
and the prices x. As a result, we get:
357.939161015
I looked it up; the actual close price of the next day was 353.56. So, our estimate with N = 5
was not that far off.
[ 88 ]
Chapter 3
What just happened?
We predicted tomorrow's stock price today. If this works in practice, we can retire early! See,
this book was a good investment, after all! We designed a linear model for the predictions.
The financial problem was reduced to a linear algebraic one. NumPy's linalg package has a
practical lstsq() function that helped us with the task at hand, estimating the coefficients
of a linear model. After obtaining a solution, we plugged the numbers in the NumPy dot()
function that presented us an estimate through linear regression (see linearmodel.py):
from __future__ import print_function
import numpy as np
N = 5
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
b = c[-N:]
b = b[::-1]
print("b", b)
A = np.zeros((N, N), float)
print("Zeros N by N", A)
for i in range(N):
A[i, ] = c[-N - 1 - i: - 1 - i]
print("A", A)
(x, residuals, rank, s) = np.linalg.lstsq(A, b)
print(x, residuals, rank, s)
print(np.dot(b, x))
Trend lines
A trend line is a line among a number of the so-called pivot points on a stock chart. As the
name suggests, the line's trend portrays the trend of the price development. In the past,
traders drew trend lines on paper but nowadays, we can let a computer draw it for us. In this
section, we shall use a very simple approach that probably won't be very useful in real life,
but should clarify the principle well.
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Getting Familiar with Commonly Used Functions
Time for action – drawing trend lines
Perform the following steps to draw trend lines:
1.
First, we need to determine the pivot points. We shall pretend they are equal to the
arithmetic mean of the high, low, and close price:
h, l, c = np.loadtxt('data.csv', delimiter=',', usecols=(4, 5,
6), unpack=True)
pivots = (h + l + c) / 3
print("Pivots", pivots)
From the pivots, we can deduce the so-called resistance and support levels.
The support level is the lowest level at which the price rebounds. The resistance
level is the highest level at which the price bounces back. These are not natural
phenomena, they are merely estimates. Based on these estimates, it is possible to
draw support and resistance trend lines. We will define the daily spread to be the
difference of the high and low price.
2.
Define a function to fit line to data to a line where y = at + b. The function should
return a and b. This is another opportunity to apply the lstsq() function of the
NumPy linalg package. Rewrite the line equation to y = Ax, where A = [t 1]
and x = [a b]. Form A with the NumPy ones_like(), which creates an array,
where all the values are equal to 1, using an input array as a template for the
array dimensions:
def fit_line(t, y):
A = np.vstack([t, np.ones_like(t)]).T
return np.linalg.lstsq(A, y)[0]
3.
Assuming that support levels are one daily spread below the pivots, and that
resistance levels are one daily spread above the pivots, fit the support and
resistance trend lines:
t = np.arange(len(c))
sa, sb = fit_line(t, pivots - (h - l))
ra, rb = fit_line(t, pivots + (h - l))
support = sa * t + sb
resistance = ra * t + rb
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Chapter 3
4.
At this juncture, we have all the necessary information to draw the trend lines;
however, it is wise to check how many points fall between the support and resistance
levels. Obviously, if only a small percentage of the data is between the trend lines,
then this setup is of no use to us. Make up a condition for points between the bands
and select with the where() function, based on the following condition:
condition = (c > support) & (c < resistance)
print("Condition", condition)
between_bands = np.where(condition)
These are the printed condition values:
Condition [False False
True False False
True
True
True
False False False True False False False
False False True True True False True]
True
True
True False False
True
True
True
Double-check the values:
print(support[between_bands])
print( c[between_bands])
print( resistance[between_bands])
The array returned by the where() function has rank 2, so call the ravel()
function before calling the len() function:
between_bands = len(np.ravel(between_bands))
print("Number points between bands", between_bands)
print("Ratio between bands", float(between_bands)/len(c))
You will get the following result:
Number points between bands 15
Ratio between bands 0.5
As an extra bonus, we gained a predictive model. Extrapolate the next day resistance
and support levels:
print("Tomorrows support", sa * (t[-1] + 1) + sb)
print("Tomorrows resistance", ra * (t[-1] + 1) + rb)
This results in the following output:
Tomorrows support 349.389157088
Tomorrows resistance 360.749340996
[ 91 ]
Getting Familiar with Commonly Used Functions
Another approach to figure out how many points are between the support and
resistance estimates is to use [] and intersect1d(). Define selection criteria
in the [] operator and intersect the results with the intersect1d() function:
a1 = c[c > support]
a2 = c[c < resistance]
print("Number of points between bands 2nd approach" ,len(np.
intersect1d(a1, a2)))
Not surprisingly, we get:
Number of points between bands 2nd approach 15
5.
Once more, plot the results:
plt.plot(t, c, label='Data')
plt.plot(t, support, '--', lw=2.0, label='Support')
plt.plot(t, resistance, '-.', lw=3.0, label='Resistance')
plt.title('Trend Lines')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.grid()
plt.legend()
plt.show()
In the following plot, we have the price data and the corresponding support and
resistance lines:
[ 92 ]
Chapter 3
What just happened?
We drew trend lines without having to mess around with rulers, pencils, and paper charts.
We defined a function that can fit data to a line with the NumPy vstack(), ones_like(),
and lstsq() functions. We fit the data in order to define support and resistance trend lines.
Then, we figured out how many points are within the support and resistance range. We did
this using two separate methods that produced the same result.
The first method used the where() function with a Boolean condition. The second method
made use of the [] operator and the intersect1d() function. The intersect1d()
function returns an array of common elements from two arrays (see trendline.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
def fit_line(t, y):
''' Fits t to a line y = at + b '''
A = np.vstack([t, np.ones_like(t)]).T
return np.linalg.lstsq(A, y)[0]
# Determine pivots
h, l, c = np.loadtxt('data.csv', delimiter=',', usecols=(4, 5, 6),
unpack=True)
pivots = (h + l + c) / 3
print("Pivots", pivots)
# Fit trend lines
t = np.arange(len(c))
sa, sb = fit_line(t, pivots - (h - l))
ra, rb = fit_line(t, pivots + (h - l))
support = sa * t + sb
resistance = ra * t + rb
condition = (c > support) & (c < resistance)
print("Condition", condition)
between_bands = np.where(condition)
print(support[between_bands])
print(c[between_bands])
print(resistance[between_bands])
between_bands = len(np.ravel(between_bands))
[ 93 ]
Getting Familiar with Commonly Used Functions
print("Number points between bands", between_bands)
print("Ratio between bands", float(between_bands)/len(c))
print("Tomorrows support", sa * (t[-1] + 1) + sb)
print("Tomorrows resistance", ra * (t[-1] + 1) + rb)
a1 = c[c > support]
a2 = c[c < resistance]
print("Number of points between bands 2nd approach" ,len(np.
intersect1d(a1, a2)))
# Plotting
plt.plot(t, c, label='Data')
plt.plot(t, support, '--', lw=2.0, label='Support')
plt.plot(t, resistance, '-.', lw=3.0, label='Resistance')
plt.title('Trend Lines')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.grid()
plt.legend()
plt.show()
Methods of ndarray
The NumPy ndarray class has a lot of methods that work on the array. Most of the time,
these methods return an array. You may have noticed that many of the functions part of the
NumPy library have a counterpart with the same name and functionality in the ndarray
class. This is mostly due to the historical development of NumPy.
The list of ndarray methods is pretty long, so we cannot cover them all. The mean(),
var(), sum(), std(), argmax(), argmin(), and mean() functions that we saw earlier
are also ndarray methods.
Time for action – clipping and compressing arrays
Here are a few examples of ndarray methods. Perform the following steps to clip and
compress arrays:
1.
The clip() method returns a clipped array, so that all values above a maximum
value are set to the maximum and values below a minimum are set to the minimum
value. Clip an array with values 0 to 4 to 1 and 2:
a = np.arange(5)
print("a =", a)
print("Clipped", a.clip(1, 2))
[ 94 ]
Chapter 3
This gives the following output:
a = [0 1 2 3 4]
Clipped [1 1 2 2 2]
2.
The ndarray compress() method returns an array based on a condition. For
instance, look at the following code:
a = np.arange(4)
print(a)
print("Compressed", a.compress(a > 2))
This returns the following output:
[0 1 2 3]
Compressed [3]
What just happened?
We created an array with values 0 to 3 and selected the last element with the compress()
function based on the a > 2 condition.
Factorial
Many programming books have an example of calculating the factorial. We should not break
with this tradition.
Time for action – calculating the factorial
The ndarray class has the prod() method, which computes the product of the elements in
an array. Perform the following steps to calculate the factorial:
1.
Calculate the factorial of 8. To do this, generate an array with values 1 to 8 and call
the prod() function on it:
b = np.arange(1, 9)
print("b =", b)
print("Factorial", b.prod())
Check the result with your pocket calculator:
b = [1 2 3 4 5 6 7 8]
Factorial 40320
This is nice, but what if we want to know all the factorials from 1 to 8?
[ 95 ]
Getting Familiar with Commonly Used Functions
2.
No problem! Call the cumprod() method, which computes the cumulative product
of an array:
print("Factorials", b.cumprod())
It's pocket calculator time again:
Factorials [
1
2
6
24
120
720
5040 40320]
What just happened?
We used the prod() and cumprod() functions to calculate factorials (see
ndarraymethods.py):
from __future__ import print_function
import numpy as np
a = np.arange(5)
print("a =", a)
print("Clipped", a.clip(1, 2))
a = np.arange(4)
print(a)
print("Compressed", a.compress(a > 2))
b = np.arange(1, 9)
print("b =", b)
print("Factorial", b.prod())
print("Factorials", b.cumprod())
Missing values and Jackknife resampling
Data often misses values because of errors or technical issues. Even if we are not missing
values, we may have cause to suspect certain values. Once we doubt data values, derived
values such as the arithmetic mean, which we learned to calculate in this chapter, become
questionable too. It is common for these reasons to try to estimate how reliable the
arithmetic mean, variance, and standard deviation are.
A simple but effective method is called Jackknife resampling (see http://en.wikipedia.
org/wiki/Jackknife_resampling). The idea behind jackknife resampling is to
systematically generate datasets from the original dataset by leaving one value out at a time.
In effect, we are trying to establish what will happen if at least one of the values is wrong.
For each new generated dataset, we recalculate the arithmetic mean, variance, and standard
deviation. This gives us an idea of how much those values can vary.
[ 96 ]
Chapter 3
Time for action – handling NaNs with the nanmean(), nanvar(),
and nanstd() functions
We will apply jackknife resampling to the stock data. Each value will be omitted by setting it
to Not a Number (NaN). The nanmean(), nanvar(), and nanstd() can then be used to
compute the arithmetic mean, variance, and standard deviation.
1.
First, initialize a 30-by-3 array for the estimates as follows:
estimates = np.zeros((len(c), 3))
2.
Loop through the values and generate a new dataset by setting one value to NaN at
each iteration of the loop. For each new set of values, compute the estimates:
for i in xrange(len(c)):
a = c.copy()
a[i] = np.nan
estimates[i,] = [np.nanmean(a), np.nanvar(a), np.nanstd(a)]
3.
Print the variance for each estimate (you can also print the mean or standard
deviation if you prefer):
print("Estimates variance", estimates.var(axis=0))
The following is printed on the screen:
Estimates variance [ 0.05960347
3.63062943
0.01868965]
What just happened?
We estimated the variances of the arithmetic mean, variance, and standard deviation of a
small dataset using jackknife resampling. This gives us an idea of how much the arithmetic
mean, variance, and standard deviation vary. The code for this example can be found in the
jackknife.py file in this book's code bundle:
from __future__ import print_function
import numpy as np
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
# Initialize estimates array
estimates = np.zeros((len(c), 3))
for i in xrange(len(c)):
[ 97 ]
Getting Familiar with Commonly Used Functions
# Create a temporary copy and omit one value
a = c.copy()
a[i] = np.nan
# Compute estimates
estimates[i,] = [np.nanmean(a), np.nanvar(a), np.nanstd(a)]
print("Estimates variance", estimates.var(axis=0))
Summary
This chapter informed us about a great number of common NumPy functions. A few
common statistics functions were also mentioned.
After this tour through the common NumPy functions, we will continue covering
convenience NumPy functions such as polyfit(), sign(), and piecewise()
in the next chapter.
[ 98 ]
4
Convenience Functions
for Your Convenience
As we have seen, NumPy has a great number of functions. Many of those
functions exist just for convenience, and knowing them will greatly increase
your productivity. This includes functions that select certain parts of your arrays
(based on a Boolean condition, for instance) or manipulate polynomials. This
chapter has an example of computing correlation to give you a taste of data
analysis with NumPy.
In this chapter, we shall cover the following topics:
Data selection and extraction
Simple data analysis
Examples of correlation of returns
Polynomials
Linear algebra functions
In Chapter 3, Getting Familiar with Commonly Used Functions, we had one data file
to play around with. Things have improved in this chapter—we now have two data files.
Let's explore the data with NumPy.
[ 99 ]
Convenience Functions for Your Convenience
Correlation
Have you noticed that the stock price of some companies will be closely followed by another,
usually a rival in the same sector? The theoretical explanation is that because these two
companies are in the same type of business, they share the same challenges, require the
same materials and resources, and compete for the same type of customers.
You could think of many possible pairs, but you need to check for a real relationship. One
way is to take a look at the correlation of the stock returns of both stocks (see https://
www.khanacademy.org/math/probability/statistical-studies/types-ofstudies/v/correlation-and-causality). A high correlation implies a relationship of
some sort. It is not proof of causality though, especially if you don't use sufficient data.
Time for action – trading correlated pairs
For this section, we will use two sample datasets, containing end-of-day price data. The first
company is BHP Billiton (BHP), which is active in mining of petroleum, metals, and diamonds.
The second is Vale (VALE), which is also a metals and mining company. So, there is some
overlap of activity, albeit not 100 percent. For evaluating correlated pairs, follow these steps:
1.
First, load the data, specifically the close price of the two securities, from the CSV
files in the example code directory of this chapter and calculate the returns. If you
don't remember how to do it, look at the examples in Chapter 3, Getting Familiar
with Commonly Used Functions.
2.
Covariance tells us how two variables vary together; which is nothing more
than unnormalized correlation (see https://www.khanacademy.org/math/
probability/regression/regression-correlation/v/covariance-andthe-regression-line):
cov ( a, b ) =
1
N
N
∑(a
i =1
j
− mean ( a ) ) ( b j − mean ( b ) )
Compute the covariance matrix from the returns with the cov() function
(it's not strictly necessary to do this, but it will allow us to demonstrate a
few matrix operations):
covariance = np.cov(bhp_returns, vale_returns)
print("Covariance", covariance)
The covariance matrix is as follows:
Covariance [[ 0.00028179
[ 0.00019766
0.00019766]
0.00030123]]
[ 100 ]
Chapter 4
3.
View the values on the diagonal with the diagonal() method:
print("Covariance diagonal", covariance.diagonal())
The diagonal values of the covariance matrix are as follows:
Covariance diagonal [ 0.00028179
0.00030123]
Notice that the values on the diagonal are not equal to each other. This is different
from the correlation matrix.
4.
Compute the trace, the sum of the diagonal values, with the trace() method:
print("Covariance trace", covariance.trace())
The trace values of the covariance matrix are as follows:
Covariance trace 0.00058302354992
5.
The correlation of two vectors is defined as the covariance, divided by the product
of the respective standard deviations of the vectors. The equation for vectors a and
b is as follows:
print(covariance/ (bhp_returns.std() * vale_returns.std()))
The correlation matrix is as follows:
[[ 1.00173366
[ 0.70264666
6.
0.70264666]
1.0708476 ]]
We will measure the correlation of our pair with the correlation coefficient. The
correlation coefficient takes values between -1 and 1. The correlation of a set of
values with itself is 1 by definition. This would be the ideal value; however, we will
also be happy with a slightly lower value. Calculate the correlation coefficient (or,
more accurately, the correlation matrix) with the corrcoef() function:
print("Correlation coefficient", np.corrcoef(bhp_returns, vale_
returns))
The coefficients are as follows:
[[ 1.
[ 0.67841747
0.67841747]
1.
]]
The values on the diagonal are just the correlations of the BHP and VALE with
themselves and are, therefore, equal to 1. In all likelihood, no real calculation takes
place. The other two values are equal to each other since correlation is symmetrical,
meaning that the correlation of BHP with VALE is equal to the correlation of VALE
with BHP. It seems that here the correlation is not that strong.
[ 101 ]
Convenience Functions for Your Convenience
7.
Another important point is whether the two stocks under consideration are in sync
or not. Two stocks are considered out of sync if their difference is two standard
deviations from the mean of the differences.
If they are out of sync, we could initiate a trade, hoping that they will eventually
get back in sync again. Compute the difference between the close prices of the
two securities to check the synchronization:
difference = bhp - vale
Check whether the last difference in price is out of sync; see the following code:
avg = np.mean(difference)
dev = np.std(difference)
print("Out of sync", np.abs(difference[-1] – avg) > 2 * dev)
Unfortunately, we cannot trade yet:
Out of sync False
8.
Plotting requires matplotlib; this will be discussed in Chapter 9, Plotting with
matplotlib. Plotting can be done as follows:
t = np.arange(len(bhp_returns))
plt.plot(t, bhp_returns, lw=1, label='BHP returns')
plt.plot(t, vale_returns, '--', lw=2, label='VALE returns')
plt.title('Correlating arrays')
plt.xlabel('Days')
plt.ylabel('Returns')
plt.grid()
plt.legend(loc='best')
plt.show()
[ 102 ]
Chapter 4
The resulting plot is shown here:
What just happened?
We analyzed the relation of the closing stock prices of BHP and VALE. To be precise, we
calculated the correlation of their stock returns. We achieved this with the corrcoef()
function. Furthermore, we saw how to compute the covariance matrix from which the
correlation can be derived. As a bonus, we demonstrated the diagonal() and trace()
methods that give us the diagonal values and the trace of a matrix, respectively. For the
source code, see the correlation.py file in this book's code bundle:
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
bhp = np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True)
bhp_returns = np.diff(bhp) / bhp[ : -1]
vale = np.loadtxt('VALE.csv', delimiter=',', usecols=(6,),
unpack=True)
vale_returns = np.diff(vale) / vale[ : -1]
covariance = np.cov(bhp_returns, vale_returns)
[ 103 ]
Convenience Functions for Your Convenience
print("Covariance", covariance)
print("Covariance diagonal", covariance.diagonal())
print("Covariance trace", covariance.trace())
print(covariance/ (bhp_returns.std() * vale_returns.std()))
print("Correlation coefficient", np.corrcoef(bhp_returns, vale_
returns))
difference = bhp - vale
avg = np.mean(difference)
dev = np.std(difference)
print("Out of sync", np.abs(difference[-1] - avg) > 2 * dev)
t = np.arange(len(bhp_returns))
plt.plot(t, bhp_returns, lw=1, label='BHP returns')
plt.plot(t, vale_returns, '--', lw=2, label='VALE returns')
plt.title('Correlating arrays')
plt.xlabel('Days')
plt.ylabel('Returns')
plt.grid()
plt.legend(loc='best')
plt.show()
Pop quiz – calculating covariance
Q1. Which function returns the covariance of two arrays?
1. covariance
2. covar
3. cov
4. cvar
Polynomials
Do you like calculus? Well I love it! One of the ideas in calculus is Taylor expansion,
that is, representing a differentiable function as an infinite series (see https://www.
khanacademy.org/math/integral-calculus/sequences_series_approx_calc/
taylor-series/v/generalized-taylor-series-approximation and
http://en.wikipedia.org/wiki/Taylor_series.).
[ 104 ]
Chapter 4
The Taylor series is defined as the following sum:
∑
f(
∞
n =0
n)
(a)
n!
( x − a)
n
f (n) ( a )
in this definition is the nth derivative of the function f computed at the
point a.
In practice, this means that we can estimate any differentiable, and therefore continuous,
function with a polynomial of a high degree. We would then assume that the terms of the
higher degrees are negligibly small.
Time for action – fitting to polynomials
The NumPy polyfit() function fits a set of data points to a polynomial, even if the
underlying function is not continuous:
1.
Continuing with the price data of BHP and VALE, look at the difference of their
close prices and fit it to a polynomial of the third power:
bhp=np.loadtxt('BHP.csv', delimiter=',', usecols=(6,),
unpack=True)
vale=np.loadtxt('VALE.csv', delimiter=',', usecols=(6,),
unpack=True)
t = np.arange(len(bhp))
poly = np.polyfit(t, bhp - vale, 3)
print("Polynomial fit", poly)
The polynomial fit (in this example, a cubic polynomial was chosen) is as follows:
Polynomial fit [
5.79791202e+01]
2.
1.11655581e-03
-5.28581762e-02
5.80684638e-01
The numbers you see are the coefficients of the polynomial. Extrapolate to the
next value with the polyval() function and the polynomial object that we got
from the fit:
print("Next value", np.polyval(poly, t[-1] + 1))
The next value we predict will be this:
Next value 57.9743076081
[ 105 ]
Convenience Functions for Your Convenience
3.
Ideally, the difference between the close prices of BHP and VALE should be as small
as possible. In an extreme case, it might be zero at some point. Find out when our
polynomial fit reaches zero with the roots() function:
print( "Roots", np.roots(poly))
The roots of the polynomial are as follows:
Roots [ 35.48624287+30.62717062j
-23.63210575 +0.j
]
4.
35.48624287-30.62717062j
Another thing you may have learned in calculus class was to find extrema—these
could be potential maxima or minima. Remember, from calculus, that these are the
points where the derivative of our function is zero. Differentiate the polynomial fit
with the polyder() function:
der = np.polyder(poly)
print("Derivative", der)
The coefficients of the derivative polynomial are as follows:
Derivative [ 0.00334967 -0.10571635
5.
0.58068464]
Get the roots of the derivative:
print("Extremas", np.roots(der))
The extremas that we get are as follows:
Extremas [ 24.47820054
7.08205278]
Let's double-check and compute the values of the fit with the polyval() function:
vals = np.polyval(poly, t)
6.
Now, find the maximum and minimum values with the argmax() and the
argmin() function:
vals = np.polyval(poly, t)
print(np.argmax(vals))
print(np.argmin(vals))
This gives us the expected results shown in the following screenshot. OK, not quite the
same results, but, if we backtrack to step 1, we can see that t was defined with the
arange() function:
7
24
[ 106 ]
Chapter 4
Plot the data and the fit it to get the following plot:
Obviously, the smooth line is the fit and the jagged line is the underlying data.
But as it's not that good a fit, you might want to try a higher order polynomial.
What just happened?
We fit data to a polynomial with the polyfit() function. We learned about the polyval()
function that computes the values of a polynomial, the roots() function that returns the
roots of the polynomial, and the polyder() function that gives back the derivative of a
polynomial (see polynomials.py):
from __future__ import print_function
import numpy as np
import sys
import matplotlib.pyplot as plt
bhp=np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True)
vale=np.loadtxt('VALE.csv', delimiter=',', usecols=(6,), unpack=True)
t = np.arange(len(bhp))
poly = np.polyfit(t, bhp - vale, 3)
print("Polynomial fit", poly)
print("Next value", np.polyval(poly, t[-1] + 1))
[ 107 ]
Convenience Functions for Your Convenience
print("Roots", np.roots(poly))
der = np.polyder(poly)
print("Derivative", der)
print("Extremas", np.roots(der))
vals = np.polyval(poly, t)
print(np.argmax(vals))
print(np.argmin(vals))
plt.plot(t, bhp - vale, label='BHP - VALE')
plt.plot(t, vals, '-—', label='Fit')
plt.title('Polynomial fit')
plt.xlabel('Days')
plt.ylabel('Difference ($)')
plt.grid()
plt.legend()
plt.show()
Have a go hero – improving the fit
You could do a number of things to improve the fit. For example, try a different power as, in
this section, a cubic polynomial was chosen. Consider smoothing the data before fitting it. One
way you could smooth the data is with a moving average. You can find examples of simple and
EMA calculations in the Chapter 3, Getting Familiar with Commonly Used Functions.
On-balance volume
Volume is a very important variable in investing; it indicates how big a price move is. The
on-balance volume indicator is one of the simplest stock price indicators. It is based on the
close price of the current and previous days and the volume of the current day. For each day,
if the close price today is higher than the close price of yesterday, then the value of the onbalance volume is equal to the volume of today. On the other hand, if today's close price is
lower than yesterday's close price, then the value of the on-balance volume indicator is the
difference between the on-balance volume and the volume of today. However, if the close
price did not change, then the value of the on-balance volume is zero.
[ 108 ]
Chapter 4
Time for action – balancing volume
In other words, we need to multiply the sign of the close price and the volume. In this
section, we look at two approaches to this problem: one using the NumPy sign() function
and the other using the NumPy piecewise() function.
1.
Load the BHP data into a close and volume array:
c, v=np.loadtxt('BHP.csv', delimiter=',', usecols=(6, 7),
unpack=True)
Compute the absolute value changes. Calculate the change of the closing price with
the diff() function. The diff() function computes the difference between two
sequential array elements and returns an array containing these differences:
change = np.diff(c)
print("Change", change)
The changes of the close price are shown as follows:
Change [ 1.92 -1.08 -1.26
0.69 -1.33 1.16
0.63 -1.54 -0.28
1.59 -0.26 -1.29 -0.13 -2.12 -3.91
1.18
-0.88
2.
1.31
0.25 -0.6
2.15
1.28 -0.57 -2.07 -2.07
2.5
1.24 -0.59]
The NumPy sign() function returns the signs for each element in an array. -1 is
returned for a negative number, 1 for a positive number, and 0, otherwise. Apply
the sign() function to the change array:
signs = np.sign(change)
print("Signs", signs)
The signs of the change array are as follows:
Signs [ 1. -1. -1.
-1. -1. -1.
-1. -1. -1.
1.
1.
1. -1. -1.
1. -1.
1.
1. -1.
1.
1. -1.
1.
1. -1. -1.
1. -1.]
Alternatively, we can calculate the signs with the piecewise() function. The
piecewise() function, as its name suggests, evaluates a function piece-by-piece.
Call the function with the appropriate return values and conditions:
pieces = np.piecewise(change, [change < 0, change > 0], [-1,
1])
print("Pieces", pieces)
[ 109 ]
Convenience Functions for Your Convenience
The signs are shown again as follows:
Pieces [ 1. -1. -1.
-1. -1. -1. -1.
1. -1. -1.
1. -1.
-1. -1. -1.
1. -1.
1. -1.]
1.
1.
1.
1.
1. -1.
1.
1. -1.
Check that the outcome is the same:
print("Arrays equal?", np.array_equal(signs, pieces))
And the outcome is as follows:
Arrays equal? True
3.
The on-balance volume depends on the change of the previous close, so we cannot
calculate it for the first day in our sample:
print("On balance volume", v[1:] * signs)
The on-balance volume is as follows:
[ 2620800. -2461300. -3270900.
7768400.
-4799100.
-2463900.
3448300.
2650200. -4667300. -5359800.
4719800. -3898900.
-3590900. -3805000. -3271700. -5507800.
-5008300.
-7809799.
3918800.
3947100.
3809700.
3727700.
3379400.
2996800. -3434800.
3098200. -3500200.
4285600.
-3632200.]
What just happened?
We computed the on-balance volume that depends on the change of the closing price.
Using the NumPy sign() and piecewise() functions, we went over two different
methods to determine the sign of the change (see obv.py) as follows:
from __future__ import print_function
import numpy as np
c, v=np.loadtxt('BHP.csv', delimiter=',', usecols=(6, 7), unpack=True)
change = np.diff(c)
print("Change", change)
signs = np.sign(change)
print("Signs", signs)
[ 110 ]
Chapter 4
pieces = np.piecewise(change, [change < 0, change > 0], [-1, 1])
print("Pieces", pieces)
print("Arrays equal?", np.array_equal(signs, pieces))
print("On balance volume", v[1:] * signs)
Simulation
Often, you would want to try something out first. Play around, experiment, but preferably
without blowing things up or getting dirty! NumPy is perfect for experimentation. We will
use NumPy to simulate a trading day, without actually losing money. Many people like to buy
on the dip or, in other words, wait for the price of stocks to drop before buying. A variant of
this is to wait for the price to drop a small percentage, say 0.1 percent below the opening
price of the day.
Time for action – avoiding loops with vectorize()
The vectorize() function is a yet another trick to reduce the number of loops in your
programs. Calculate the profit of a single trading day following these steps:
1.
First, load the data:
o, h, l, c = np.loadtxt('BHP.csv', delimiter=',', usecols=(3, 4,
5, 6), unpack=True)
2.
The vectorize() function is the NumPy equivalent of the Python map() function.
Call the vectorize() function, giving it as an argument the calc_profit()
function:
func = np.vectorize(calc_profit)
3.
We can now apply func() as if it is a function. Apply the func() function result
that we got to the price arrays:
profits = func(o, h, l, c)
4.
The calc_profit() function is pretty simple. First, we try to buy slightly below
the open price. If this is outside of the daily range, then, obviously, our attempt
failed and no profit was made, or we incurred a loss, therefore, will return 0.
Otherwise, we sell at the close price and the profit is simply the difference between
the buy price and the close price. Actually, it is, in fact, more interesting to have a
look at the relative profit:
def calc_profit(open, high, low, close):
#buy just below the open
buy = open * 0.999
[ 111 ]
Convenience Functions for Your Convenience
# daily range
if low < buy < high:
return (close - buy)/buy
else:
return 0
print("Profits", profits)
5.
Assume that there are two days with zero profits, where there was either no net
gain or a loss. Select the days with trades and calculate the averages:
real_trades = profits[profits != 0]
print("Number of trades", len(real_trades), round(100.0 *
len(real_trades)/len(c), 2), "%")
print("Average profit/loss %", round(np.mean(real_trades) * 100,
2))
The trades summary is shown as follows:
Number of trades 28 93.33 %
Average profit/loss % 0.02
6.
As optimists, we are interested in winning trades with a gain greater than zero.
Select the days with winning trades and calculate the averages:
winning_trades = profits[profits > 0]
print("Number of winning trades", len(winning_trades),
round(100.0 * len(winning_trades)/len(c), 2), "%")
print("Average profit %", round(np.mean(winning_trades) * 100,
2))
The winning trades statistics are as follows:
Number of winning trades 16 53.33 %
Average profit % 0.72
7.
Alternatively, as pessimists, we are interested in losing trades with a profit less than
zero. Select the days with losing trades and calculate the averages:
losing_trades = profits[profits < 0]
print("Number of losing trades", len(losing_trades), round(100.0 *
len(losing_trades)/len(c), 2), "%")
print("Average loss %", round(np.mean(losing_trades) * 100, 2))
The losing trades statistics are as follows:
Number of losing trades 12 40.0 %
Average loss % -0.92
[ 112 ]
Chapter 4
What just happened?
We vectorized a function, which is just another way to avoid using loops. We simulated
a trading day with a function, which returned the relative profit of each day's trade. We
printed a statistics summary of the losing and winning trades (see simulation.py):
from __future__ import print_function
import numpy as np
o, h, l, c = np.loadtxt('BHP.csv', delimiter=',', usecols=(3, 4, 5,
6), unpack=True)
def calc_profit(open, high, low, close):
#buy just below the open
buy = open * 0.999
# daily range
if low < buy < high:
return (close - buy)/buy
else:
return 0
func = np.vectorize(calc_profit)
profits = func(o, h, l, c)
print("Profits", profits)
real_trades = profits[profits != 0]
print("Number of trades", len(real_trades), round(100.0 * len(real_
trades)/len(c), 2), "%")
print("Average profit/loss %", round(np.mean(real_trades) * 100, 2))
winning_trades = profits[profits > 0]
print("Number of winning trades", len(winning_trades), round(100.0 *
len(winning_trades)/len(c), 2), "%")
print("Average profit %", round(np.mean(winning_trades) * 100, 2))
losing_trades = profits[profits < 0]
print("Number of losing trades", len(losing_trades), round(100.0 *
len(losing_trades)/len(c), 2), "%")
print("Average loss %", round(np.mean(losing_trades) * 100, 2))
[ 113 ]
Convenience Functions for Your Convenience
Have a go hero – analyzing consecutive wins and losses
Although the average profit is positive, it is also important to know whether we had to
endure a long streak of consecutive losses. If this is the case, we might be left with little or
no capital, and then the average profit would not matter.
Find out if there was such a losing streak. If you want, you can also find out if there was a
prolonged winning streak.
Smoothing
Noisy data is difficult to deal with, so we often need to do some smoothing. Besides
calculating moving averages, we can use one of the NumPy functions to smooth data.
The hanning() function is a window function formed by a weighted cosine
(see http://en.wikipedia.org/wiki/Hann_function):
2π n
w ( n ) = 0.5 − 0.5cos
N −1
In the preceding formula, N corresponds to the size of the window. We will cover the other
window functions in later chapters.
Time for action – smoothing with the hanning() function
We will use the hanning() function to smooth arrays of stock returns, as shown in the
following steps:
1.
Call the hanning() function to compute weights for a certain length window (in
this example 8) as follows:
N = 8
weights = np.hanning(N)
print("Weights", weights)
The weights are as follows:
Weights [ 0.
0.1882551
0.95048443 0.61126047
0.1882551
0.
]
[ 114 ]
0.61126047
0.95048443
Chapter 4
2.
Calculate the stock returns for the BHP and VALE quotes using convolve() with
normalized weights:
bhp = np.loadtxt('BHP.csv', delimiter=',', usecols=(6,),
unpack=True)
bhp_returns = np.diff(bhp) / bhp[ : -1]
smooth_bhp = np.convolve(weights/weights.sum(),
bhp_returns)[N-1:-N+1]
vale = np.loadtxt('VALE.csv', delimiter=',', usecols=(6,),
unpack=True)
vale_returns = np.diff(vale) / vale[ : -1]
smooth_vale = np.convolve(weights/weights.sum(),
vale_returns)[N-1:-N+1]
3.
Plot with matplotlib using this code:
t = np.arange(N - 1, len(bhp_returns))
plt.plot(t, bhp_returns[N-1:], lw=1.0)
plt.plot(t, smooth_bhp, lw=2.0)
plt.plot(t, vale_returns[N-1:], lw=1.0)
plt.plot(t, smooth_vale, lw=2.0)
plt.show()
The chart would appear as follows:
[ 115 ]
Convenience Functions for Your Convenience
The thin lines on the preceding chart are the stock returns and the thick lines are the
result of smoothing. As you can see, the lines cross a few times. These points might
be important because the trend might have changed there. Or, at least, the relation
of BHP to VALE might have changed. These turning inflection points probably occur
often, so we might want to project into the future.
4.
Fit the result of the smoothing step to polynomials as follows:
K = 8
t = np.arange(N - 1, len(bhp_returns))
poly_bhp = np.polyfit(t, smooth_bhp, K)
poly_vale = np.polyfit(t, smooth_vale, K)
5.
Next, we need to evaluate the situation, where the polynomials we found in
the previous step were equal to each other. This boils down to subtracting the
polynomials and finding the roots of the resulting polynomial. Subtract the
polynomials using polysub():
poly_sub = np.polysub(poly_bhp, poly_vale)
xpoints = np.roots(poly_sub)
print("Intersection points", xpoints)
The points are shown as follows:
Intersection points [ 27.73321597+0.j
24.32064343+0.j
18.86423973+0.j
1.73218179j
12.43797190+1.73218179j
6.34613053+0.62519463j
6.
27.51284094+0.j
12.43797190-
6.34613053-0.62519463j]
The numbers we get are complex, and that is not good for us (unless there
is such a thing as imaginary time). Check which numbers are real with the
isreal() function:
reals = np.isreal(xpoints)
print("Real number?", reals)
The result is as follows:
Real number? [ True
True
True
True False False False False]
Some of the numbers are real, so select them with the select() function. The
select() function forms an array by taking elements from a list of choices, based
on a list of conditions:
xpoints = np.select([reals], [xpoints])
xpoints = xpoints.real
print("Real intersection points", xpoints)
[ 116 ]
Chapter 4
The real intersection points are as follows:
Real intersection points [ 27.73321597 27.51284094
18.86423973
0.
0.
0. 0.]
7.
24.32064343
We managed to pick up some zeroes. The trim_zeros() function strips the
leading and trailing zeroes from a one-dimensional array. Get rid of the zeroes
with the trim_zeros() function:
print("Sans 0s", np.trim_zeros(xpoints))
The zeroes are gone, and the output is shown as follows:
Sans 0s [ 27.73321597
27.51284094
24.32064343
18.86423973]
What just happened?
We applied the hanning() function to smooth arrays containing stock returns. We
subtracted two polynomials with the polysub() function. We then checked for real
numbers with the isreal() function and selected the real numbers with the select()
function. Finally, we stripped zeroes from an array with the trim_zeros() function
(see smoothing.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
N = 8
weights = np.hanning(N)
print("Weights", weights)
bhp = np.loadtxt('BHP.csv', delimiter=',', usecols=(6,), unpack=True)
bhp_returns = np.diff(bhp) / bhp[ : -1]
smooth_bhp = np.convolve(weights/weights.sum(), bhp_returns)[N-1:-N+1]
vale = np.loadtxt('VALE.csv', delimiter=',', usecols=(6,),
unpack=True)
vale_returns = np.diff(vale) / vale[ : -1]
smooth_vale = np.convolve(weights/weights.sum(), vale_returns)
[N-1:-N+1]
K = 8
t = np.arange(N - 1, len(bhp_returns))
poly_bhp = np.polyfit(t, smooth_bhp, K)
[ 117 ]
Convenience Functions for Your Convenience
poly_vale = np.polyfit(t, smooth_vale, K)
poly_sub = np.polysub(poly_bhp, poly_vale)
xpoints = np.roots(poly_sub)
print("Intersection points", xpoints)
reals = np.isreal(xpoints)
print("Real number?", reals)
xpoints = np.select([reals], [xpoints])
xpoints = xpoints.real
print("Real intersection points", xpoints)
print("Sans 0s", np.trim_zeros(xpoints))
plt.plot(t, bhp_returns[N-1:], lw=1.0, label='BHP returns')
plt.plot(t, smooth_bhp, lw=2.0, label='BHP smoothed')
plt.plot(t, vale_returns[N-1:], '--', lw=1.0, label='VALE returns')
plt.plot(t, smooth_vale, '-.', lw=2.0, label='VALE smoothed')
plt.title('Smoothing')
plt.xlabel('Days')
plt.ylabel('Returns')
plt.grid()
plt.legend(loc='best')
plt.show()
Have a go hero – smoothing variations
Experiment with the other smoothing functions—hamming(), blackman(), bartlett(),
and kaiser(). They work in more or less the same way as the hanning() function.
Initialization
So far in this book, we encountered several convenient functions for intializing arrays. The
full() and full_like() functions were recently added to NumPy to make initialization
even easier.
[ 118 ]
Chapter 4
The following short Python session shows (abbreviated) documentation for these
two functions:
$ python
>>> import numpy as np
>>> help(np.full)
Return a new array of given shape and type, filled with `fill_value`.
>>> help(np.full_like)
Return a full array with the same shape and type as a given array.
Time for action – creating value initialized arrays with the full()
and full_like() functions
Let's demonstrate how the full() and full_like() functions work. If you are not in a
Python shell already, type the following:
$ python
>>> import numpy as np
1.
Create a one-by-two array with the full() function filled with the number 42
as follows:
>>> np.full((1, 2), 42)
array([[ 42.,
42.]])
As you can deduce from the output, the array elements are floating-point numbers,
which is the default data type for NumPy arrays. Specify an integer data type
as follows:
>>> np.full((1, 2), 42, dtype=np.int)
array([[42, 42]])
2.
The full_like() function looks at the metadata of an input array and uses that
information to create a new array, filled with a specified value. For instance, after
creating an array with the linspace() function, use that as a template for the
full_like() function:
>>> a = np.linspace(0, 1, 5)
>>> a
array([ 0.
,
0.25,
0.5 ,
0.75,
1.
>>> np.full_like(a, 42)
array([ 42.,
42.,
42.,
42.,
[ 119 ]
42.])
])
Convenience Functions for Your Convenience
Again we have an array filled with 42. To change the data type to integer, type the
following:
>>> np.full_like(a, 42, dtype=np.int)
array([42, 42, 42, 42, 42])
What just happened?
We created arrays using the full() and full_like() functions. The full() function
filled the array with the number 42. The full_like() function uses the metadata of an
input array to create a new array. Both functions allow you to specify the data type.
Summary
We calculated the correlation of the stock returns of two stocks with the corrcoef()
function. As a bonus, we demonstrated the diagonal() and trace() functions,
which can give us the diagonal and trace of a matrix.
We fit data to a polynomial with the polyfit() function. We learned about the
polyval() function that computes the values of a polynomial, the roots() function
that returns the roots of the polynomial, and the polyder() function, which gives back
the derivative of a polynomial.
We saw that the full() function fills an array with a number, and the full_like()
function uses the metadata of an input array to create a new array. Both functions allow
you to specify the data type.
Hopefully, you have increased your productivity, so that we can continue in the next chapter
with matrices and Universal Functions (ufuncs).
[ 120 ]
5
Working with Matrices and ufuncs
This chapter covers matrices and Universal functions (ufuncs). Matrices are
well known in mathematics and have their representation in NumPy as well.
Universal functions work on arrays, element by element, or on scalars. ufuncs
expect a set of scalars as input and produce a set of scalars as output. Universal
functions can typically be mapped to their mathematical counterparts such as
add, subtract, divide, multiply, and so on. We will also introduce trigonometric,
bitwise, and comparison universal functions.
In this chapter, we will cover the following topics:
Matrix creation
Matrix operations
Basic ufuncs
Trigonometric functions
Bitwise functions
Comparison functions
[ 121 ]
Working with Matrices and ufuncs
Matrices
Matrices in NumPy are subclasses of ndarray. We can create matrices using a special
string format. They are, just like in mathematics, two-dimensional (see https://www.
khanacademy.org/math/precalculus/precalc-matrices). Matrix multiplication is, as
you would expect, different from the normal NumPy multiplication. The same is true for the
power operator. We can create matrices with the mat(), matrix(), and bmat() functions.
Time for action – creating matrices
The mat() function does not make a copy if the input is already a matrix or an ndarray.
Calling this function is equivalent to calling matrix(data, copy=False). We will also
demonstrate transposing and inverting matrices.
1.
Rows are delimited by a semicolon and values by a space. Call the mat() function
with the following string to create a matrix:
A = np.mat('1 2 3; 4 5 6; 7 8 9')
print("Creation from string", A)
The matrix output should be the following matrix:
Creation from string [[1 2 3]
[4 5 6]
[7 8 9]]
2.
Transpose the matrix with the T attribute as follows:
print("transpose A", A.T)
The following is the transposed matrix:
transpose A [[1 4 7]
[2 5 8]
[3 6 9]]
3.
The matrix can be inverted with the I attribute as follows (see https://www.
khanacademy.org/math/precalculus/precalc-matrices/inverting_
matrices/v/inverse-matrix-part-1):
print("Inverse A", A.I)
The inverse matrix is printed as follows (be warned that this is a O(n3) operation,
meaning that it takes on average cubic time):
Inverse A [[ -4.50359963e+15
[
9.00719925e+15
-4.50359963e+15]
9.00719925e+15
-1.80143985e+16
9.00719925e+15]
[ -4.50359963e+15
9.00719925e+15
-4.50359963e+15]]
[ 122 ]
Chapter 5
4.
Instead of using a string to create a matrix, do it with an array:
print("Creation from array", np.mat(np.arange(9).reshape(3, 3)))
The newly created array is printed as follows:
Creation from array [[0 1 2]
[3 4 5]
[6 7 8]]
What just happened?
We created matrices with the mat() function. We transposed the matrices with the T
attribute and inverted them with the I attribute (see matrixcreation.py):
from __future__ import print_function
import numpy as np
A = np.mat('1 2 3; 4 5 6; 7 8 9')
print("Creation from string", A)
print("transpose A", A.T)
print("Inverse A", A.I)
print("Check Inverse", A * A.I)
print("Creation from array", np.mat(np.arange(9).reshape(3, 3)))
Creating a matrix from other matrices
Sometimes, we want to create a matrix from other smaller matrices. We can do this with the
bmat() function. The b here stands for block matrix.
Time for action – creating a matrix from other matrices
We will create a matrix from two smaller matrices as follows:
1.
First, create a 2-by-2 identity matrix:
A = np.eye(2)
print("A", A)
The identity matrix looks like the following:
A [[ 1.
[ 0.
0.]
1.]]
[ 123 ]
Working with Matrices and ufuncs
2.
Create another matrix like A and multiply it by 2:
B = 2 * A
print("B", B)
The second matrix is as follows:
B [[ 2.
[ 0.
3.
0.]
2.]]
Create the compound matrix from a string. The string uses the same format as the
mat() function—use matrices instead of numbers:
print("Compound matrix\n", np.bmat("A B; A B"))
The compound matrix is shown as follows:
Compound matrix
[[ 1.
0.
2.
0.]
[ 0.
1.
0.
2.]
[ 1.
0.
2.
0.]
[ 0.
1.
0.
2.]]
What just happened?
We created a block matrix from two smaller matrices with the bmat() function. We
gave the function a string containing the names of matrices instead of numbers (see
bmatcreation.py):
from __future__ import print_function
import numpy as np
A = np.eye(2)
print("A", A)
B = 2 * A
print("B", B)
print("Compound matrix\n", np.bmat("A B; A B"))
Pop quiz – defining a matrix with a string
Q1. What is the row delimiter in a string accepted by the mat() and bmat() functions?
1. Semicolon
2. Colon
3. Comma
4. Space
[ 124 ]
Chapter 5
Universal functions
Universal functions (ufuncs) expect a set of scalars as input and produce a set of scalars as
output. They are actually Python objects that encapsulate the behavior of a function. We
can typically map ufuncs to their mathematical counterparts such as add, subtract, divide,
multiply, and so on. Universal functions are, in general, faster because of their special
optimizations and because they run on the native level.
Time for action – creating universal functions
We can create a ufunc from a Python function with the NumPy the frompyfunc() function
as follows:
1.
Define a Python function that answers the ultimate question to the universe,
existence, and the rest (it's from The Hitchhiker's Guide to the Galaxy, Douglas
Adam, Pan Books, if you haven't read it, you can safely ignore this!):
def ultimate_answer(a):
So far, nothing special; we gave the function the name ultimate_answer() and
defined one parameter, a.
2.
Create a result consisting of all zeros that has the same shape as a, with the
zeros_like() function:
result = np.zeros_like(a)
3.
Now, set the elements of the initialized array to the answer 42 and return the
result. The complete function should appear as shown in the following code snippet.
The flat attribute gives us access to a flat iterator that allows us to set the value of
the array.
def ultimate_answer(a):
result = np.zeros_like(a)
result.flat = 42
return result
4.
Create a ufunc with frompyfunc(); specify 1 as the number of input parameter
followed by 1 as the number of output parameters:
ufunc = np.frompyfunc(ultimate_answer, 1, 1)
print("The answer", ufunc(np.arange(4)))
The result for a one-dimensional array is shown as follows:
The answer [42 42 42 42]
[ 125 ]
Working with Matrices and ufuncs
Do the same for a two-dimensional array with the following code:
print("The answer", ufunc(np.arange(4).reshape(2, 2)))
The output for a two dimensional array is shown as follows:
The answer [[42 42]
[42 42]]
What just happened?
We defined a Python function. In this function, we initialized to zero the elements of an
array, based on the shape of an input argument, with the zeros_like() function. Then,
with the flat attribute of ndarray, we set the array elements to the ultimate answer, 42
(see answer42.py):
from __future__ import print_function
import numpy as np
def ultimate_answer(a):
result = np.zeros_like(a)
result.flat = 42
return result
ufunc = np.frompyfunc(ultimate_answer, 1, 1)
print("The answer", ufunc(np.arange(4)))
print("The answer", ufunc(np.arange(4).reshape(2, 2)))
Universal function methods
How can functions have methods? As we said earlier, universal functions are not functions
but Python objects representing functions. Universal functions have five important methods
listed as follows:
1. ufunc.reduce(a[, axis, dtype, out, keepdims])
2. ufunc.accumulate(array[, axis, dtype, out])
3. ufunc.reduceat(a, indices[, axis, dtype, out])
4. ufunc.outer(A, B)
5. ufunc.at(a, indices[, b])])])
[ 126 ]
Chapter 5
Time for action – applying the ufunc methods to the
add function
Let's call the first four methods on the add() function:
1.
The universal function reduces the input array recursively along a specified axis on
consecutive elements. For the add() function, the result of reducing is similar to
calculating the sum of an array. Call the reduce() method:
a = np.arange(9)
print("Reduce", np.add.reduce(a))
The reduced array should be as follows:
Reduce 36
2.
The accumulate() method also recursively goes through the input array. But,
contrary to the reduce() method, it stores the intermediate results in an array and
returns that. The result, in the case of the add() function, is equivalent to calling
the cumsum() function. Call the accumulate() method on the add() function:
print("Accumulate", np.add.accumulate(a))
The accumulated array is as follows:
Accumulate [ 0
3.
1
3
6 10 15 21 28 36]
The reduceat() method is a bit complicated to explain, so let's call it and go
through its algorithm, step by step. The reduceat() method requires as arguments
an input array and a list of indices:
print("Reduceat", np.add.reduceat(a, [0, 5, 2, 7]))
The result is shown as follows:
Reduceat [10
5 20 15]
The first step concerns the indices 0 and 5. This step results in a reduce operation of
the array elements between indices 0 and 5:
print("Reduceat step I", np.add.reduce(a[0:5]))
The output of step 1 is as follows:
Reduceat step I 10
The second step concerns indices 5 and 2. Since 2 is less than 5, the array element
at index 5 is returned:
print("Reduceat step II", a[5])
[ 127 ]
Working with Matrices and ufuncs
The second step results in the following output:
Reduceat step II 5
The third step concerns indices 2 and 7. This step results in a reduce operation of
the array elements between indices 2 and 7:
print("Reduceat step III", np.add.reduce(a[2:7]))
The result of the third step is shown as follows:
Reduceat step III 20
The fourth step concerns index 7. This step results in a reduce operation of the array
elements from index 7 to the end of the array:
print("Reduceat step IV", np.add.reduce(a[7:]))
The fourth step result is shown as follows:
Reduceat step IV 15
4.
The outer() method returns an array that has a rank, which is the sum of the ranks
of its two input arrays. The method is applied to all possible pairs of the input array
elements. Call the outer() method on the add() function:
print("Outer", np.add.outer(np.arange(3), a))
The outer sum output result is as follows:
Outer [[ 0
1
2
3
4
5
6
7
8]
[ 1
2
3
4
5
6
7
8
9]
[ 2
3
4
5
6
7
8
9 10]]
What just happened?
We applied the first four methods, reduce(), accumulate(), reduceat(), and outer(),
of universal functions to the add() function (see ufuncmethods.py):
from __future__ import print_function
import numpy as np
a = np.arange(9)
print("Reduce", np.add.reduce(a))
print("Accumulate", np.add.accumulate(a))
print("Reduceat", np.add.reduceat(a, [0, 5, 2, 7]))
print("Reduceat step I", np.add.reduce(a[0:5]))
print("Reduceat step II", a[5])
[ 128 ]
Chapter 5
print("Reduceat step III", np.add.reduce(a[2:7]))
print("Reduceat step IV", np.add.reduce(a[7:]))
print("Outer", np.add.outer(np.arange(3), a))
Arithmetic functions
The common arithmetic operators +, -, and * are implicitly linked to the add, subtract,
and multiply universal functions, respectively. This means that when you use one of these
operators on a NumPy array, the corresponding universal function will get called. Division
involves a slightly more complex process. The three universal functions that have to do with
array division are divide(), true_divide(), and floor_division(). Two operators
correspond to division: / and //.
Time for action – dividing arrays
Let's see the array division in action:
1.
The divide() function does truncated integer division and normal floating-point
division:
a = np.array([2, 6, 5])
b = np.array([1, 2, 3])
print("Divide", np.divide(a, b), np.divide(b, a))
The result of the divide() function is shown as follows:
Divide [2 3 1] [0 0 0]
As you can see, truncation took place.
2.
The true_divide() function comes closer to the mathematical definition of
division. Integer division returns a floating-point result and no truncation occurs:
print("True Divide", np.true_divide(a, b), np.true_divide(b, a))
The result of the true_divide() function is as follows:
True Divide [ 2.
0.33333333 0.6
3.
3.
1.66666667] [ 0.5
]
The floor_divide() function always returns an integer result. It is equivalent to
calling the floor() function after calling the divide() function. The floor()
function discards the decimal part of a floating-point number and returns an integer:
print("Floor Divide", np.floor_divide(a, b), np.floor_divide(b, a))
c = 3.14 * b
print("Floor Divide 2", np.floor_divide(c, b),
np.floor_divide(b, c))
[ 129 ]
Working with Matrices and ufuncs
The floor_divide() function call results in:
Floor Divide [2 3 1] [0 0 0]
Floor Divide 2 [ 3.
4.
3.
3.] [ 0.
0.
0.]
By default, the / operator is equivalent to calling the divide() function:
from __future__ import division
However, if this line is found at the beginning of a Python program, the
true_divide() function is called instead. So, this code will appear as follows:
print("/ operator", a/b, b/a)
The result is shown as follows:
/ operator [ 2.
0.33333333 0.6
5.
3.
1.66666667] [ 0.5
]
The // operator is equivalent to calling the floor_divide() function. For
example, look at the following code snippet:
print("// operator", a//b, b//a)
print("// operator 2", c//b, b//c)
The // operator result is shown as follows:
// operator [2 3 1] [0 0 0]
// operator 2 [ 3.
3.
3.] [ 0.
0.
0.]
What just happened?
The divide() function truncates the integer division and normal floating-point division. The
true_divide() function always returns a floating-point result without any truncation. The
floor_divide() function always returns an integer result; the result is the same that you
will get by calling the divide() and floor() functions consecutively (see dividing.py):
from __future__ import print_function
from __future__ import division
import numpy as np
a = np.array([2, 6, 5])
b = np.array([1, 2, 3])
print("Divide", np.divide(a, b), np.divide(b, a))
print("True Divide", np.true_divide(a, b), np.true_divide(b, a))
[ 130 ]
Chapter 5
print("Floor Divide", np.floor_divide(a, b), np.floor_divide(b, a))
c = 3.14 * b
print("Floor Divide 2", np.floor_divide(c, b), np.floor_divide(b, c))
print("/ operator", a/b, b/a)
print("// operator", a//b, b//a)
print("// operator 2", c//b, b//c)
Have a go hero – experimenting with __future__.division
Experiment to confirm the impact of importing __future__.division.
Modulo operation
We can calculate the modulo or remainder using the NumPy mod(), remainder(), and
fmod() functions. Also, we can use the % operator. The main difference among these
functions is how they deal with negative numbers. The odd one out in this group is the
fmod() function.
Time for action – computing the modulo
Let's call the previously mentioned functions:
1.
The remainder() function returns the remainder of the two arrays, element-wise.
0 is returned if the second number is 0:
a = np.arange(-4, 4)
print("Remainder", np.remainder(a, 2))
The result of the remainder() function is shown as follows:
Remainder [0 1 0 1 0 1 0 1]
2.
The mod() function does exactly the same as the remainder() function:
print("Mod", np.mod(a, 2))
The result of the mod() function is shown as follows:
Mod [0 1 0 1 0 1 0 1]
3.
The % operator is just shorthand for the remainder() function:
print("% operator", a % 2)
The result of the % operator is shown as follows:
% operator [0 1 0 1 0 1 0 1]
[ 131 ]
Working with Matrices and ufuncs
4.
The fmod() function handles negative numbers differently than mod(), fmod(),
and % do. The sign of the remainder is the sign of the dividend, and the sign of the
divisor has no influence on the results:
print("Fmod", np.fmod(a, 2))
The fmod() result is printed as follows:
Fmod [ 0 -1
0 -1
0
1
0
1]
What just happened?
We demonstrated the NumPy the mod(), remainder(), and fmod() functions, which
compute the modulo or remainder (see modulo.py):
from __future__ import print_function
import numpy as np
a = np.arange(-4, 4)
print("Remainder", np.remainder(a, 2))
print("Mod", np.mod(a, 2))
print("% operator", a % 2)
print("Fmod", np.fmod(a, 2))
Fibonacci numbers
The Fibonacci numbers (see http://en.wikipedia.org/wiki/Fibonacci_number)
are based on a recurrence relation:
Fn = Fn −1 + Fn − 2
It is difficult to express this relation directly with NumPy code. However, we can express this
relation in a matrix form or use the following golden ratio formula:
Fn =
ϕ n − ( −ϕ )
5
with
ϕ=
1+ 5
2
[ 132 ]
−n
Chapter 5
This will introduce the matrix() and rint() functions. The matrix() function creates
matrices and the rint() function rounds numbers to the closest integer, but the result is
not an integer.
Time for action – computing Fibonacci numbers
A matrix can represent the Fibonacci recurrence relation. We can express the calculation of
Fibonacci numbers as a repeated matrix multiplication:
1.
Create the Fibonacci matrix as follows:
F = np.matrix([[1, 1], [1, 0]])
print("F", F)
The Fibonacci matrix appears as follows:
F [[1 1]
[1 0]]
2.
Calculate the 8th Fibonacci number (ignoring 0), by subtracting 1 from 8 and taking
the power of the matrix. The Fibonacci number then appears on the diagonal:
print("8th Fibonacci", (F ** 7)[0, 0])
The Fibonacci number is as follows:
8th Fibonacci 21
3.
The golden ratio formula, better known as Binet's formula, allows us to calculate
Fibonacci numbers with a rounding step at the end. Calculate the first eight
Fibonacci numbers:
n = np.arange(1, 9)
sqrt5 = np.sqrt(5)
phi = (1 + sqrt5)/2
fibonacci = np.rint((phi**n - (-1/phi)**n)/sqrt5)
print("Fibonacci", fibonacci)
The first eight Fibonacci numbers are as follows:
Fibonacci [
1.
1.
2.
3.
[ 133 ]
5.
8.
13.
21.]
Working with Matrices and ufuncs
What just happened?
We computed Fibonacci numbers in two ways. In the process, we learned about the
matrix() function that creates matrices. We also learned about the rint() function
that rounds numbers to the closest integer but does not change the type to integer
(see fibonacci.py):
from __future__ import print_function
import numpy as np
F = np.matrix([[1, 1], [1, 0]])
print("F", F)
print("8th Fibonacci", (F ** 7)[0, 0])
n = np.arange(1, 9)
sqrt5 = np.sqrt(5)
phi = (1 + sqrt5)/2
fibonacci = np.rint((phi**n - (-1/phi)**n)/sqrt5)
print("Fibonacci", fibonacci)
Have a go hero – timing the calculations
You are probably wondering which approach is faster, so go ahead and time it. Create a
universal Fibonacci function with frompyfunc() and time that too.
Lissajous curves
All the standard trigonometric functions such as sin, cos, tan, and so on are represented
by universal functions in NumPy (see https://www.khanacademy.org/math/
trigonometry). Lissajous curves are a fun way of using trigonometry. I remember
producing Lissajous figures on an oscilloscope in the physics lab. Two parametric equations
describe the figures:
x = A sin(at + π/2)
y = B sin(bt)
[ 134 ]
Chapter 5
Time for action – drawing Lissajous curves
The Lissajous figures are determined by four parameters: A, B, a, and b. Let's set A and B to 1
for simplicity:
1.
Initialize t with the linspace() function from -pi to pi with 201 points:
a = 9
b = 8
t = np.linspace(-np.pi, np.pi, 201)
2.
Calculate x with the sin() function and np.pi:
x = np.sin(a * t + np.pi/2)
3.
Calculate y with the sin() function:
y = np.sin(b * t)
4.
Plot as shown in the following:
plt.plot(x, y)
plt.title('Lissajous curves')
plt.grid()
plt.show()
The result for a = 9 and b = 8 is as follows:
[ 135 ]
Working with Matrices and ufuncs
What just happened?
We plotted the Lissajous curve with the aforementioned parametric equations where A=B=1,
a=9, and b=8. We used the sin() and linspace() functions, as well as the NumPy pi
constant (see lissajous.py):
import numpy as np
import matplotlib.pyplot as plt
a = 9
b = 8
t = np.linspace(-np.pi, np.pi, 201)
x = np.sin(a * t + np.pi/2)
y = np.sin(b * t)
plt.plot(x, y)
plt.title('Lissajous curves')
plt.grid()
plt.show()
Square waves
Square waves are also one of those neat things that you can view on an oscilloscope. They
can be approximated pretty well with sine waves; after all, a square wave is a signal that can
be represented by an infinite Fourier series.
A Fourier series is the sum of a series of sine and cosine terms named after the
famous mathematician Jean-Baptiste Fourier (see http://en.wikipedia.
org/wiki/Fourier_series).
The formula of this particular series representing the square wave is as follows:
∞
∑
k =1
4sin ( 2π ( 2k − 1) ft )
( 2k − 1) π
[ 136 ]
Chapter 5
Time for action – drawing a square wave
We will initialize t just as in the previous section. We need to sum a number of terms. The
higher the number of terms, the more accurate the result; k = 99 should be sufficient. In
order to draw a square wave, follow these steps:
1.
We will start by initializing t and k. Set the initial values for the function to 0:
t
k
k
f
2.
=
=
=
=
np.linspace(-np.pi, np.pi, 201)
np.arange(1, 99)
2 * k - 1
np.zeros_like(t)
Compute the function values with the sin() and sum() functions:
for i, ti in enumerate(t):
f[i] = np.sum(np.sin(k * ti)/k)
f = (4 / np.pi) * f
3.
The code to plot is almost identical to the one in the previous section:
plt.plot(t, f)
plt.title('Square wave')
plt.grid()
plt.show()
The resulting square wave generated with k = 99 is as follows:
[ 137 ]
Working with Matrices and ufuncs
What just happened?
We generated a square wave or, at least, a fair approximation of it, using the sin() function.
The input values were assembled with the linspace() function and the k values with the
arange() function (see squarewave.py):
import numpy as np
import matplotlib.pyplot as plt
t
k
k
f
=
=
=
=
np.linspace(-np.pi, np.pi, 201)
np.arange(1, 99)
2 * k - 1
np.zeros_like(t)
for i, ti in enumerate(t):
f[i] = np.sum(np.sin(k * ti)/k)
f = (4 / np.pi) * f
plt.plot(t, f)
plt.title('Square wave')
plt.grid()
plt.show()
Have a go hero – getting rid of the loop
You may have noticed that there is one loop in the code. Get rid of it with NumPy functions
and make sure the performance is also improved.
Sawtooth and triangle waves
Sawtooth and triangle waves are also a phenomenon easily viewed on an oscilloscope. Just
as with square waves, we can define an infinite Fourier series. The triangle waves can be
found by taking the absolute value of a sawtooth wave. The formula for the representation
of a series of sawtooth waves is as follows:
−2sin ( 2π kft )
kπ
k =1
∞
∑
[ 138 ]
Chapter 5
Time for action – drawing sawtooth and triangle waves
We will initialize t just like in the previous section. Again, k = 99 should be sufficient. In
order to draw sawtooth and triangle waves, follow these steps:
1.
Set initial values for the function to zero:
t = np.linspace(-np.pi, np.pi, 201)
k = np.arange(1, 99)
f = np.zeros_like(t)
2.
Compute the function values with the sin() and sum() functions:
for i, ti in enumerate(t):
f[i] = np.sum(np.sin(2 * np.pi * k * ti)/k)
f = (-2 / np.pi) * f
3.
It's easy to plot the sawtooth and triangle waves since the value of the triangle
wave should be equal to the absolute value of the sawtooth wave. Plot the waves
as shown in the following:
plt.plot(t, f, lw=1.0, label='Sawtooth')
plt.plot(t, np.abs(f), '--', lw=2.0, label='Triangle')
plt.title('Triangle and sawtooth waves')
plt.grid()
plt.legend(loc='best')
plt.show()
In the following figure, the triangle wave is the one with the dashed line:
[ 139 ]
Working with Matrices and ufuncs
What just happened?
We drew a sawtooth wave using the sin() function. We assembled the input values with
the linspace() function and the k values with the arange() function. A triangle wave
was derived from the sawtooth wave by taking the absolute value (see sawtooth.py):
import numpy as np
import matplotlib.pyplot as plt
t = np.linspace(-np.pi, np.pi, 201)
k = np.arange(1, 99)
f = np.zeros_like(t)
for i, ti in enumerate(t):
f[i] = np.sum(np.sin(2 * np.pi * k * ti)/k)
f = (-2 / np.pi) * f
plt.plot(t, f, lw=1.0, label='Sawtooth')
plt.plot(t, np.abs(f), '--', lw=2.0, label='Triangle')
plt.title('Triangle and sawtooth waves')
plt.grid()
plt.legend(loc='best')
plt.show()
Have a go hero – getting rid of the loop
Your challenge, should you choose to accept it, is to get rid of the loop in the program. It
should be doable with NumPy functions and the performance should improve.
Bitwise and comparison functions
Bitwise functions operate on the bits of integers or integer arrays since they are universal
functions. The operators ^, &, |, <<, >>, and so on have their NumPy counterparts. The same
goes for comparison operators such as <, >, ==, and so on. These operators allow you to do
clever tricks, which should be good for performance; however, they can make your code
quite unreadable, so use them with care.
[ 140 ]
Chapter 5
Time for action – twiddling bits
We will now cover three tricks—checking whether the signs of integers are different,
checking whether a number is a power of 2, and calculating the modulus of a number that
is a power of 2. We will show an operators-only notation and one using the corresponding
NumPy functions:
1.
The first trick depends on the XOR or ^ operator. The XOR operator is also called
the inequality operator; so, if the sign bit of the two operands is different, the
XOR operation will lead to a negative number (see https://www.khanacademy.
org/computing/computer-science/cryptography/ciphers/a/xorbitwise-operation).
The following truth table illustrates the XOR operator:
Input 1
Input 2
XOR
True
True
False
False
True
True
True
False
True
False
False
False
The ^ operator corresponds to the bitwise_xor() function, and the < operator
corresponds to the less() function:
x = np.arange(-9, 9)
y = -x
print("Sign different?", (x ^ y) < 0)
print("Sign different?", np.less(np.bitwise_xor(x, y), 0))
The result is shown as follows:
Sign different? [ True
True False True True
True
True
True
True
True
True
Sign different? [ True
True False True True
True
True
True
True
True
True
True
True
True
True
True
True
True]
True
True
True
True
True
True
True]
As expected, all the signs differ, except for zero.
[ 141 ]
Working with Matrices and ufuncs
2.
A power of 2 is represented by a 1, followed by a series of trailing zeroes in binary
notation. For instance, 10, 100, or 1000. A number one less than a power of 2 will
be represented by a row of ones in binary. For instance, 11, 111, or 1111 (or 3, 7,
and 15 in the decimal system). Now, if we bitwise AND a power of 2, and the integer
that is one less than that, then we should get 0.
The truth table for the AND operator looks like the following:
Input 1
Input 2
AND
True
True
True
False
True
False
True
False
False
False
False
False
The NumPy counterpart of & is bitwise_and(), and the counterpart of == is the
equal() universal function:
print("Power of 2?\n", x, "\n", (x & (x - 1)) == 0)
print("Power of 2?\n", x, "\n", np.equal(np.bitwise_and(x,
(x - 1)), 0))
The result is shown as follows:
Power of 2?
[-9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
8]
[False False False False False False False False False
True
False
True False False False
True
True
True
True
True]
Power of 2?
[-9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
8]
[False False False False False False False False False
True
False
3.
True False False False
True]
The trick of computing the modulus of 4 actually works when taking the modulus
of integers that are a power of 2 such as 4, 8, 16, and so on. A bitwise left
shift leads to doubling of values (see https://wiki.python.org/moin/
BitwiseOperators). We saw in the previous step that subtracting one from a
power of 2 leads to a number in binary notation that has a row of ones such as 11,
111, or 1111. This basically gives us a mask. Bitwise-ANDing with such a number
gives you the remainder with a power of 2. The NumPy equivalent of << is the
left_shift() universal function:
print("Modulus 4\n", x, "\n", x & ((1 << 2) - 1))
[ 142 ]
Chapter 5
print("Modulus 4\n", x, "\n", np.bitwise_and(x,
np.left_shift(1, 2) - 1))
The result is shown as follows:
Modulus 4
[-9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
8]
3
4
5
6
7
8]
[3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0]
Modulus 4
[-9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
[3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0]
What just happened?
We covered three bit twiddling hacks—checking whether the signs of integers are different,
checking whether a number is a power of 2, and calculating the modulus of a number
that is a power of 2. We saw the NumPy counterparts of the operators ^, &, <<, and <
(see bittwidling.py):
from __future__ import print_function
import numpy as np
x = np.arange(-9, 9)
y = -x
print("Sign different?", (x ^ y) < 0)
print("Sign different?", np.less(np.bitwise_xor(x, y), 0))
print("Power of 2?\n", x, "\n", (x & (x - 1)) == 0)
print("Power of 2?\n", x, "\n", np.equal(np.bitwise_and(x, (x - 1)),
0))
print("Modulus 4\n", x, "\n", x & ((1 << 2) - 1))
print("Modulus 4\n", x, "\n", np.bitwise_and(x, np.left_shift(1, 2)
- 1))
Fancy indexing
The at() method was added in NumPy 1.8. This method allows fancy indexing in-place.
Fancy indexing is indexing that does not involve integers or slices, which is normal indexing.
In-place means that the array we operate on will be modified.
The signature for the at() method is ufunc.at(a, indices[, b]). The indices array
specifies the elements to operate on. We need the b array only for universal functions with
two operands. The following Time for action section gives examples of the at() method.
[ 143 ]
Working with Matrices and ufuncs
Time for action – fancy indexing in-place for ufuncs with the at()
method
To demonstrate how the at() method works, start a Python or IPython shell and import
NumPy. You should know how to do this by now.
1.
Create an array with seven random integers from -3 to 3 with a seed of 42:
>>> a = np.random.random_integers(-3, 3, 7)
>>> a
array([ 1,
0, -1,
2,
1, -2,
0])
When we talk about random numbers in programming, we usually talk about
pseudo-random numbers (see https://www.khanacademy.org/computing/
computer-science/cryptography/crypt/v/random-vs-pseudorandomnumber-generators). The numbers appear random, but in fact are calculated
using a seed.
2.
Apply the at() method of the sign() universal function to the fourth and sixth
array elements:
>>> np.sign.at(a, [3, 5])
>>> a
array([ 1, 0, -1,
1,
1, -1,
0])
What just happened?
We used the at() method to select array elements and performed an in-place
operation—determining the sign. We also learned how to create random integers.
Summary
In this chapter, you learned, about matrices and universal functions. We covered how to
create matrices and looked at how universal functions work. You had a brief introduction
to arithmetic, trigonometric, bitwise, and comparison universal functions.
In the next chapter, you will cover the NumPy modules.
[ 144 ]
6
Moving Further with NumPy Modules
NumPy has a number of modules inherited from its predecessor, Numeric.
Some of these packages have a SciPy counterpart, which may have fuller
functionality. We will discuss SciPy in a later chapter.
In this chapter, we will cover the following topics:
The linalg package
The fft package
Random numbers
Continuous and discrete distributions
Linear algebra
Linear algebra is an important branch of mathematics. The numpy.linalg package
contains linear algebra functions. With this module, you can invert matrices, calculate
eigenvalues, solve linear equations, and determine determinants, among other things
(see http://docs.scipy.org/doc/numpy/reference/routines.linalg.html).
[ 145 ]
Moving Further with NumPy Modules
Time for action – inverting matrices
The inverse of a matrix A in linear algebra is the matrix A-1, which, when multiplied with the
original matrix, is equal to the identity matrix I. This can be written as follows:
A A-1 = I
The inv() function in the numpy.linalg package can invert an example matrix with the
following steps:
1.
Create the example matrix with the mat() function we used in the previous chapters:
A = np.mat("0 1 2;1 0 3;4 -3 8")
print("A\n", A)
The A matrix appears as follows:
A
[[ 0
1
2]
[ 1
0
3]
[ 4 -3
2.
8]]
Invert the matrix with the inv() function:
inverse = np.linalg.inv(A)
print("inverse of A\n", inverse)
The inverse matrix appears as follows:
inverse of A
[[-4.5
7.
-1.5]
[-2.
4.
-1. ]
[ 1.5 -2.
0.5]]
If the matrix is singular, or not square, a LinAlgError is
raised. If you want, you can check the result manually with a
pen and paper. This is left as an exercise for the reader.
3.
Check the result by multiplying the original matrix with the result of the
inv() function:
print("Check\n", A * inverse)
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The result is the identity matrix, as expected:
Check
[[ 1.
0.
0.]
[ 0.
1.
0.]
[ 0.
0.
1.]]
What just happened?
We calculated the inverse of a matrix with the inv() function of the numpy.linalg
package. We checked, with matrix multiplication, whether this is indeed the inverse matrix
(see inversion.py):
from __future__ import print_function
import numpy as np
A = np.mat("0 1 2;1 0 3;4 -3 8")
print("A\n", A)
inverse = np.linalg.inv(A)
print("inverse of A\n", inverse)
print("Check\n", A * inverse)
Pop quiz – creating a matrix
Q1. Which function can create matrices?
1. array
2. create_matrix
3. mat
4. vector
Have a go hero – inverting your own matrix
Create your own matrix and invert it. The inverse is only defined for square matrices. The
matrix must be square and invertible; otherwise, a LinAlgError exception is raised.
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Solving linear systems
A matrix transforms a vector into another vector in a linear way. This transformation
mathematically corresponds to a system of linear equations. The numpy.linalg function
solve() solves systems of linear equations of the form Ax = b, where A is a matrix, b can
be a one-dimensional or two-dimensional array, and x is an unknown variable. We will see the
dot() function in action. This function returns the dot product of two floating-point arrays.
The dot() function calculates the dot product (see https://www.khanacademy.org/
math/linear-algebra/vectors_and_spaces/dot_cross_products/v/vectordot-product-and-vector-length). For a matrix A and vector b, the dot product is equal
to the following sum:
∑A B
i
ij
i
Time for action – solving a linear system
Solve an example of a linear system with the following steps:
1.
Create A and b:
A = np.mat("1 -2 1;0 2 -8;-4 5 9")
print("A\n", A)
b = np.array([0, 8, -9])
print("b\n", b)
A and b appear as follows:
2.
Solve this linear system with the solve() function:
x = np.linalg.solve(A, b)
print("Solution", x)
The solution of the linear system is as follows:
Solution [ 29.
16.
3.]
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3.
Check whether the solution is correct with the dot() function:
print("Check\n", np.dot(A , x))
The result is as expected:
Check
[[ 0.
8. -9.]]
What just happened?
We solved a linear system using the solve() function from the NumPy linalg module and
checked the solution with the dot() function. Please refer to the solution.py file in this
book's code bundle:
from __future__ import print_function
import numpy as np
A = np.mat("1 -2 1;0 2 -8;-4 5 9")
print("A\n", A)
b = np.array([0, 8, -9])
print("b\n", b)
x = np.linalg.solve(A, b)
print("Solution", x)
print("Check\n", np.dot(A , x))
Finding eigenvalues and eigenvectors
Eigenvalues are scalar solutions to the equation Ax = ax, where A is a two-dimensional
matrix and x is a one-dimensional vector. Eigenvectors are vectors corresponding to
eigenvalues (see https://www.khanacademy.org/math/linear-algebra/
alternate_bases/eigen_everything/v/linear-algebra-introduction-toeigenvalues-and-eigenvectors). The eigvals() function in the numpy.linalg
package calculates eigenvalues. The eig() function returns a tuple containing eigenvalues
and eigenvectors.
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Time for action – determining eigenvalues and eigenvectors
Let's calculate the eigenvalues of a matrix:
1.
Create a matrix as shown in the following:
A = np.mat("3 -2;1 0")
print("A\n", A)
The matrix we created looks like the following:
A
[[ 3 -2]
[ 1
2.
0]]
Call the eigvals() function:
print("Eigenvalues", np.linalg.eigvals(A))
The eigenvalues of the matrix are as follows:
Eigenvalues [ 2.
3.
1.]
Determine eigenvalues and eigenvectors with the eig() function. This function
returns a tuple, where the first element contains eigenvalues and the second
element contains corresponding eigenvectors, arranged column-wise:
eigenvalues, eigenvectors = np.linalg.eig(A)
print("First tuple of eig", eigenvalues)
print("Second tuple of eig\n", eigenvectors)
The eigenvalues and eigenvectors appear as follows:
First tuple of eig [ 2.
1.]
Second tuple of eig
4.
[[ 0.89442719
0.70710678]
[ 0.4472136
0.70710678]]
Check the result with the dot() function by calculating the right and left side of the
eigenvalues equation Ax = ax:
for i, eigenvalue in enumerate(eigenvalues):
print("Left", np.dot(A, eigenvectors[:,i]))
print("Right", eigenvalue * eigenvectors[:,i])
print()
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The output is as follows:
Left [[ 1.78885438]
[ 0.89442719]]
Right [[ 1.78885438]
[ 0.89442719]]
What just happened?
We found the eigenvalues and eigenvectors of a matrix with the eigvals() and eig()
functions of the numpy.linalg module. We checked the result using the dot() function
(see eigenvalues.py):
from __future__ import print_function
import numpy as np
A = np.mat("3 -2;1 0")
print("A\n", A)
print("Eigenvalues", np.linalg.eigvals(A) )
eigenvalues, eigenvectors = np.linalg.eig(A)
print("First tuple of eig", eigenvalues)
print("Second tuple of eig\n", eigenvectors)
for i, eigenvalue in enumerate(eigenvalues):
print("Left", np.dot(A, eigenvectors[:,i]))
print("Right", eigenvalue * eigenvectors[:,i])
print()
Singular value decomposition
Singular value decomposition (SVD) is a type of factorization that decomposes a matrix
into a product of three matrices. The SVD is a generalization of the previously discussed
eigenvalue decomposition. SVD is very useful for algorithms such as the pseudo inverse,
which we will discuss in the next section. The svd() function in the numpy.linalg package
can perform this decomposition. This function returns three matrices U, ∑, and V such that U
and V are unitary and ∑ contains the singular values of the input matrix:
M = U ∑V ∗
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The asterisk denotes the Hermitian conjugate or the conjugate transpose. The complex
conjugate changes the sign of the imaginary part of a complex number and is therefore not
relevant for real numbers.
A complex square matrix A is unitary if A*A = AA* = I (the identity matrix).
We can interpret SVD as a sequence of three operations—rotation, scaling, and
another rotation.
We already transposed matrices in this book. The transpose flips matrices, turning rows into
columns, and columns into rows.
Time for action – decomposing a matrix
It's time to decompose a matrix with the SVD using the following steps:
1.
First, create a matrix as shown in the following:
A = np.mat("4 11 14;8 7 -2")
print("A\n", A)
The matrix we created looks like the following:
A
[[ 4 11 14]
[ 8
2.
7 -2]]
Decompose the matrix with the svd() function:
U, Sigma, V = np.linalg.svd(A, full_matrices=False)
print("U")
print(U)
print("Sigma")
print(Sigma)
print("V")
print(V)
Because of the full_matrices=False specification, NumPy performs a reduced
SVD decomposition, which is faster to compute. The result is a tuple containing the
two unitary matrices U and V on the left and right, respectively, and the singular
values of the middle matrix:
U
[[-0.9486833
[-0.31622777
-0.31622777]
0.9486833 ]]
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Sigma
[ 18.97366596
9.48683298]
V
[[-0.33333333 -0.66666667 -0.66666667]
[ 0.66666667
3.
0.33333333 -0.66666667]]
We do not actually have the middle matrix—we only have the diagonal values. The
other values are all 0. Form the middle matrix with the diag() function. Multiply
the three matrices as follows:
print("Product\n", U * np.diag(Sigma) * V)
The product of the three matrices is equal to the matrix we created in the first step:
Product
[[
4.
11.
[
8.
7.
14.]
-2.]]
What just happened?
We decomposed a matrix and checked the result by matrix multiplication. We used the
svd() function from the NumPy linalg module (see decomposition.py):
from __future__ import print_function
import numpy as np
A = np.mat("4 11 14;8 7 -2")
print("A\n", A)
U, Sigma, V = np.linalg.svd(A, full_matrices=False)
print("U")
print(U)
print("Sigma")
print(Sigma)
print("V")
print(V)
print("Product\n", U * np.diag(Sigma) * V)
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Pseudo inverse
The Moore-Penrose pseudo inverse of a matrix can be computed with the pinv()
function of the numpy.linalg module (see http://en.wikipedia.org/wiki/
Moore%E2%80%93Penrose_pseudoinverse). The pseudo inverse is calculated using
the SVD (see previous example). The inv() function only accepts square matrices; the
pinv() function does not have this restriction and is therefore considered a generalization
of the inverse.
Time for action – computing the pseudo inverse of a matrix
Let's compute the pseudo inverse of a matrix:
1.
First, create a matrix:
A = np.mat("4 11 14;8 7 -2")
print("A\n", A)
The matrix we created looks like the following:
A
[[ 4 11 14]
[ 8
2.
7 -2]]
Calculate the pseudo inverse matrix with the pinv() function:
pseudoinv = np.linalg.pinv(A)
print("Pseudo inverse\n", pseudoinv)
The pseudo inverse result is as follows:
Pseudo inverse
[[-0.00555556
0.07222222]
[ 0.02222222
0.04444444]
[ 0.05555556 -0.05555556]]
3.
Multiply the original and pseudo inverse matrices:
print("Check", A * pseudoinv)
What we get is not an identity matrix, but it comes close to it:
Check [[
[
1.00000000e+00
8.32667268e-17
0.00000000e+00]
1.00000000e+00]]
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What just happened?
We computed the pseudo inverse of a matrix with the pinv() function of the numpy.
linalg module. The check by matrix multiplication resulted in a matrix that is
approximately an identity matrix (see pseudoinversion.py):
from __future__ import print_function
import numpy as np
A = np.mat("4 11 14;8 7 -2")
print("A\n", A)
pseudoinv = np.linalg.pinv(A)
print("Pseudo inverse\n", pseudoinv)
print("Check", A * pseudoinv)
Determinants
The determinant is a value associated with a square matrix. It is used throughout
mathematics; for more details, please refer to http://en.wikipedia.org/wiki/
Determinant. For a n x n real value matrix, the determinant corresponds to the scaling
a n-dimensional volume undergoes when transformed by the matrix. The positive sign of
the determinant means the volume preserves its orientation (clockwise or anticlockwise),
while a negative sign means reversed orientation. The numpy.linalg module has a det()
function that returns the determinant of a matrix.
Time for action – calculating the determinant of a matrix
To calculate the determinant of a matrix, follow these steps:
1.
Create the matrix:
A = np.mat("3 4;5 6")
print("A\n", A)
The matrix we created appears as follows:
A
[[ 3.
[ 5.
4.]
6.]]
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2.
Compute the determinant with the det() function:
print("Determinant", np.linalg.det(A))
The determinant appears as follows:
Determinant -2.0
What just happened?
We calculated the determinant of a matrix with the det() function from the numpy.
linalg module (see determinant.py):
from __future__ import print_function
import numpy as np
A = np.mat("3 4;5 6")
print("A\n", A)
print("Determinant", np.linalg.det(A))
Fast Fourier transform
The Fast Fourier transform (FFT) is an efficient algorithm to calculate the discrete Fourier
transform (DFT).
The Fourier transform is related to the Fourier series, which was mentioned in
the previous chapter—Chapter 5, Working with Matrices and ufuncs. The Fourier
series represents a signal as a sum of sine and cosine terms.
FFT improves on more naïve algorithms and is of order O(N log N). DFT has applications in
signal processing, image processing, solving partial differential equations, and more. NumPy
has a module called fft that offers FFT functionality. Many functions in this module are
paired; for those functions, another function does the inverse operation. For instance, the
fft() and ifft() function form such a pair.
Time for action – calculating the Fourier transform
First, we will create a signal to transform. Calculate the Fourier transform with the
following steps:
1.
Create a cosine wave with 30 points as follows:
x = np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
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Chapter 6
2.
Transform the cosine wave with the fft() function:
transformed = np.fft.fft(wave)
3.
Apply the inverse transform with the ifft() function. It should approximately
return the original signal. Check with the following line:
print(np.all(np.abs(np.fft.ifft(transformed) - wave)
< 10 ** -9))
The result appears as follows:
True
4.
Plot the transformed signal with matplotlib:
plt.plot(transformed)
plt.title('Transformed cosine')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.show()
The following resulting diagram shows the FFT result:
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What just happened?
We applied the fft() function to a cosine wave. After applying the ifft() function, we
got our signal back (see fourier.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
transformed = np.fft.fft(wave)
print(np.all(np.abs(np.fft.ifft(transformed) - wave) < 10 ** -9))
plt.plot(transformed)
plt.title('Transformed cosine')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.show()
Shifting
The fftshift() function of the numpy.linalg module shifts zero-frequency components
to the center of a spectrum. The zero-frequency component corresponds to the mean of the
signal. The ifftshift() function reverses this operation.
Time for action – shifting frequencies
We will create a signal, transform it, and then shift the signal. Shift the frequencies with the
following steps:
1.
Create a cosine wave with 30 points:
x = np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
2.
Transform the cosine wave with the fft() function:
transformed = np.fft.fft(wave)
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Chapter 6
3.
Shift the signal with the fftshift() function:
shifted = np.fft.fftshift(transformed)
4.
Reverse the shift with the ifftshift() function. This should undo the shift. Check
with the following code snippet:
print(np.all((np.fft.ifftshift(shifted) - transformed)
< 10 ** -9))
The result appears as follows:
True
5.
Plot the signal and transform it with matplotlib:
plt.plot(transformed, lw=2, label="Transformed")
plt.plot(shifted, '--', lw=3, label="Shifted")
plt.title('Shifted and transformed cosine wave')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.legend(loc='best')
plt.show()
The following diagram shows the effect of the shift and the FFT:
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What just happened?
We applied the fftshift() function to a cosine wave. After applying the ifftshift()
function, we got our signal back (see fouriershift.py):
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2 * np.pi, 30)
wave = np.cos(x)
transformed = np.fft.fft(wave)
shifted = np.fft.fftshift(transformed)
print(np.all(np.abs(np.fft.ifftshift(shifted) - transformed) < 10 **
-9))
plt.plot(transformed, lw=2, label="Transformed")
plt.plot(shifted, '--', lw=3, label="Shifted")
plt.title('Shifted and transformed cosine wave')
plt.xlabel('Frequency')
plt.ylabel('Amplitude')
plt.grid()
plt.legend(loc='best')
plt.show()
Random numbers
Random numbers are used in Monte Carlo methods, stochastic calculus, and more. Real
random numbers are hard to generate, so, in practice, we use pseudo random numbers,
which are random enough for most intents and purposes, except for some very special
cases. These numbers appear random, but if you analyze them more closely, you will realize
that they follow a certain pattern. The random numbers-related functions are in the NumPy
random module. The core random number generator is based on the Mersenne Twister
algorithm—a standard and well-known algorithm (see https://en.wikipedia.org/
wiki/Mersenne_Twister). We can generate random numbers from discrete or continuous
distributions. The distribution functions have an optional size parameter, which tells NumPy
how many numbers to generate. You can specify either an integer or a tuple as size. This will
result in an array filled with random numbers of appropriate shape. Discrete distributions
include the geometric, hypergeometric, and binomial distributions.
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Time for action – gambling with the binomial
The binomial distribution models the number of successes in an integer number of
independent trials of an experiment, where the probability of success in each experiment is
a fixed number (see https://www.khanacademy.org/math/probability/randomvariables-topic/binomial_distribution).
Imagine a 17th century gambling house where you can bet on flipping pieces of eight. Nine
coins are flipped. If less than five are heads, then you lose one piece of eight, otherwise
you win one. Let's simulate this, starting with 1,000 coins in our possession. Use the
binomial() function from the random module for that purpose.
To understand the binomial() function, look at the following section:
1.
Initialize an array, which represents the cash balance, to zeros. Call the
binomial() function with a size of 10000. This represents 10,000 coin
flips in our casino:
cash = np.zeros(10000)
cash[0] = 1000
outcome = np.random.binomial(9, 0.5, size=len(cash))
2.
Go through the outcomes of the coin flips and update the cash array. Print the
minimum and maximum of the outcome, just to make sure we don't have any
strange outliers:
for i in range(1, len(cash)):
if outcome[i] < 5:
cash[i] = cash[i - 1] - 1
elif outcome[i] < 10:
cash[i] = cash[i - 1] + 1
else:
raise AssertionError("Unexpected outcome " + outcome)
print(outcome.min(), outcome.max())
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As expected, the values are between 0 and 9. In the following diagram, you can see
the cash balance performing a random walk:
What just happened?
We did a random walk experiment using the binomial() function from the NumPy random
module (see headortail.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
cash = np.zeros(10000)
cash[0] = 1000
np.random.seed(73)
outcome = np.random.binomial(9, 0.5, size=len(cash))
for i in range(1, len(cash)):
if outcome[i] < 5:
cash[i] = cash[i - 1] - 1
elif outcome[i] < 10:
cash[i] = cash[i - 1] + 1
else:
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Chapter 6
raise AssertionError("Unexpected outcome " + outcome)
print(outcome.min(), outcome.max())
plt.plot(np.arange(len(cash)), cash)
plt.title('Binomial simulation')
plt.xlabel('# Bets')
plt.ylabel('Cash')
plt.grid()
plt.show()
Hypergeometric distribution
The hypergeometric distribution models a jar with two types of objects in it. The model tells
us how many objects of one type we can get if we take a specified number of items out of the
jar without replacing them (see https://en.wikipedia.org/wiki/Hypergeometric_
distribution). The NumPy random module has a hypergeometric() function that
simulates this situation.
Time for action – simulating a game show
Imagine a game show where every time the contestants answer a question correctly, they
get to pull three balls from a jar and then put them back. Now, there is a catch, one ball in
the jar is bad. Every time it is pulled out, the contestants lose six points. If, however, they
manage to get out 3 of the 25 normal balls, they get one point. So, what is going to happen
if we have 100 questions in total? Look at the following section for the solution:
1.
Initialize the outcome of the game with the hypergeometric() function. The
first parameter of this function is the number of ways to make a good selection,
the second parameter is the number of ways to make a bad selection, and the
third parameter is the number of items sampled:
points = np.zeros(100)
outcomes = np.random.hypergeometric(25, 1, 3, size=len(points))
2.
Set the scores based on the outcomes from the previous step:
for i in range(len(points)):
if outcomes[i] == 3:
points[i] = points[i - 1] + 1
elif outcomes[i] == 2:
points[i] = points[i - 1] - 6
else:
print(outcomes[i])
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The following diagram shows how the scoring evolved:
What just happened?
We simulated a game show using the hypergeometric() function from the NumPy
random module. The game scoring depends on how many good and how many bad balls
the contestants pulled out of a jar in each session (see urn.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
points = np.zeros(100)
np.random.seed(16)
outcomes = np.random.hypergeometric(25, 1, 3, size=len(points))
for i in range(len(points)):
if outcomes[i] == 3:
points[i] = points[i - 1] + 1
elif outcomes[i] == 2:
points[i] = points[i - 1] - 6
else:
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Chapter 6
print(outcomes[i])
plt.plot(np.arange(len(points)), points)
plt.title('Game show simulation')
plt.xlabel('# Rounds')
plt.ylabel('Score')
plt.grid()
plt.show()
Continuous distributions
We usually model continuous distributions with probability density functions (PDF). The
probability that a value is in a certain interval is determined by integration of the PDF
(see https://www.khanacademy.org/math/probability/random-variablestopic/random_variables_prob_dist/v/probability-density-functions).
The NumPy random module has functions that represent continuous distributions—
beta(), chisquare(), exponential(), f(), gamma(), gumbel(), laplace(),
lognormal(), logistic(), multivariate_normal(), noncentral_chisquare(),
noncentral_f(), normal(), and others.
Time for action – drawing a normal distribution
We can generate random numbers from a normal distribution and visualize their distribution
with a histogram (see https://www.khanacademy.org/math/probability/
statistics-inferential/normal_distribution/v/introduction-to-thenormal-distribution). Draw a normal distribution with the following steps:
1.
Generate random numbers for a given sample size using the normal() function
from the random NumPy module:
N=10000
normal_values = np.random.normal(size=N)
2.
Draw the histogram and theoretical PDF with a center value of 0 and standard
deviation of 1. Use matplotlib for this purpose:
_, bins, _ = plt.hist(normal_values,
np.sqrt(N), normed=True, lw=1)
sigma = 1
mu = 0
plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi))
* np.exp( - (bins - mu)**2 / (2 * sigma**2) ),lw=2)
plt.show()
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In the following diagram, we see the familiar bell curve:
What just happened?
We visualized the normal distribution using the normal() function from the random NumPy
module. We did this by drawing the bell curve and a histogram of randomly generated values
(see normaldist.py):
import numpy as np
import matplotlib.pyplot as plt
N=10000
np.random.seed(27)
normal_values = np.random.normal(size=N)
_, bins, _ = plt.hist(normal_values, np.sqrt(N), normed=True, lw=1,
label="Histogram")
sigma = 1
mu = 0
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Chapter 6
plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * np.exp( - (bins mu)**2 / (2 * sigma**2) ), '--', lw=3, label="PDF")
plt.title('Normal distribution')
plt.xlabel('Value')
plt.ylabel('Normalized Frequency')
plt.grid()
plt.legend(loc='best')
plt.show()
Lognormal distribution
A lognormal distribution is a distribution of a random variable whose natural logarithm
is normally distributed. The lognormal() function of the random NumPy module models
this distribution.
Time for action – drawing the lognormal distribution
Let's visualize the lognormal distribution and its PDF with a histogram:
1.
Generate random numbers using the normal() function from the random
NumPy module:
N=10000
lognormal_values = np.random.lognormal(size=N)
2.
Draw the histogram and theoretical PDF with a center value of 0 and standard
deviation of 1:
_, bins, _ = plt.hist(lognormal_values,
np.sqrt(N), normed=True, lw=1)
sigma = 1
mu = 0
x = np.linspace(min(bins), max(bins), len(bins))
pdf = np.exp(-(numpy.log(x) - mu)**2 / (2 * sigma**2))/ (x *
sigma * np.sqrt(2 * np.pi))
plt.plot(x, pdf,lw=3)
plt.show()
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The fit of the histogram and theoretical PDF is excellent, as you can see in the
following diagram:
What just happened?
We visualized the lognormal distribution using the lognormal() function from the random
NumPy module. We did this by drawing the curve of the theoretical PDF and a histogram of
randomly generated values (see lognormaldist.py):
import numpy as np
import matplotlib.pyplot as plt
N=10000
np.random.seed(34)
lognormal_values = np.random.lognormal(size=N)
_, bins, _ = plt.hist(lognormal_values,
np.sqrt(N), normed=True, lw=1, label="Histogram")
sigma = 1
mu = 0
x = np.linspace(min(bins), max(bins), len(bins))
pdf = np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))/ (x * sigma *
np.sqrt(2 * np.pi))
plt.xlim([0, 15])
plt.plot(x, pdf,'--', lw=3, label="PDF")
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Chapter 6
plt.title('Lognormal distribution')
plt.xlabel('Value')
plt.ylabel('Normalized frequency')
plt.grid()
plt.legend(loc='best')
plt.show()
Bootstrapping in statistics
Bootstrapping is a method used to estimate variance, accuracy, and other metrics of sample
estimates, such as the arithmetic mean. The simplest bootstrapping procedure consists of
the following steps:
1. Generate a large number of samples from the original data sample having the
same size N. You can think of the original data as a jar containing numbers. We
create the new samples by N times randomly picking a number from the jar. Each
time we return the number into the jar, so a number can occur multiple times in a
generated sample.
2. With the new samples, we calculate the statistical estimate under investigation for
each sample (for example, the arithmetic mean). This gives us a sample of possible
values for the estimator.
Time for action – sampling with numpy.random.choice()
We will use the numpy.random.choice() function to perform bootstrapping.
1.
Start the IPython or Python shell and import NumPy:
$ ipython
In [1]: import numpy as np
2.
Generate a data sample following the normal distribution:
In [2]: N = 500
In [3]: np.random.seed(52)
In [4]: data = np.random.normal(size=N)
3.
Calculate the mean of the data:
In [5]: data.mean()
Out[5]: 0.07253250605445645
[ 169 ]
Moving Further with NumPy Modules
Generate 100 samples from the original data and calculate their means (of course,
more samples may lead to a more accurate result):
In [6]: bootstrapped = np.random.choice(data, size=(N, 100))
In [7]: means = bootstrapped.mean(axis=0)
In [8]: means.shape
Out[8]: (100,)
4.
Calculate the mean, variance, and standard deviation of the arithmetic means
we obtained:
In [9]: means.mean()
Out[9]: 0.067866373318115278
In [10]: means.var()
Out[10]: 0.001762807104774598
In [11]: means.std()
Out[11]: 0.041985796464692651
If we are assuming a normal distribution for the means, it may be relevant to know
the z-score, which is defined as follows:
z=
x−µ
σ
In [12]: (data.mean() - means.mean())/means.std()
Out[12]: 0.11113598238549766
From the z-score value, we get an idea of how probable the actual mean is.
[ 170 ]
Chapter 6
What just happened?
We bootstrapped a data sample by generating samples and calculating the means of each
sample. Then we computed the mean, standard deviation, variance, and z-score of the
means. We used the numpy.random.choice() function for bootstrapping.
Summary
You learned a lot in this chapter about NumPy modules. We covered linear algebra, the Fast
Fourier transform, continuous and discrete distributions, and random numbers.
In the next chapter, we will cover specialized routines. These are functions that you probably
will not use often, but are very useful when you do need them.
[ 171 ]
7
Peeking into Special Routines
As NumPy users, we sometimes find ourselves having special needs, for
instance, financial calculations or signal processing. Fortunately, NumPy
provides for most of our needs. This chapter describes some of the more
specialized NumPy functions.
In this chapter, we will cover the following topics:
Sorting and searching
Special functions
Financial utilities
Window functions
Sorting
NumPy has several data sorting routines:
The sort() function returns a sorted array
The lexsort() function performs sorting with a list of keys
The argsort() function returns the indices that will sort an array
The ndarray class has a sort() method that performs in-place sorting
The msort() function sorts an array along the first axis
The sort_complex() function sorts complex numbers by their real part and then
their imaginary part
[ 173 ]
Peeking into Special Routines
From this list, the argsort() and sort() functions are available as methods on NumPy
arrays as well.
Time for action – sorting lexically
The NumPy lexsort() function returns an array of indices of the input array elements
corresponding to lexically sorting an array. We need to give the function an array or tuple
of sort keys:
1.
Let's go back to Chapter 3, Getting Familiar with Commonly Used Functions. In that
chapter, we used stock price data of AAPL. We will load the close prices and the
(always complex) dates. In fact, create a converter function just for the dates:
def datestr2num(s):
return datetime.datetime.strptime(s, "%d-%m-%Y").toordinal()
dates, closes=np.loadtxt('AAPL.csv', delimiter=',',
usecols=(1, 6), converters={1:datestr2num}, unpack=True)
2.
Sort the names lexically with the lexsort() function. The data is already sorted
by date, but sort it by close as well:
indices = np.lexsort((dates, closes))
print("Indices", indices)
print(["%s %s" % (datetime.date.fromordinal(dates[i]),
closes[i]) for i in indices])
The code prints the following:
Indices [ 0 16 1 17 18 4 3 2
7 25 26 10 8 14 11 23 12 24 13]
5 28 19 21 15
6 29 22 27 20
['2011-01-28 336.1', '2011-02-22 338.61', '2011-01-31 339.32',
'2011-02-23 342.62', '2011-02-24 342.88', '2011-02-03 343.44',
'2011-02-02 344.32', '2011-02-01 345.03', '2011-02-04 346.5',
'2011-03-10 346.67', '2011-02-25 348.16', '2011-03-01 349.31',
'2011-02-18 350.56', '2011-02-07 351.88', '2011-03-11 351.99',
'2011-03-02 352.12', '2011-03-09 352.47', '2011-02-28 353.21',
'2011-02-10 354.54', '2011-02-08 355.2', '2011-03-07 355.36',
'2011-03-08 355.76', '2011-02-11 356.85', '2011-02-09 358.16',
'2011-02-17 358.3', '2011-02-14 359.18', '2011-03-03 359.56',
'2011-02-15 359.9', '2011-03-04 360.0', '2011-02-16 363.13']
[ 174 ]
9
Chapter 7
What just happened?
We sorted the close prices of AAPL lexically using the NumPy lexsort() function. The
function returned the indices corresponding with sorting the array (see lex.py):
from __future__ import print_function
import numpy as np
import datetime
def datestr2num(s):
return datetime.datetime.strptime(s, "%d-%m-%Y").toordinal()
dates, closes=np.loadtxt('AAPL.csv', delimiter=',',
usecols=(1, 6), converters={1:datestr2num}, unpack=True)
indices = np.lexsort((dates, closes))
print("Indices", indices)
print(["%s %s" % (datetime.date.fromordinal(int(dates[i])),
closes[i])
for i in indices])
Have a go hero – trying a different sort order
We sorted using the dates and the close price sort order. Try a different order. Generate
random numbers using the random module we learned about in the previous chapter and
sort those using lexsort().
Time for action – partial sorting via selection for a fast median
with the partition() function
The partition() function does partial sorting, which should be faster than full sorting,
because it's less work.
For more information, please refer to http://en.wikipedia.org/
wiki/Partial_sorting. A common use case is getting the top 10
elements of a collection. Partial sorting doesn't guarantee the correct order
within the group of top elements itself.
[ 175 ]
Peeking into Special Routines
The first argument of the function is the array to partially sort. The second argument
is an integer or a sequence of integers corresponding to indices of array elements. The
partition() function sorts elements in those indices correctly. With one specified index,
we get two partitions; with multiple indices, we get more than one partition. The sorting
algorithm makes sure that elements in partitions, which are smaller than a correctly sorted
element, come before this element. Otherwise, they are placed behind this element. Let's
illustrate this explanation with an example. Start a Python or IPython shell and import NumPy:
$ ipython
In [1]: import numpy as np
Create an array with random elements to sort:
In [2]: np.random.seed(20)
In [3]: a = np.random.random_integers(0, 9, 9)
In [4]: a
Out[4]: array([3, 9, 4, 6, 7, 2, 0, 6, 8])
Partially sort the array by partitioning it in two roughly equal parts:
In [5]: np.partition(a, 4)
Out[5]: array([0, 2, 3, 4, 6, 6, 7, 9, 8])
We get an almost perfect sorting except for the last two elements.
What just happened?
We partially sorted a nine-element array. The sorting only guaranteed that one element in
the middle at index 4 is at the correct position. This corresponds to trying to get the top five
elements of the array without caring about the order within the top five group. Since the
correctly sorted element is in the middle, this also gives the median of the array.
Complex numbers
Complex numbers are numbers that have a real and imaginary part. As you remember from
previous chapters, NumPy has special complex data types that represent complex numbers by
two floating-point numbers. These numbers can be sorted using the NumPy sort_complex()
function. This function sorts the real part first and then the imaginary part.
[ 176 ]
Chapter 7
Time for action – sorting complex numbers
We will create an array of complex numbers and sort it:
1.
Generate five random numbers for the real part of the complex numbers and five
numbers for the imaginary part. Seed the random generator to 42:
np.random.seed(42)
complex_numbers = np.random.random(5) + 1j *
np.random.random(5)
print("Complex numbers\n", complex_numbers)
2.
Call the sort_complex() function to sort the complex numbers we generated in
the previous step:
print("Sorted\n", np.sort_complex(complex_numbers))
The sorted numbers would be:
Sorted
[ 0.39342751+0.34955771j
0.41516850+0.26221878j
0.86631422+0.74612422j
0.40597665+0.77477433j
0.92293095+0.81335691j]
What just happened?
We generated random complex numbers and sorted them using the sort_complex()
function (see sortcomplex.py):
from __future__ import print_function
import numpy as np
np.random.seed(42)
complex_numbers = np.random.random(5) + 1j * np.random.random(5)
print("Complex numbers\n", complex_numbers)
print("Sorted\n", np.sort_complex(complex_numbers))
Pop quiz – generating random numbers
Q1. Which NumPy module deals with random numbers?
1. Randnum
2. random
3. randomutil
4. rand
[ 177 ]
Peeking into Special Routines
Searching
NumPy has several functions that can search through arrays:
The argmax() function gives the indices of the maximum values of an array:
>>> a = np.array([2, 4, 8])
>>> np.argmax(a)
2
The nanargmax() function does the same, but ignores NaN values:
>>> b = np.array([np.nan, 2, 4])
>>> np.nanargmax(b)
2
The argmin() and nanargmin() functions provide similar functionality but
pertaining to minimum values. The argmax() and nanargmax() functions are
also available as methods of the ndarray class.
The argwhere() function searches for non-zero values and returns the
corresponding indices grouped by element:
>>> a = np.array([2, 4, 8])
>>> np.argwhere(a <= 4)
array([[0],
[1]])
The searchsorted() function tells you the index in an array where a specified
value belongs to maintain the sort order. It uses binary search (see https://www.
khanacademy.org/computing/computer-science/algorithms/binarysearch/a/binary-search), which is a O(log n) algorithm. We will see this
function in action shortly.
The extract() function retrieves values from an array based on a condition.
Time for action – using searchsorted
The searchsorted() function gets the index of a value in a sorted array. An example
should make this clear:
1.
To demonstrate, create an array with arange(), which of course is sorted:
a = np.arange(5)
2.
Time to call the searchsorted() function:
indices = np.searchsorted(a, [-2, 7])
print("Indices", indices)
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Chapter 7
The indices, which should maintain the sort order:
Indices [0 5]
3.
Construct the full array with the insert() function:
print("The full array", np.insert(a, indices, [-2, 7]))
This gives us the full array:
The full array [-2
0
1
2
3
4
7]
What just happened?
The searchsorted() function gave us indices 5 and 0 for 7 and -2. With these indices,
we made the array [-2, 0, 1, 2, 3, 4, 7], so the array remains sorted (see
sortedsearch.py):
from __future__ import print_function
import numpy as np
a = np.arange(5)
indices = np.searchsorted(a, [-2, 7])
print("Indices", indices)
print("The full array", np.insert(a, indices, [-2, 7]))
Array elements extraction
The NumPy extract() function allows us to extract items from an array based on a
condition. This function is similar to the where() function we encountered in Chapter 3,
Getting Familiar with Commonly Used Functions. The special nonzero() function selects
non-zero elements.
Time for action – extracting elements from an array
Let's extract the even elements of an array:
1.
Create the array with the arange() function:
a = np.arange(7)
2.
Create the condition that selects the even elements:
condition = (a % 2) == 0
[ 179 ]
Peeking into Special Routines
3.
Extract the even elements using our condition with the extract() function:
print("Even numbers", np.extract(condition, a))
This gives us the even numbers as required (np.extract(condition, a) is
equivalent to a[np.where(condition)[0]]):
Even numbers [0 2 4 6]
4.
Select non-zero values with the nonzero() function:
print("Non zero", np.nonzero(a))
This prints all the non-zero values of the array:
Non zero (array([1, 2, 3, 4, 5, 6]),)
What just happened?
We extracted the even elements from an array using a Boolean condition with the NumPy
extract() function (see extracted.py):
from __future__ import print_function
import numpy as np
a = np.arange(7)
condition = (a % 2) == 0
print("Even numbers", np.extract(condition, a))
print("Non zero", np.nonzero(a))
Financial functions
NumPy has a number of financial functions:
The fv() function calculates the so-called future value. The future value gives the
value of a financial instrument at a future date, based on certain assumptions.
The pv() function computes the present value (see https://www.khanacademy.
org/economics-finance-domain/core-finance/interest-tutorial/
present-value/v/time-value-of-money). The present value is the value of
an asset today.
The npv() function returns the net present value. The net present value is defined
as the sum of all the present value cash flows.
The pmt() function computes the payment against loan principal plus interest.
[ 180 ]
Chapter 7
The irr() function calculates the internal rate of return. The internal rate of return
is the effective interested rate, which does not take into account inflation.
The mirr() function calculates the modified internal rate of return. The modified
internal rate of return is an improved version of the internal rate of return.
The nper() function returns the number of periodic payments.
The rate() function calculates the rate of interest.
Time for action – determining the future value
The future value gives the value of a financial instrument at a future date, based on certain
assumptions. The future value depends on four parameters—the interest rate, the number
of periods, a periodic payment, and the present value.
Read more about future value at http://en.wikipedia.org/
wiki/Future_value. The formula for future value with compound
interest is as follows:
PV (1 + r )
n
In the preceding formula, PV is the present value, r is the interest rate,
and n is the number of periods.
In this section, let's take an interest rate of 3 percent, a quarterly payment of 10 for 5 years,
and a present value of 1000. Call the fv() function with the appropriate values (negative
values represent outgoing cash flow):
print("Future value", np.fv(0.03/4, 5 * 4, -10, -1000))
The future value is as follows:
Future value 1376.09633204
[ 181 ]
Peeking into Special Routines
If we vary the number of years we save and keep the other parameters constant, we get the
following plot:
What just happened?
We calculated the future value using the NumPy fv() function starting with a present value
of 1000, an interest rate of 3 percent, and quarterly payments of 10 for 5 years. We plotted
the future value for various saving periods (see futurevalue.py):
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
print("Future value", np.fv(0.03/4, 5 * 4, -10, -1000))
fvals = []
for i in xrange(1, 10):
fvals.append(np.fv(.03/4, i * 4, -10, -1000))
plt.plot(range(1, 10), fvals, 'bo')
plt.title('Future value, 3 % interest,\n Quarterly payment of 10')
plt.xlabel('Saving periods in years')
plt.ylabel('Future value')
plt.grid()
plt.legend(loc='best')
plt.show()
[ 182 ]
Chapter 7
Present value
The present value is the value of an asset today. The NumPy pv() function can calculate the
present value. This function mirrors the fv() function and requires the interest rate, number
of periods, and the periodic payment as well, but here we start with the future value.
Read more about the present value at http://en.wikipedia.org/wiki/Present_
value. It should be easy to derive the formula for the present value from the formula for
the future value, if you want.
Time for action – getting the present value
Let's reverse compute the present value with the numbers from the Time for action –
determining the future value section:
Plug in the figures from the Time for action – determining the future value section:
print("Present value", np.pv(0.03/4, 5 * 4, -10, 1376.09633204))
This gives us 1000 as expected apart from a tiny numerical error. Actually, it is not an error
but a representation issue. We are dealing here with outgoing cash flow, that is the reason
for the negative value:
Present value -999.999999999
What just happened?
We did the reverse computation of the Time for action – determining the future value section
to get the present value from the future value. This was done with the NumPy pv() function.
Net present value
The net present value is defined as the sum of all the present value cash flows. The NumPy
npv() function returns the net present value of cash flows. The function requires two
arguments: the rate and an array representing the cash flows.
Read more about the net present value at http://en.wikipedia.org/wiki/Net_
present_value. In the formula of the net present value, Rt is the cash flow of a time
period, r is the discount rate, and t is the index of the time period:
N
∑
t =0
Rt
(1 + r )
[ 183 ]
t
Peeking into Special Routines
Time for action – calculating the net present value
We will calculate the net present value for a random generated cash flow series:
1.
Generate five random values for the cash flow series. Insert -100 as the start value:
cashflows = np.random.randint(100, size=5)
cashflows = np.insert(cashflows, 0, -100)
print("Cashflows", cashflows)
The cash flows would be as follows:
Cashflows [-100
2.
38
48
90
17
36]
Call the npv() function to calculate the net present value from the cash flow series
we generated in the previous step. Use a rate of 3 percent:
print("Net present value", np.npv(0.03, cashflows))
The net present value:
Net present value 107.435682443
What just happened?
We computed the net present value from a random generated cash flow series with the
NumPy npv() function (see netpresentvalue.py):
from __future__ import print_function
import numpy as np
cashflows = np.random.randint(100, size=5)
cashflows = np.insert(cashflows, 0, -100)
print("Cashflows", cashflows)
print("Net present value", np.npv(0.03, cashflows))
Internal rate of return
The internal rate of return is the effective interested rate, which does not take into account
inflation. The NumPy irr() function returns the internal rate of return for a given cash
flow series.
[ 184 ]
Chapter 7
Time for action – determining the internal rate of return
Let's reuse the cash flow series from the Time for action – calculating the net present value
section. Call the irr() function with the cash flow series from the Time for action section:
print("Internal rate of return", np.irr([-100, 38, 48, 90,
17, 36]))
The internal rate of return:
Internal rate of return 0.373420226888
What just happened?
We calculated the internal rate of return from the cash flow series of the Time for action –
calculating the net present value section. The value was given by the NumPy irr() function.
Periodic payments
The NumPy pmt() function allows you to compute periodic payments for a loan, based on
an interest rate and the number of periodic payments.
Time for action – calculating the periodic payments
Suppose you have a loan of 10 million with an interest rate of 1 percent. You have 30 years
to pay the loan back. How much do you have to pay each month? Let's find out.
Call the pmt() function with the aforementioned values:
print("Payment", np.pmt(0.01/12, 12 * 30, 10000000))
The monthly payment:
Payment -32163.9520447
What just happened?
We calculated the monthly payment for a loan of 10 million at an annual rate of 1 percent.
Given that we have 30 years to repay the loan the pmt() function tells us that we need
to pay 32163.95 per month.
[ 185 ]
Peeking into Special Routines
Number of payments
The NumPy nper() function tells us how many periodic payments are necessary to pay off
a loan. The required parameters are the interest rate of the loan, the fixed amount periodic
payment, and the present value.
Time for action – determining the number of periodic payments
Consider a loan of 9000 at a rate of 10 percent with fixed monthly payments of 100.
Find out how many payments are required with the NumPy nper() function:
print("Number of payments", np.nper(0.10/12, -100, 9000))
The number of payments:
Number of payments 167.047511801
What just happened?
We determined the number of payments needed to pay off a loan of 9000 with an interest
rate of 10 percent and monthly payments of 100. The number of payments returned
was 167.
Interest rate
The NumPy rate() function calculates the interest rate given the number of periodic
payments, the payment amount or amounts, the present value, and the future value.
Time for action – figuring out the rate
Let's take the values from the Time for action – determining the number of periodic
payments section and reverse compute the interest rate from the other parameters.
Fill in the numbers from the previous Time for action section:
print("Interest rate", 12 * np.rate(167, -100, 9000, 0))
The interest rate is approximately 10 percent as expected:
Interest rate 0.0999756420664
[ 186 ]
Chapter 7
What just happened?
We used the NumPy rate() function and the values from the Time for action – determining
the number of periodic payments section to compute the interest rate of the loan. Ignoring
the rounding errors, we got the initial 10 percent we started with.
Window functions
Window functions are mathematical functions commonly used in signal processing.
Applications include spectral analysis and filter design. These functions are defined to be
0 outside a specified domain. NumPy has a number of window functions: bartlett(),
blackman(), hamming(), hanning(), and kaiser(). You can find an example of the
hanning() function in Chapter 4, Convenience Functions for Your Convenience, and
Chapter 3, Getting Familiar with Commonly Used Functions.
Time for action – plotting the Bartlett window
The Bartlett window is a triangular smoothing window:
w(n) = 1−
1.
Call the NumPy bartlett() function:
window = np.bartlett(42)
2.
N −1
2
N −1
2
n−
Plotting is easy with matplotlib:
plt.plot(window)
plt.show()
[ 187 ]
Peeking into Special Routines
The following is the Bartlett window, which is triangular, as expected:
What just happened?
We plotted the Bartlett window with the NumPy bartlett() function.
Blackman window
The Blackman window is the sum of the following cosines:
2π n
4π n
w ( n ) = 0.42 − 0.5cos
+ 0.08cos
M
M
The NumPy blackman() function returns the Blackman window. The only parameter is the
number of points M in the output window. If this number is 0 or less than 0, the function
returns an empty array.
[ 188 ]
Chapter 7
Time for action – smoothing stock prices with the Blackman
window
Let's smooth the close prices from the small AAPL stock prices data file:
1.
Load the data into a NumPy array. Call the NumPy blackman() function to form a
window, and then use this window to smooth the price signal:
closes=np.loadtxt('AAPL.csv', delimiter=',', usecols=(6,),
converters={1:datestr2num}, unpack=True)
N = 5
window = np.blackman(N)
smoothed = np.convolve(window/window.sum(),
closes, mode='same')
2.
Plot the smoothed prices with matplotlib. In this example, we will omit the first five
data points and the last five data points. The reason for this is that there is a strong
boundary effect:
plt.plot(smoothed[N:-N], lw=2, label="smoothed")
plt.plot(closes[N:-N], label="closes")
plt.legend(loc='best')
plt.show()
The closing prices of AAPL smoothed with the Blackman window should appear
as follows:
[ 189 ]
Peeking into Special Routines
What just happened?
We plotted the closing price of AAPL from our sample data file that was smoothed using the
Blackman window with the NumPy blackman() function (see plot_blackman.py):
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.dates import datestr2num
closes=np.loadtxt('AAPL.csv', delimiter=',', usecols=(6,),
converters={1:datestr2num}, unpack=True)
N = 5
window = np.blackman(N)
smoothed = np.convolve(window/window.sum(), closes, mode='same')
plt.plot(smoothed[N:-N], lw=2, label="smoothed")
plt.plot(closes[N:-N], '--', label="closes")
plt.title('Blackman window')
plt.xlabel('Days')
plt.ylabel('Price ($)')
plt.grid()
plt.legend(loc='best')
plt.show()
Hamming window
The Hamming window is formed by a weighted cosine. The formula is as follows:
2π n
w ( n ) = 0.54 + 0.46 cos
M −1
0 ≤ n ≤ M −1
The NumPy hamming() function returns the Hamming window. The only parameter is the
number of points M in the output window. If this number is 0 or less than 0, an empty array
is returned.
Time for action – plotting the Hamming window
Let's plot the Hamming window:
1.
Call the NumPy hamming() function:
window = np.hamming(42)
2.
Plot the window with matplotlib:
plt.plot(window)
plt.show()
[ 190 ]
Chapter 7
The Hamming window plot appears as follows:
What just happened?
We plotted the Hamming window with the NumPy hamming() function.
Kaiser window
The Kaiser window is formed by the Bessel function.
Bessel functions are solutions of the Bessel differential equations (see
http://en.wikipedia.org/wiki/Bessel_function).
The formula is as follows:
4n 2
w ( n ) = I0 β 1 −
2
( M − 1)
/ I0 ( β )
Here I0 is the zero order Bessel function. The NumPy kaiser() function returns the Kaiser
window. The first parameter is the number of points in the output window. If this number is
0 or less than 0, the function returns an empty array. The second parameter is the beta.
[ 191 ]
Peeking into Special Routines
Time for action – plotting the Kaiser window
Let's plot the Kaiser window:
1.
Call the NumPy kaiser() function:
window = np.kaiser(42, 14)
2.
Plot the window with matplotlib:
plt.plot(window)
plt.show()
The Kaiser window appears as follows:
What just happened?
We plotted the Kaiser window with the NumPy kaiser() function.
Special mathematical functions
We will end this chapter with some special mathematical functions. The modified Bessel
function of the first kind 0th order is represented in NumPy by i0(). The sinc function is
represented in NumPy by a function with the same name, and there is also a two-dimensional
version of this function. Sinc is a trigonometric function; for more details, see http://
en.wikipedia.org/wiki/Sinc_function. The sinc() function has two definitions.
[ 192 ]
Chapter 7
The NumPy sinc() function complies with the following definition:
sin (π x )
πx
Time for action – plotting the modified Bessel function
Let's see what the modified Bessel function of the first kind 0th order looks like:
1.
Compute evenly spaced values with the NumPy linspace() function:
x = np.linspace(0, 4, 100)
2.
Call the NumPy i0() function:
vals = np.i0(x)
3.
Plot the modified Bessel function with matplotlib:
plt.plot(x, vals)
plt.show()
The modified Bessel function will have the following output:
What just happened?
We plotted the modified Bessel function of the first kind 0th order with the NumPy i0()
function.
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Peeking into Special Routines
sinc
The sinc() function is widely used in mathematics and signal processing. NumPy has a
function with the same name. A two-dimensional function exists as well.
Time for action – plotting the sinc function
We will plot the sinc() function:
1.
Compute evenly spaced values with the NumPy linspace() function:
x = np.linspace(0, 4, 100)
2.
Call the NumPy sinc() function:
vals = np.sinc(x)
3.
Plot the sinc() function with matplotlib:
plt.plot(x, vals)
plt.show()
The sinc() function will have the following output:
[ 194 ]
Chapter 7
The sinc2d() function requires a two-dimensional array. We can create it with the
outer() function, resulting in this plot (code is in the following section):
What just happened?
We plotted the well-known sinc function with the NumPy sinc() function
(see plot_sinc.py):
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 4, 100)
vals = np.sinc(x)
plt.plot(x, vals)
plt.title('Sinc function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid()
plt.show()
[ 195 ]
Peeking into Special Routines
We did the same for two dimensions (see sinc2d.py):
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 4, 100)
xx = np.outer(x, x)
vals = np.sinc(xx)
plt.imshow(vals)
plt.title('Sinc 2D')
plt.xlabel('x')
plt.ylabel('y')
plt.grid()
plt.show()
Summary
This was a special chapter covering more specialized NumPy topics. We covered sorting and
searching, special functions, financial utilities, and window functions.
The next chapter is about the very important subject of testing.
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8
Assuring Quality with Testing
Some programmers test only in production. If you are not one of them,
then you're probably familiar with the concept of unit testing. Unit tests are
automated tests written by a programmer to test his or her code. These tests
could, for example, test a function or part of a function in isolation. Each test
covers only a small unit of code. The benefits are increased confidence in the
quality of the code, reproducible tests, and, as a side effect, clearer code.
Python has good support for unit testing. Additionally, NumPy adds the
numpy.testing package to that for NumPy code unit testing.
Test-driven development (TDD) is one of the most important things that happened to
software development. TDD focuses a lot on automated unit testing. The goal is to test
automatically the code as much as possible. The next time we change the code, we can run
the tests and catch potential regressions. In other words, any functionality already present
will still work.
The topics in this chapter include the following:
Unit testing
Asserts
Floating-point precision
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Assuring Quality with Testing
Assert functions
Unit tests usually use functions, which assert something as part of the test. When doing
numerical calculations, often we have the fundamental issue of trying to compare floatingpoint numbers that are almost equal. For integers, comparison is a trivial operation, but for
floating-point numbers it is not, because of the inexact representation by computers. The
NumPy testing package has a number of utility functions that test whether a precondition
is true or not, taking into account the problem of floating-point comparisons. The following
table shows the different utility functions:
Function
Description
assert_almost_equal()
This function raises an exception if two numbers are not equal
up to a specified precision
assert_approx_equal()
This function raises an exception if two numbers are not equal
up to a certain significance
assert_array_almost_
equal()
This function raises an exception if two arrays are not equal up
to a specified precision
assert_array_equal()
This function raises an exception if two arrays are not equal.
assert_array_less()
This function raises an exception if two arrays do not have the
same shape, and the elements of the first array are strictly less
than the elements of the second array
assert_equal()
This function raises an exception if two objects are not equal
assert_raises()
This function fails if a specified exception is not raised by a
callable invoked with defined arguments
assert_warns()
This function fails if a specified warning is not thrown
assert_string_equal()
This function asserts that two strings are equal
assert_allclose()
This function raise an assertion if two objects are not equal up
to desired tolerance
Time for action – asserting almost equal
Imagine that you have two numbers that are almost equal. Let's use the assert_almost_
equal() function to check whether they are equal:
1.
Call the function with low precision (up to 7 decimal places):
print("Decimal 6", np.testing.assert_almost_equal(0.123456789,
0.123456780, decimal=7))
Note that no exception is raised, as you can see in the following result:
Decimal 6 None
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Chapter 8
2.
Call the function with higher precision (up to 8 decimal places):
print("Decimal 7", np.testing.assert_almost_equal(0.123456789,
0.123456780, decimal=8))
The result is as follows:
Decimal 7
Traceback (most recent call last):
…
raise AssertionError(msg)
AssertionError:
Arrays are not almost equal
ACTUAL: 0.123456789
DESIRED: 0.12345678
What just happened?
We used the assert_almost_equal() function from the NumPy testing package to
check whether 0.123456789 and 0.123456780 are equal for different decimal precisions.
Pop quiz – specifying decimal precision
Q1. Which parameter of the assert_almost_equal() function specifies the
decimal precision?
1. decimal
2. precision
3. tolerance
4. significant
Approximately equal arrays
The assert_approx_equal() function raises an exception if two numbers are not equal
up to a certain number of significant digits. The function raises an exception triggered by the
following condition:
abs(actual - expected) >= 10**-(significant - 1)
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Assuring Quality with Testing
Time for action – asserting approximately equal
Let's take the numbers from the previous Time for action section and let the
assert_approx_equal() function work on them:
1.
Call the function with low significance:
print("Significance 8",
np.testing.assert_approx_equal
(0.123456789, 0.123456780,significant=8))
The result is as follows:
Significance 8 None
2.
Call the function with high significance:
print("Significance 9",
np.testing.assert_approx_equal
(0.123456789, 0.123456780, significant=9))
The function raises an AssertionError:
Significance 9
Traceback (most recent call last):
...
raise AssertionError(msg)
AssertionError:
Items are not equal to 9 significant digits:
ACTUAL: 0.123456789
DESIRED: 0.12345678
What just happened?
We used the assert_approx_equal() function from the NumPy testing package to
check whether 0.123456789 and 0.123456780 are equal for different decimal precisions.
Almost equal arrays
The assert_array_almost_equal() function raises an exception if two arrays are not
equal up to a specified precision. The function checks whether the two arrays have the same
shape. Then, the values of the arrays are compared element by element with the following:
|expected - actual| < 0.5 10-decimal
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Chapter 8
Time for action – asserting arrays almost equal
Let's form arrays with the values from the previous Time for action section by adding a 0 to
each array:
1.
Call the function with lower precision:
print("Decimal 8", np.testing.assert_array_almost_equal([0,
0.123456789], [0, 0.123456780], decimal=8))
The result is as follows:
Decimal 8 None
2.
Call the function with higher precision:
print("Decimal 9", np.testing.assert_array_almost_equal([0,
0.123456789], [0, 0.123456780], decimal=9))
The test raises an AssertionError:
Decimal 9
Traceback (most recent call last):
…
assert_array_compare
raise AssertionError(msg)
AssertionError:
Arrays are not almost equal
(mismatch 50.0%)
x: array([ 0.
,
0.12345679])
y: array([ 0.
,
0.12345678])
What just happened?
We compared two arrays with the NumPy array_almost_equal() function.
Have a go hero – comparing arrays with different shapes
Use the NumPy array_almost_equal() function to compare two arrays with
different shapes.
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Assuring Quality with Testing
Equal arrays
The assert_array_equal() function raises an exception if two arrays are not equal. The
shapes of the arrays have to be equal and the elements of each array must be equal. NaNs are
allowed in the arrays. Alternatively, arrays can be compared with the array_allclose()
function. This function has the parameters absolute tolerance (atol) and relative tolerance
(rtol). For two arrays a and b, these parameters satisfy the following equation:
|a - b| <= (atol + rtol * |b|)
Time for action – comparing arrays
Let's compare two arrays with the functions we just mentioned. We will reuse the arrays
from the previous Time for action section and add a NaN to them:
1.
Call the array_allclose() function:
print("Pass", np.testing.assert_allclose([0, 0.123456789,
np.nan], [0, 0.123456780, np.nan], rtol=1e-7, atol=0))
The result is as follows:
Pass None
2.
Call the array_equal() function:
print("Fail", np.testing.assert_array_equal([0, 0.123456789,
np.nan], [0, 0.123456780, np.nan]))
The test fails with an AssertionError:
Fail
Traceback (most recent call last):
…
assert_array_compare
raise AssertionError(msg)
AssertionError:
Arrays are not equal
(mismatch 50.0%)
x: array([ 0.
,
0.12345679,
nan])
y: array([ 0.
,
0.12345678,
nan])
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Chapter 8
What just happened?
We compared two arrays with the array_allclose() function and the array_equal()
function.
Ordering arrays
The assert_array_less() function raises an exception if two arrays do not have the
same shape, and the elements of the first array are strictly less than the elements of the
second array.
Time for action – checking the array order
Let's check whether one array is strictly greater than another array:
1.
Call the assert_array_less() function with two strictly ordered arrays:
print("Pass", np.testing.assert_array_less([0, 0.123456789,
np.nan], [1, 0.23456780, np.nan]))
The result is as follows:
Pass None
2.
Call the assert_array_less() function:
print("Fail", np.testing.assert_array_less([0, 0.123456789,
np.nan], [0, 0.123456780, np.nan]))
The test raises an exception:
Fail
Traceback (most recent call last):
...
raise AssertionError(msg)
AssertionError:
Arrays are not less-ordered
(mismatch 100.0%)
x: array([ 0.
,
0.12345679,
nan])
y: array([ 0.
,
0.12345678,
nan])
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Assuring Quality with Testing
What just happened?
We checked the ordering of two arrays with the assert_array_less() function.
Object comparison
The assert_equal() function raises an exception if two objects are not equal. The objects
do not have to be NumPy arrays—they can also be lists, tuples, or dictionaries.
Time for action – comparing objects
Suppose you need to compare two tuples. We can use the assert_equal() function to
do that.
Call the assert_equal() function:
print("Equal?", np.testing.assert_equal((1, 2), (1, 3)))
The call raises an error because the items are not equal:
Equal?
Traceback (most recent call last):
...
raise AssertionError(msg)
AssertionError:
Items are not equal:
item=1
ACTUAL: 2
DESIRED: 3
What just happened?
We compared two tuples with the assert_equal() function—an exception was raised
because the tuples were not equal to each other.
String comparison
The assert_string_equal() function asserts that two strings are equal. If the test fails,
the function throws an exception and shows the difference between the strings. The case of
the string characters matters.
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Chapter 8
Time for action – comparing strings
Let's compare strings. Both strings are the word "NumPy":
1.
Call the assert_string_equal() function to compare a string with itself. This
test, of course, should pass:
print("Pass", np.testing.assert_string_equal("NumPy", "NumPy"))
The test passes:
Pass None
2.
Call the assert_string_equal() function to compare a string with another
string with the same letters, but different casing. This test should throw
an exception:
print("Fail", np.testing.assert_string_equal("NumPy", "Numpy"))
The test raises an error:
Fail
Traceback (most recent call last):
…
raise AssertionError(msg)
AssertionError: Differences in strings:
- NumPy?
^
+ Numpy?
^
What just happened?
We compared two strings with the assert_string_equal() function. The test threw an
exception when the casing did not match.
Floating-point comparisons
The representation of floating-point numbers in computers is not exact. This leads to issues
when comparing floating-point numbers. The assert_array_almost_equal_nulp()
and assert_array_max_ulp() NumPy functions provide consistent floating-point
comparisons. Unit of Least Precision (ULP) of floating-point numbers, according to the IEEE
754 specification, a half ULP precision is required for elementary arithmetic operations.
You can compare this to a ruler. A metric system ruler usually has ticks for millimeters,
but beyond that you can only estimate half millimeters.
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Assuring Quality with Testing
Machine epsilon is the largest relative rounding error in floating-point arithmetic. Machine
epsilon is equal to ULP relative to 1. The NumPy finfo() function allows us to determine
the machine epsilon. The Python standard library also can give you the machine epsilon
value. The value should be the same as that given by NumPy.
Time for action – comparing with assert_array_almost_equal_
nulp
Let's see the assert_array_almost_equal_nulp() function in action:
1.
Determine the machine epsilon with the finfo() function:
eps = np.finfo(float).eps
print("EPS", eps)
The epsilon would be as follows:
EPS 2.22044604925e-16
2.
Compare 1.0 with 1 + epsilon using the assert_almost_equal_nulp()
function. Do the same for 1 + 2 * epsilon:
print("1",
np.testing.assert_array_almost_equal_nulp(1.0, 1.0 + eps))
print("2",
np.testing.assert_array_almost_equal_nulp(1.0, 1.0 + 2 * eps))
The result is as follows:
1 None
2
Traceback (most recent call last):
…
assert_array_almost_equal_nulp
raise AssertionError(msg)
AssertionError: X and Y are not equal to 1 ULP (max is 2)
What just happened?
We determined the machine epsilon with the finfo() function. We then compared 1.0
with 1 + epsilon with the assert_almost_equal_nulp() function. This test passed
however, adding another epsilon resulted in an exception.
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Chapter 8
Comparison of floats with more ULPs
The assert_array_max_ulp() function allows you to specify an upper bound for the
number of ULPs you would allow. The maxulp parameter accepts an integer value for the
limit. The value of this parameter is 1 by default.
Time for action – comparing using maxulp of 2
Let's do the same comparisons as in the previous Time for action section, but specify a
maxulp of 2 when necessary:
1.
Determine the machine epsilon with the finfo() function:
eps = np.finfo(float).eps
print("EPS", eps)
The epsilon would be as follows:
EPS 2.22044604925e-16
2.
Do the comparisons as done in the previous Time for action section, but use the
assert_array_max_ulp() function with the appropriate maxulp value:
print("1", np.testing.assert_array_max_ulp(1.0, 1.0 + eps))
print("2", np.testing.assert_array_max_ulp(1.0, 1 + 2 * eps,
maxulp=2))
The output is as follows:
1 1.0
2 2.0
What just happened?
We compared the same values as the previous Time for action section, but specified a
maxulp of 2 in the second comparison. Using the assert_array_max_ulp() function
with the appropriate maxulp value, these tests passed with a return value of the number
of ULPs.
Unit tests
Unit tests are automated tests, which test a small piece of code, usually a function or
method. Python has the PyUnit API for unit testing. As NumPy users, we can make
use of the assert functions we saw in action before.
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Assuring Quality with Testing
Time for action – writing a unit test
We will write tests for a simple factorial function. The tests will check for the so-called happy
path and abnormal conditions.
1.
Start by writing the factorial function:
import numpy as np
import unittest
def factorial(n):
if n == 0:
return 1
if n < 0:
raise ValueError, "Unexpected negative value"
return np.arange(1, n+1).cumprod()
The code uses the arange() and cumprod() functions to create arrays
and calculate the cumulative product, but we added a few checks for
boundary conditions.
2.
Now we will write the unit test. Let's write a class that will contain the unit tests.
It extends the TestCase class from the unittest module, which is part of
standard Python. Test for calling the factorial function with the following three
attributes:
a positive number, the happy path
boundary condition 0
negative numbers, which should result in an error
class FactorialTest(unittest.TestCase):
def test_factorial(self):
#Test for the factorial of 3 that should pass.
self.assertEqual(6, factorial(3)[-1])
np.testing.assert_equal(np.array([1, 2, 6]),
factorial(3))
def test_zero(self):
#Test for the factorial of 0 that should pass.
self.assertEqual(1, factorial(0))
def test_negative(self):
#Test for the factorial of negative numbers that
should fail.
# It should throw a ValueError, but we expect
IndexError
self.assertRaises(IndexError, factorial(-10))
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Chapter 8
We rigged one of the tests to fail, as you can see in the following output:
$ python unit_test.py
.E.
==================================================================
====
ERROR: test_negative (__main__.FactorialTest)
--------------------------------------------------------------------Traceback (most recent call last):
File "unit_test.py", line 26, in test_negative
self.assertRaises(IndexError, factorial(-10))
File "unit_test.py", line 9, in factorial
raise ValueError, "Unexpected negative value"
ValueError: Unexpected negative value
--------------------------------------------------------------------Ran 3 tests in 0.003s
FAILED (errors=1)
What just happened?
We made some happy path tests for the factorial function code. We let the boundary
condition test fail on purpose (see unit_test.py):
import numpy as np
import unittest
def factorial(n):
if n == 0:
return 1
if n < 0:
raise ValueError, "Unexpected negative value"
return np.arange(1, n+1).cumprod()
class FactorialTest(unittest.TestCase):
def test_factorial(self):
#Test for the factorial of 3 that should pass.
self.assertEqual(6, factorial(3)[-1])
np.testing.assert_equal(np.array([1, 2, 6]), factorial(3))
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Assuring Quality with Testing
def test_zero(self):
#Test for the factorial of 0 that should pass.
self.assertEqual(1, factorial(0))
def test_negative(self):
#Test for the factorial of negative numbers that should fail.
# It should throw a ValueError, but we expect IndexError
self.assertRaises(IndexError, factorial(-10))
if __name__ == '__main__':
unittest.main()
Nose test decorators
A nose is an organ above the mouth that is used by humans and animals to breathe and
smell. It is also a Python framework that makes (unit) testing easier. Nose helps you organize
tests. According to the nose documentation:
"Any python source file, directory or package that matches the testMatch regular
expression (by default: (?:^|[b_.-])[Tt]est) will be collected as a test."
Nose makes extensive use of decorators. Python decorators are annotations that indicate
something about a method or a function (see http://thecodeship.com/patterns/
guide-to-python-function-decorators/). The numpy.testing module has a
number of decorators. The following table shows the different decorators in the
numpy.testing module:
Decorator
numpy.testing.decorators.deprecated
Description
numpy.testing.decorators.
knownfailureif
This function raises KnownFailureTest
exception based on a condition
numpy.testing.decorators.setastest
This decorator marks a function as being a
test or not being a test
numpy.testing.decorators.skipif
This function raises a SkipTest exception
based on a condition
numpy.testing.decorators.slow
This function labels test functions or
methods as slow
This function filters deprecation warnings
when running tests
Additionally, we can call the decorate_methods() function to apply decorators on
methods of a class matching a regular expression or a string.
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Chapter 8
Time for action – decorating tests
We will apply the @setastest decorator directly to test functions. Then we will apply the
same decorator to a method to disable it. Also, we will skip one of the tests and fail another.
First, install nose in case you don't have it yet.
1.
Install nose with setuptools:
$ [sudo] easy_install nose
Or pip:
$ [sudo] pip install nose
2.
Apply one function as being a test and another as not being a test:
@setastest(False)
def test_false():
pass
@setastest(True)
def test_true():
pass
3.
Skip tests with the @skipif decorator. Let's use a condition that always leads to a
test being skipped:
@skipif(True)
def test_skip():
pass
4.
Add a test function that always passes. Then, decorate it with the
@knownfailureif decorator so that the test always fails:
@knownfailureif(True)
def test_alwaysfail():
pass
5.
Define some test classes with methods that normally should be executed by nose:
class TestClass():
def test_true2(self):
pass
class TestClass2():
def test_false2(self):
pass
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Assuring Quality with Testing
6.
Let's disable the second test method from the previous step:
decorate_methods(TestClass2, setastest(False), 'test_false2')
7.
Run the tests with the following command:
$ nosetests -v decorator_setastest.py
decorator_setastest.TestClass.test_true2 ... ok
decorator_setastest.test_true ... ok
decorator_test.test_skip ... SKIP: Skipping test: test_skipTest
skipped due to test condition
decorator_test.test_alwaysfail ... ERROR
==================================================================
====
ERROR: decorator_test.test_alwaysfail
--------------------------------------------------------------------Traceback (most recent call last):
File "…/nose/case.py", line 197, in runTest
self.test(*self.arg)
File …/numpy/testing/decorators.py", line 213, in knownfailer
raise KnownFailureTest(msg)
KnownFailureTest: Test skipped due to known failure
--------------------------------------------------------------------Ran 4 tests in 0.001s
FAILED (SKIP=1, errors=1)
What just happened?
We decorated some functions and methods as not being tests so that they were ignored by
nose. We skipped one test and failed another too. We did this by applying decorators directly
and with the decorate_methods() function (see decorator_test.py):
from numpy.testing.decorators import setastest
from numpy.testing.decorators import skipif
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Chapter 8
from numpy.testing.decorators import knownfailureif
from numpy.testing import decorate_methods
@setastest(False)
def test_false():
pass
@setastest(True)
def test_true():
pass
@skipif(True)
def test_skip():
pass
@knownfailureif(True)
def test_alwaysfail():
pass
class TestClass():
def test_true2(self):
pass
class TestClass2():
def test_false2(self):
pass
decorate_methods(TestClass2, setastest(False), 'test_false2')
Docstrings
Doctests are strings embedded in Python code that resemble interactive sessions. These
strings can be used to test certain assumptions or just to provide examples. The numpy.
testing module has a function to run these tests.
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Assuring Quality with Testing
Time for action – executing doctests
Let's write a simple example that is supposed to calculate the well-known factorial, but
doesn't cover all of the possible boundary conditions. In other words, some tests will fail.
1.
The docstring will look like text you would see in a Python shell (including a
prompt). Rig one of the tests to fail, just to see what will happen:
"""
Test for the factorial of 3 that should pass.
>>> factorial(3)
6
Test for the factorial of 0 that should fail.
>>> factorial(0)
1
"""
2.
Write the following line of NumPy code:
return np.arange(1, n+1).cumprod()[-1]
We want this code to fail from time to time for demonstration purposes.
3.
Run the doctest by calling the rundocs() function of the numpy.testing
module, for instance, in the Python shell:
>>> from numpy.testing import rundocs
>>> rundocs('docstringtest.py')
Traceback (most recent call last):
File "", line 1, in
File "…/numpy/testing/utils.py", line 998, in rundocs
raise AssertionError("Some doctests failed:\n%s" % "\n".
join(msg))
AssertionError: Some doctests failed:
******************************************************************
****
File "docstringtest.py", line 10, in docstringtest.factorial
Failed example:
factorial(0)
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Chapter 8
Exception raised:
Traceback (most recent call last):
File "…/doctest.py", line 1254, in __run
compileflags, 1) in test.globs
File "", line 1, in
factorial(0)
File "docstringtest.py", line 13, in factorial
return np.arange(1, n+1).cumprod()[-1]
IndexError: index -1 is out of bounds for axis 0 with size 0
What just happened?
We wrote a docstring test, which didn't take into account 0 and negative numbers. We ran
the test with the rundocs() function from the numpy.testing module and got an index
error as a result (see docstringtest.py):
import numpy as np
def factorial(n):
"""
Test for the factorial of 3 that should pass.
>>> factorial(3)
6
Test for the factorial of 0 that should fail.
>>> factorial(0)
1
"""
return np.arange(1, n+1).cumprod()[-1]
Summary
You learned about testing and NumPy testing utilities in this chapter. We covered unit
testing, docstring tests, assert functions, and floating-point precision. Most of the NumPy
assert functions take care of the complexities of floating-point numbers. We demonstrated
NumPy decorators that can be used by nose. Decorators make testing easier and document
the developer intention.
The topic of the next chapter is matplotlib—the Python scientific visualization and graphing
open source library.
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9
Plotting with matplotlib
matplotlib is a very useful Python plotting library. It integrates nicely with
NumPy but is a separate open source project. You can find a gallery of beautiful
examples at http://matplotlib.org/gallery.html.
matplotlib also has utility functions to download and manipulate data from
Yahoo Finance. We will see several examples of stock charts.
This chapter features extended coverage of the following topics:
Simple plots
Subplots
Histograms
Plot customization
Three-dimensional plots
Contour plots
Animation
Logplots
Simple plots
The matplotlib.pyplot package contains functionality for simple plots. It is important
to remember that each subsequent function call changes the state of the current plot.
Eventually, we will want to either save the plot in a file or display it with the show()
function. However, if we are in IPython running on a Qt or Wx backend, the figure updates
interactively without waiting for the show() function. This is comparable to the way text
output is printed on the fly.
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Plotting with matplotlib
Time for action – plotting a polynomial function
To illustrate how plotting works, let's display some polynomial graphs. We will use the
NumPy polynomial function poly1d() to create a polynomial.
1.
Take the standard input values as polynomial coefficients. Use the NumPy
poly1d() function to create a polynomial:
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
2.
Create the x values with the NumPy the linspace() function. Use the range -10
to 10 and create 30 even spaced values:
x = np.linspace(-10, 10, 30)
3.
Calculate the polynomial values using the polynomial we created in the first step:
y = func(x)
4.
Call the plot() function; this does not immediately display the graph:
plt.plot(x, y)
5.
Add a label to the x axis with the xlabel() function:
plt.xlabel('x')
6.
Add a label to the y axis with the ylabel() function:
plt.ylabel('y(x)')
7.
Call the show() function to display the graph:
plt.show()
The following is a plot with polynomial coefficients 1, 2, 3, and 4:
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Chapter 9
What just happened?
We displayed a polynomial graph on our screen. We added labels to the x and y axes
(see polyplot.py):
import numpy as np
import matplotlib.pyplot as plt
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
x = np.linspace(-10, 10, 30)
y = func(x)
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y(x)')
plt.show()
Pop quiz – the plot() function
Q1. What does the plot() function do?
1. It displays two-dimensional plots on screen.
2. It saves an image of a two-dimensional plot in a file.
3. It does both a and b.
4. It does neither a, b, or c.
Plot format string
The plot() function accepts an unlimited number of arguments. In the previous section,
we gave it two arrays as arguments. We could also specify the line color and style with an
optional format string. By default, it is a solid blue line denoted as b-, but you can specify
a different color and style, such as red dashes.
Time for action – plotting a polynomial and its derivatives
Let's plot a polynomial and its first-order derivative using the deriv() function with m as 1.
We already did the first part in the previous Time for action section. We want two different
line styles to discern what is what.
1.
Create and differentiate the polynomial:
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
func1 = func.deriv(m=1)
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Plotting with matplotlib
x = np.linspace(-10, 10, 30)
y = func(x)
y1 = func1(x)
2.
Plot the polynomial and its derivative in two styles: red circles and green dashes. You
cannot see the colors in a print copy of this book, so you will have to try the code
out for yourself:
plt.plot(x, y, 'ro', x, y1, 'g--')
plt.xlabel('x')
plt.ylabel('y')
plt.show()
The graph with polynomial coefficients 1, 2, 3, and 4 is as follows:
What just happened?
We plotted a polynomial and its derivative using two different line styles and one call of the
plot() function (see polyplot2.py):
import numpy as np
import matplotlib.pyplot as plt
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
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Chapter 9
func1 = func.deriv(m=1)
x = np.linspace(-10, 10, 30)
y = func(x)
y1 = func1(x)
plt.plot(x, y, 'ro', x, y1, 'g--')
plt.xlabel('x')
plt.ylabel('y')
plt.show()
Subplots
At a certain point, you will have too many lines in one plot. However, you would still like
everything grouped together. We can do this with the subplot() function. This function
creates multiple plots in a grid.
Time for action – plotting a polynomial and its derivatives
Let's plot a polynomial and its first and second derivative. We will make three subplots for
the sake of clarity:
1.
Create a polynomial and its derivatives using the following code:
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
x = np.linspace(-10, 10, 30)
y = func(x)
func1 = func.deriv(m=1)
y1 = func1(x)
func2 = func.deriv(m=2)
y2 = func2(x)
2.
Create the first subplot of the polynomial with the subplot() function. The first
parameter of this function is the number of rows, the second parameter is the
number of columns, and the third parameter is an index number starting with 1.
Alternatively, combine the three parameters into a single number, such as 311. The
subplots will be organized in three rows and one column. Give the subplot the title
Polynomial. Make a solid red line:
plt.subplot(311)
plt.plot(x, y, 'r-')
plt.title("Polynomial")
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Plotting with matplotlib
3.
Create the third subplot of the first derivative with the subplot() function. Give
the subplot the title First Derivative. Use a line of blue triangles:
plt.subplot(312)
plt.plot(x, y1, 'b^')
plt.title("First Derivative")
4.
Create the second subplot of the second derivative with the subplot() function.
Give the subplot the title Second Derivative. Use a line of green circles:
plt.subplot(313)
plt.plot(x, y2, 'go')
plt.title("Second Derivative")
plt.xlabel('x')
plt.ylabel('y')
plt.show()
The three subplots with polynomial coefficients 1, 2, 3, and 4 are as follows:
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Chapter 9
What just happened?
We plotted a polynomial and its first and second derivatives using three different line styles
and three subplots in three rows and one column (see polyplot3.py):
import numpy as np
import matplotlib.pyplot as plt
func = np.poly1d(np.array([1, 2, 3, 4]).astype(float))
x = np.linspace(-10, 10, 30)
y = func(x)
func1 = func.deriv(m=1)
y1 = func1(x)
func2 = func.deriv(m=2)
y2 = func2(x)
plt.subplot(311)
plt.plot(x, y, 'r-')
plt.title("Polynomial")
plt.subplot(312)
plt.plot(x, y1, 'b^')
plt.title("First Derivative")
plt.subplot(313)
plt.plot(x, y2, 'go')
plt.title("Second Derivative")
plt.xlabel('x')
plt.ylabel('y')
plt.show()
Finance
matplotlib can help monitor our stock investments. The matplotlib.finance
package has utilities with which we can download stock quotes from Yahoo Finance at
http://finance.yahoo.com/. We can then plot the data as candlesticks.
Time for action – plotting a year's worth of stock quotes
We can plot a year's worth of stock quotes data with the matplotlib.finance package.
This requires a connection to Yahoo Finance, which is the data source.
1.
Determine the start date by subtracting one year from today:
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
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Plotting with matplotlib
from matplotlib.finance import quotes_historical_yahoo
from matplotlib.finance import candlestick
import sys
from datetime import date
import matplotlib.pyplot as plt
today = date.today()
start = (today.year - 1, today.month, today.day)
2.
We need to create the so-called locators. These objects from the matplotlib.
dates package locate months and days on the x axis:
alldays = DayLocator()
months = MonthLocator()
3.
Create a date formatter to format the dates on the x axis. This formatter creates a
string containing the short name of a month and the year:
month_formatter = DateFormatter("%b %Y")
4.
Download the stock quote data from Yahoo finance with the following code:
quotes = quotes_historical_yahoo(symbol, start, today)
5.
Create a matplotlib Figure object—this is a top-level container for plot
components:
fig = plt.figure()
6.
Add a subplot to the figure:
ax = fig.add_subplot(111)
7.
Set the major locator on the x axis to the months locator. This locator is responsible
for the big ticks on the x axis:
ax.xaxis.set_major_locator(months)
8.
Set the minor locator on the x axis to the days locator. This locator is responsible for
the small ticks on the x axis:
ax.xaxis.set_minor_locator(alldays)
9.
Set the major formatter on the x axis to the months formatter. This formatter is
responsible for the labels of the big ticks on the x axis:
ax.xaxis.set_major_formatter(month_formatter)
10.
A function in the matplotlib.finance package allows us to display candlesticks.
Create the candlesticks using the quotes data. It is possible to specify the width of
the candlesticks. For now, use the default value:
candlestick(ax, quotes)
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Chapter 9
11.
Format the labels on the x axis as dates. This rotates the labels on the x axis so that
they fit better:
fig.autofmt_xdate()
plt.show()
The candlestick chart for DISH (Dish Network Corp) appears as follows:
What just happened?
We downloaded a year's worth of data from Yahoo Finance. We charted this data using
candlesticks (see candlesticks.py):
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
from matplotlib.finance import quotes_historical_yahoo
from matplotlib.finance import candlestick
import sys
from datetime import date
import matplotlib.pyplot as plt
today = date.today()
start = (today.year - 1, today.month, today.day)
[ 225 ]
Plotting with matplotlib
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
symbol = 'DISH'
if len(sys.argv) == 2:
symbol = sys.argv[1]
quotes = quotes_historical_yahoo(symbol, start, today)
fig = plt.figure()
ax = fig.add_subplot(111)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_formatter(month_formatter)
candlestick(ax, quotes)
fig.autofmt_xdate()
plt.show()
Histograms
Histograms visualize the distribution of numerical data. matplotlib has the handy hist()
function that graphs histograms. The hist() function has two main arguments—the array
containing the data and the number of bars.
Time for action – charting stock price distributions
Let's chart the stock price distribution of quotes from Yahoo Finance.
1.
Download the data going back one year:
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo(symbol, start, today)
2.
The quotes data in the previous step is stored in a Python list. Convert this to a
NumPy array and extract the close prices:
quotes = np.array(quotes)
close = quotes.T[4]
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Chapter 9
3.
Draw the histogram with a reasonable number of bars:
plt.hist(close, np.sqrt(len(close)))
plt.show()
The histogram for DISH appears as follows:
What just happened?
We charted the stock price distribution of DISH as a histogram (see stockhistogram.py):
from matplotlib.finance import quotes_historical_yahoo
import sys
from datetime import date
import matplotlib.pyplot as plt
import numpy as np
today = date.today()
start = (today.year - 1, today.month, today.day)
symbol = 'DISH'
if len(sys.argv) == 2:
symbol = sys.argv[1]
[ 227 ]
Plotting with matplotlib
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
close = quotes.T[4]
plt.hist(close, np.sqrt(len(close)))
plt.show()
Have a go hero – drawing a bell curve
Overlay a bell curve (related to the Gaussian or normal distribution) using the average price
and standard deviation. This is, of course, only an exercise.
Logarithmic plots
Logarithmic plots are useful when the data has a wide range of values. matplotlib has
the functions semilogx() (logarithmic x axis), semilogy() (logarithmic y axis), and
loglog() (x and y axes logarithmic).
Time for action – plotting stock volume
Stock volume varies a lot, so let's plot it on a logarithmic scale. First, we need to download
historical data from Yahoo Finance, extract the dates and volume, create locators and a date
formatter, and create the figure and add it to a subplot. We already went through these
steps in the previous Time for action section, so we will skip them here.
Plot the volume using a logarithmic scale:
plt.semilogy(dates, volume)
Now, set the locators and format the x axis as dates. Instructions for these steps can be
found in the previous Time for action section as well.
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Chapter 9
The stock volume using a logarithmic scale for DISH appears as follows:
What just happened?
We plotted stock volume using a logarithmic scale (see logy.py):
from matplotlib.finance import quotes_historical_yahoo
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
import sys
from datetime import date
import matplotlib.pyplot as plt
import numpy as np
today = date.today()
start = (today.year - 1, today.month, today.day)
symbol = 'DISH'
if len(sys.argv) == 2:
symbol = sys.argv[1]
[ 229 ]
Plotting with matplotlib
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
dates = quotes.T[0]
volume = quotes.T[5]
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
fig = plt.figure()
ax = fig.add_subplot(111)
plt.semilogy(dates, volume)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_formatter(month_formatter)
fig.autofmt_xdate()
plt.show()
Scatter plots
A scatter plot displays values for two numerical variables in the same dataset. The matplotlib
scatter() function creates a scatter plot. Optionally, we can specify the color and size of
the data points, as well as alpha transparency, in the plot.
Time for action – plotting price and volume returns with a
scatter plot
We can easily make a scatter plot of the stock price and volume returns. Again, let's
download the necessary data from Yahoo Finance.
1.
The quotes data in the previous step is stored in a Python list. Convert this to a
NumPy array and extract the close and volume values:
dates = quotes.T[4]
volume = quotes.T[5]
2.
Calculate the close price and volume returns:
ret = np.diff(close)/close[:-1]
volchange = np.diff(volume)/volume[:-1]
3.
Create a matplotlib figure object:
fig = plt.figure()
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Chapter 9
4.
Add a subplot to the figure:
ax = fig.add_subplot(111)
5.
Create the scatter plot with the color of the data points linked to the close return,
and the size linked to the volume change:
ax.scatter(ret, volchange, c=ret * 100,
s=volchange * 100, alpha=0.5)
6.
Set the title of the plot and put a grid on it:
ax.set_title('Close and volume returns')
ax.grid(True)
plt.show()
The scatter plot for DISH appears as follows:
What just happened?
We made a scatter plot of the close price and volume returns for DISH
(see scatterprice.py):
from matplotlib.finance import quotes_historical_yahoo
import sys
from datetime import date
import matplotlib.pyplot as plt
import numpy as np
[ 231 ]
Plotting with matplotlib
today = date.today()
start = (today.year - 1, today.month, today.day)
symbol = 'DISH'
if len(sys.argv) == 2:
symbol = sys.argv[1]
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
close = quotes.T[4]
volume = quotes.T[5]
ret = np.diff(close)/close[:-1]
volchange = np.diff(volume)/volume[:-1]
fig = plt.figure()
ax = fig.add_subplot(111)
ax.scatter(ret, volchange, c=ret * 100, s=volchange * 100, alpha=0.5)
ax.set_title('Close and volume returns')
ax.grid(True)
plt.show()
Fill between
The fill_between() function fills a plot region with a specified color. We can choose an
optional alpha channel value. The function also has a where parameter so that we can shade
a region based on a condition.
Time for action – shading plot regions based on a condition
Imagine that you want to shade a region of a stock chart, where the closing price is below
average, with a different color than when it is above the mean. The fill_between()
function is the best choice for the job. We will, again, omit the steps of downloading
historical data going back one year, extracting dates and close prices, and creating locators
and date formatter.
1.
Create a matplotlib Figure object:
fig = plt.figure()
2.
Add a subplot to the figure:
ax = fig.add_subplot(111)
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Chapter 9
3.
Plot the closing price:
ax.plot(dates, close)
4.
Shade the regions of the plot below the closing price using different colors
depending on whether the values are below or above the average price:
plt.fill_between(dates, close.min(), close,
where=close>close.mean(), facecolor="green", alpha=0.4)
plt.fill_between(dates, close.min(), close,
where=closeclose.mean(),
facecolor="green", alpha=0.4)
plt.fill_between(dates, close.min(), close, where=close"))
4.
Add a legend and let matplotlib decide where to put it:
leg = ax.legend(loc='best', fancybox=True))
[ 235 ]
Plotting with matplotlib
5.
Make the legend transparent by setting the alpha channel value:
leg.get_frame().set_alpha(0.5)
The stock price and moving averages with a legend and annotations appears
as follows:
What just happened?
We plotted the close price of a stock and three of its EMAs. We added a legend to the
plot. We annotated the crossover points of the first two averages with annotations
(see emalegend.py):
from matplotlib.finance import quotes_historical_yahoo
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
import sys
from datetime import date
import matplotlib.pyplot as plt
import numpy as np
today = date.today()
start = (today.year - 1, today.month, today.day)
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Chapter 9
symbol = 'DISH'
if len(sys.argv) == 2:
symbol = sys.argv[1]
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
dates = quotes.T[0]
close = quotes.T[4]
fig = plt.figure()
ax = fig.add_subplot(111)
emas = []
for i in range(9, 18, 3):
weights = np.exp(np.linspace(-1., 0., i))
weights /= weights.sum()
ema = np.convolve(weights, close)[i-1:-i+1]
idx = (i - 6)/3
ax.plot(dates[i-1:], ema, lw=idx, label="EMA(%s)" % (i))
data = np.column_stack((dates[i-1:], ema))
emas.append(np.rec.fromrecords(data, names=["dates", "ema"]))
first = emas[0]["ema"].flatten()
second = emas[1]["ema"].flatten()
bools = np.abs(first[-len(second):] - second)/second < 0.0001
xpoints = np.compress(bools, emas[1])
for xpoint in xpoints:
ax.annotate('x', xy=xpoint, textcoords='offset points',
xytext=(-50, 30),
arrowprops=dict(arrowstyle="->"))
leg = ax.legend(loc='best', fancybox=True)
leg.get_frame().set_alpha(0.5)
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
ax.plot(dates, close, lw=1.0, label="Close")
ax.xaxis.set_major_locator(months)
ax.xaxis.set_minor_locator(alldays)
[ 237 ]
Plotting with matplotlib
ax.xaxis.set_major_formatter(month_formatter)
ax.grid(True)
fig.autofmt_xdate()
plt.show()
Three-dimensional plots
Three-dimensional plots are pretty spectacular, so we have to cover them here too. For
three-dimensional plots, we need an Axes3D object associated with a 3D projection.
Time for action – plotting in three dimensions
We will plot a simple three-dimensional function:
z = x2 + y 2
1.
Use the 3D keyword to specify a three-dimensional projection for the plot:
ax = fig.add_subplot(111, projection='3d')
2.
To create a square two-dimensional grid, use the meshgrid() function to initialize
the x and y values:
u = np.linspace(-1, 1, 100)
x, y = np.meshgrid(u, u)
3.
We will specify the row strides, column strides, and the color map for the surface
plot. The strides determine the size of the tiles in the surface. The choice for color
map is a matter of taste:
ax.plot_surface(x, y, z,
cmap=cm.YlGnBu_r)
rstride=4, cstride=4,
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Chapter 9
The result is the following three-dimensional plot:
What just happened?
We created a plot of a three-dimensional function (see three_d.py):
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import cm
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
u = np.linspace(-1, 1, 100)
x, y = np.meshgrid(u, u)
z = x ** 2 + y ** 2
ax.plot_surface(x, y, z, rstride=4,
cstride=4, cmap=cm.YlGnBu_r)
plt.show()
[ 239 ]
Plotting with matplotlib
Contour plots
matplotlib contour three-dimensional plots come in two flavors—filled and unfilled.
Contour plots use the so-called contour lines. You may be familiar with contour lines from
geographic maps. In such maps, contour lines connect points of the same elevation above
sea level. We can create normal contour plots with the contour() function. For filled
contour plots, we use the contourf() function.
Time for action – drawing a filled contour plot
We will draw a filled contour plot of the three-dimensional mathematical function in the
previous Time for action section. The code is also pretty similar. One key difference is that we
don't need the 3D projection parameter any more. To draw the filled contour plot, use the
following line of code:
ax.contourf(x, y, z)
This gives us the following filled contour plot:
What just happened?
We created a filled contour plot of a three-dimensional mathematical function (see
contour.py):
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import cm
[ 240 ]
Chapter 9
fig = plt.figure()
ax = fig.add_subplot(111)
u = np.linspace(-1, 1, 100)
x, y = np.meshgrid(u, u)
z = x ** 2 + y ** 2
ax.contourf(x, y, z)
plt.show()
Animation
matplotlib offers fancy animation capabilities via a special animation module. We need
to define a callback function that is used to regularly update the screen. We also need a
function to generate data to be plotted.
Time for action – animating plots
We will plot three random datasets and display them as circles, dots, and triangles. However,
we will only update two of those datasets with random values.
1.
Plot three random datasets as circles, dots, and triangles in different colors:
circles, triangles, dots = ax.plot(x, 'ro', y, 'g^', z, 'b.')
2.
This function gets called to update the screen regularly. Update two of the plots with
new y values:
def update(data):
circles.set_ydata(data[0])
triangles.set_ydata(data[1])
return circles, triangles
3.
Generate random data with NumPy:
def generate():
while True: yield np.random.rand(2, N)
[ 241 ]
Plotting with matplotlib
The following is a snapshot of the animation in action:
What just happened?
We created an animation of random data points (see animation.py):
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
fig = plt.figure()
ax = fig.add_subplot(111)
N = 10
x = np.random.rand(N)
y = np.random.rand(N)
z = np.random.rand(N)
circles, triangles, dots = ax.plot(x, 'ro', y, 'g^', z, 'b.')
ax.set_ylim(0, 1)
plt.axis('off')
def update(data):
circles.set_ydata(data[0])
triangles.set_ydata(data[1])
return circles, triangles
[ 242 ]
Chapter 9
def generate():
while True: yield np.random.rand(2, N)
anim = animation.FuncAnimation(fig, update,
generate, interval=150)
plt.show()
Summary
This chapter was about matplotlib—a Python plotting library. We covered simple plots,
histograms, plot customization, subplots, three-dimensional plots, contour plots, and
logarithmic plots. You also saw a few examples of displaying stock charts. Obviously, we
only scratched the surface and just saw the tip of the iceberg. matplotlib is very feature
rich, so we didn't have space to cover Latex support, polar coordinates support, and other
functionality.
The author of matplotlib, John Hunter, passed away in August 2012. One of the technical
reviewers of this book suggested mentioning the John Hunter Memorial Fund (http://
numfocus.org/news/2012/08/28/johnhunter/). The memorial fund set up by the
NumFocus Foundation is an opportunity for us, fans of John Hunter's work, to "give back".
Again, for more details, check out the preceding link to the NumFocus website.
The next chapter is about SciPy—a scientific Python framework that is built on top of NumPy.
[ 243 ]
10
When NumPy Is Not Enough – SciPy
and Beyond
SciPy is a world famous Python open source scientific computing library
built on top of NumPy. It adds functionalitties such as numerical integration,
optimization, statistics, and special functions.
In this chapter, we will cover the following topics:
File I/O
Statistics
Signal processing
Optimization
Interpolation
Image and audio processing
MATLAB and Octave
MATLAB and its open source alternative, Octave, are popular mathematical programs. The
scipy.io package has functions that let you load MATLAB or Octave matrices and arrays
of numbers or strings in Python programs, and vice versa. The loadmat() function loads a
.mat file. The savemat() function saves a dictionary of names and arrays into a .mat file.
[ 245 ]
When NumPy Is Not Enough – SciPy and Beyond
Time for action – saving and loading a .mat file
If we start with NumPy arrays and decide to use said arrays within a MATLAB or Octave
environment, the easiest thing to do is create a .mat file. We can, then, load the file within
MATLAB or Octave. Let's go through the necessary steps:
1.
Create a NumPy array and call the savemat() function to create a .mat file. This
function has two parameters: a file name and a dictionary containing variable names
and values:
a = np.arange(7)
io.savemat("a.mat", {"array": a})
2.
Within a MATLAB or Octave environment, load the .mat file and check the
stored array:
octave-3.4.0:7> load a.mat
octave-3.4.0:8> a
octave-3.4.0:8> array
array =
0
1
2
3
4
5
6
What just happened?
We created a .mat file from NumPy code and loaded it within Octave. We checked the
NumPy array that was created (see scipyio.py):
import numpy as np
from scipy import io
a = np.arange(7)
io.savemat("a.mat", {"array": a})
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Chapter 10
Pop quiz – loading .mat files
Q1. Which function loads .mat files?
1.
2.
3.
4.
Loadmatlab
loadmat
loadoct
frommat
Statistics
The SciPy statistics module is called scipy.stats. There is one class that implements
continuous distributions and one class that implements discrete distributions. Also, in this
module, functions that perform a great number of statistical tests can be found.
Time for action – analyzing random values
We will generate random values that mimic a normal distribution and analyze the generated
data with statistical functions from the scipy.stats package.
1.
Generate random values from a normal distribution using the scipy.stats package:
generated = stats.norm.rvs(size=900)
2.
Fit the generated values to a normal distribution. This basically gives the mean and
standard deviation of the dataset:
print("Mean", "Std", stats.norm.fit(generated))
The mean and standard deviation appear as follows:
Mean Std (0.0071293257063200707, 0.95537708218972528)
3.
Skewness tells us how skewed (asymmetric) a probability distribution is (see
http://en.wikipedia.org/wiki/Skewness). Perform a skewness test. This
test returns two values. The second value is the p-value—the probability that the
skewness of the dataset does not correspond to a normal distribution.
Generally speaking, the p-value is the probability of an outcome
different than what was expected given the null hypothesis—in this
case, the probability of getting a skewness different from that of a
normal distribution (which is 0 because of symmetry).
P-values range from 0 to 1:
print("Skewtest", "pvalue", stats.skewtest(generated))
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The result of the skewness test appears as follows:
Skewtest pvalue (-0.62120640688766893, 0.5344638245033837)
So, there is a 53 percent chance we are not dealing with a normal distribution. It is
instructive to see what happens if we generate more points, because if we generate
more points, we should have a more normal distribution. For 900,000 points, we get
a p-value of 0.16. For 20 generated values, the p-value is 0.50.
4.
Kurtosis tells us how curved a probability distribution is. Perform a kurtosis test. This
test is set up similarly to the skewness test, but, of course, applies to kurtosis:
print("Kurtosistest", "pvalue",
stats.kurtosistest(generated))
The result of the kurtosis test appears as follows:
Kurtosistest pvalue (1.3065381019536981, 0.19136963054975586)
The p-value for 900,000 values is 0.028. For 20 generated values, the p-values
is 0.88.
5.
A normality test tells us how likely it is that a dataset complies the normal
distribution. Perform a normality test. This test also returns two values,
of which the second is a p-value:
print("Normaltest", "pvalue", stats.normaltest(generated))
The result of the normality test appears as follows:
Normaltest pvalue (2.09293921181506, 0.35117535059841687)
The p-value for 900,000 generated values is 0.035. For 20 generated values,
the p-value is 0.79.
6.
We can find the value at a certain percentile easily with SciPy:
print("95 percentile",
stats.scoreatpercentile(generated, 95))
The value at the 95th percentile appears as follows:
95 percentile 1.54048860252
7.
Do the opposite of the previous step to find the percentile at 1:
print("Percentile at 1",
stats.percentileofscore(generated, 1))
The percentile at 1 appears as follows:
Percentile at 1 85.5555555556
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8.
Plot the generated values in a histogram with matplotlib (more information about
matplotlib can be found in the previous Chapter 9, Plotting with matplotlib):
plt.hist(generated)
The histogram of the generated random values is as follows:
What just happened?
We created a dataset from a normal distribution and analyzed it with the scipy.stats
module (see statistics.py):
from __future__ import print_function
from scipy import stats
import matplotlib.pyplot as plt
generated = stats.norm.rvs(size=900)
print("Mean", "Std", stats.norm.fit(generated))
print("Skewtest", "pvalue", stats.skewtest(generated))
print("Kurtosistest", "pvalue", stats.kurtosistest(generated))
print("Normaltest", "pvalue", stats.normaltest(generated))
print("95 percentile", stats.scoreatpercentile(generated, 95))
print("Percentile at 1", stats.percentileofscore(generated, 1))
plt.title('Histogram of 900 random normally distributed values')
plt.hist(generated)
plt.grid()
plt.show()
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Have a go hero – improving the data generation
Judging from the histogram in the previous Time for action section, there is room for
improvement when it comes to generating the data. Try using NumPy or different
parameters of the scipy.stats.norm.rvs() function.
Sample comparison and SciKits
Often we have two data samples, maybe from different experiments, that are somehow
related. Statistical tests exist that can compare the samples. Some of these are implemented
in the scipy.stats module.
Another statistical test that I like is the Jarque–Bera normality test from scikits.
statsmodels.stattools. SciKits are small experimental Python software toolkits. They
are not part of SciPy. There is also pandas, which is an offshoot of scikits.statsmodels.
A list of SciKits can be found at https://scikits.appspot.com/scikits. You can
install statsmodels using setuptools with:
$ [sudo] easy_install statsmodels
Time for action – comparing stock log returns
We will download the stock quotes for the last year of two trackers using matplotlib. As
mentioned in the previous Chapter 9, Plotting with matplotlib, we can retrieve quotes from
Yahoo Finance. We will compare the log returns of the close price of DIA and SPY (DIA tracks
the Dow Jones index; SPY tracks the S & P 500 index). We will also perform the Jarque–Bera
test on the difference of the log returns.
1.
Write a function that can return the close price for a specified stock:
def get_close(symbol):
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
return quotes.T[4]
2.
Calculate the log returns for DIA and SPY. Compute the log returns by taking the
natural logarithm of the close price and then taking the difference of consecutive
values:
spy =
dia =
np.diff(np.log(get_close("SPY")))
np.diff(np.log(get_close("DIA")))
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3.
The means comparison test checks whether two different samples could have the
same mean value. Two values are returned, of which the second is the p-value from
0 to 1:
print("Means comparison", stats.ttest_ind(spy, dia))
The result of the means comparison test appears as follows:
Means comparison (-0.017995865641886155, 0.98564930169871368)
So, there is about a 98 percent chance that the two samples have the same mean
log return. Actually, the documentation has the following to say:
If we observe a large p-value, for example, larger than 0.05 or 0.1,
then we cannot reject the null hypothesis of identical average
scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or
10%, then we reject the null hypothesis of equal averages.
4.
The Kolmogorov–Smirnov two samples test tells us how likely it is that two samples
are drawn from the same distribution:
print("Kolmogorov smirnov test", stats.ks_2samp(spy, dia))
Again, two values are returned, of which the second value is the p-value:
Kolmogorov smirnov test (0.063492063492063516,
0.67615647616238039)
5.
Unleash the Jarque–Bera normality test on the difference of the log returns:
print("Jarque Bera test",
jarque_bera(spy – dia)[1])
The p-value of the Jarque–Bera normality test appears as follows:
Jarque Bera test 0.596125711042
6.
Plot the histograms of the log returns and the difference thereof with matplotlib:
plt.hist(spy, histtype="step", lw=1, label="SPY")
plt.hist(dia, histtype="step", lw=2, label="DIA")
plt.hist(spy - dia, histtype="step", lw=3,
label="Delta")
plt.legend()
plt.show()
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The histograms of the log returns and difference are shown as follows:
What just happened?
We compared samples of log returns for DIA and SPY. Also, we performed the Jarque-Bera
test on the difference of the log returns (see pair.py):
from __future__ import print_function
from matplotlib.finance import quotes_historical_yahoo
from datetime import date
import numpy as np
from scipy import stats
from statsmodels.stats.stattools import jarque_bera
import matplotlib.pyplot as plt
def get_close(symbol):
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
return quotes.T[4]
spy =
dia =
np.diff(np.log(get_close("SPY")))
np.diff(np.log(get_close("DIA")))
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print("Means comparison", stats.ttest_ind(spy, dia))
print("Kolmogorov smirnov test", stats.ks_2samp(spy, dia))
print("Jarque Bera test", jarque_bera(spy - dia)[1])
plt.title('Log returns of SPY and DIA')
plt.hist(spy, histtype="step", lw=1, label="SPY")
plt.hist(dia, histtype="step", lw=2, label="DIA")
plt.hist(spy - dia, histtype="step", lw=3, label="Delta")
plt.xlabel('Log returns')
plt.ylabel('Counts')
plt.grid()
plt.legend(loc='best')
plt.show()
Signal processing
The scipy.signal module contains filter functions and B-spline interpolation algorithms.
Spline interpolation uses a polynomial called a spline for interpolation (see
http://en.wikipedia.org/wiki/Spline_interpolation).
The interpolation then tries to glue splines together to fit the data. B-spline
is a type of spline.
A SciPy signal is defined as an array of numbers. An example of a filter is the detrend()
function. This function takes a signal and does a linear fit on it. This trend is then subtracted
from the original input data.
Time for action – detecting a trend in QQQ
Often we are more interested in the trend of a data sample than in detrending it. We can still
get the trend back easily after detrending. Let's do that for one year of price data for QQQ.
1.
Write code that gets the close price and corresponding dates for QQQ:
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo("QQQ", start, today)
quotes = np.array(quotes)
dates = quotes.T[0]
qqq = quotes.T[4]
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2.
Detrend the signal:
y = signal.detrend(qqq)
3.
Create month and day locators for the dates:
alldays = DayLocator()
months = MonthLocator()
4.
Create a date formatter that creates a string of month name and year:
month_formatter = DateFormatter("%b %Y")
5.
Create a figure and subplot:
fig = plt.figure()
ax = fig.add_subplot(111)
6.
Plot the data and underlying trend by subtracting the detrended signal:
plt.plot(dates, qqq, 'o', dates, qqq - y, '-')
7.
Set the locators and formatter:
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_major_formatter(month_formatter)
8.
Format the x-axis labels as dates:
fig.autofmt_xdate()
plt.show()
The following figure shows the QQQ prices with a trend line:
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What just happened?
We plotted the closing price for QQQ with a trend line (see trend.py):
from matplotlib.finance import quotes_historical_yahoo
from datetime import date
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo("QQQ", start, today)
quotes = np.array(quotes)
dates = quotes.T[0]
qqq = quotes.T[4]
y = signal.detrend(qqq)
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
fig = plt.figure()
ax = fig.add_subplot(111)
plt.title('QQQ close price with trend')
plt.ylabel('Close price')
plt.plot(dates, qqq, 'o', dates, qqq - y, '-')
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_major_formatter(month_formatter)
fig.autofmt_xdate()
plt.grid()
plt.show()
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Fourier analysis
Signals in the real world often have a periodic nature. A commonly used tool to deal
with these signals is the Discrete Fourier transform (see https://en.wikipedia.
org/wiki/Discrete-time_Fourier_transform). The Discrete Fourier transform
is a transformation from the time domain into the frequency domain, that is, the linear
decomposition of a periodic signal into sine and cosine functions with various frequencies:
Functions for Fourier transforms can be found in the scipy.fftpack module (NumPy
also has its own Fourier package numpy.fft). Included in the package are Fast Fourier
transforms, differential and pseudo-differential operators, as well as several helper functions.
MATLAB users will be pleased to know that a number of functions in the scipy.fftpack
module have the same name as their MATLAB counterparts, and a similar function as their
MATLAB equivalents.
Time for action – filtering a detrended signal
We learned in the previous Time for action section how to detrend a signal. This detrended
signal could have a cyclical component. Let's try to visualize this. Some of the steps are a
repetition of steps in the previous Time for action section, such as downloading the data and
setting up matplotlib objects. These steps are omitted here.
1.
Apply the Fourier transform, giving us the frequency spectrum:
amps = np.abs(fftpack.fftshift(fftpack.rfft(y)))
2.
Filter out the noise. Let's say, if the magnitude of a frequency component is below
10 percent of the strongest component, throw it out:
amps[amps < 0.1 * amps.max()] = 0
3.
Transform the filtered signal back to the original domain and plot it together with
the detrended signal:
plt.plot(dates, y, 'o', label="detrended")
plt.plot(dates,
-fftpack.irfft(fftpack.ifftshift(amps)),
label="filtered")
4.
Format the x-axis labels as dates and add a legend with extra large size:
fig.autofmt_xdate()
plt.legend(prop={'size':'x-large'})
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5.
Add a second subplot and plot the frequency spectrum after filtering:
ax2 = fig.add_subplot(212)
N = len(qqq)
plt.plot(np.linspace(-N/2, N/2, N), amps,
label="transformed")
6.
Display the legend and plot:
plt.legend(prop={'size':'x-large'})
plt.show()
The following plots are of the signal and frequency spectrum:
What just happened?
We detrended a signal and applied a simple filter on it using the scipy.fftpack module
(see frequencies.py):
from matplotlib.finance import quotes_historical_yahoo
from datetime import date
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
from scipy import fftpack
from matplotlib.dates import DateFormatter
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from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
today = date.today()
start = (today.year - 1, today.month, today.day)
quotes = quotes_historical_yahoo("QQQ", start, today)
quotes = np.array(quotes)
dates = quotes.T[0]
qqq = quotes.T[4]
y = signal.detrend(qqq)
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
fig = plt.figure()
fig.subplots_adjust(hspace=.3)
ax = fig.add_subplot(211)
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_major_formatter(month_formatter)
# make font size bigger
ax.tick_params(axis='both', which='major', labelsize='x-large')
amps = np.abs(fftpack.fftshift(fftpack.rfft(y)))
amps[amps < 0.1 * amps.max()] = 0
plt.title('Detrended and filtered signal')
plt.plot(dates, y, 'o', label="detrended")
plt.plot(dates, -fftpack.irfft(fftpack.ifftshift(amps)),
label="filtered")
fig.autofmt_xdate()
plt.legend(prop={'size':'x-large'})
plt.grid()
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ax2 = fig.add_subplot(212)
plt.title('Transformed signal')
ax2.tick_params(axis='both', which='major', labelsize='x-large')
N = len(qqq)
plt.plot(np.linspace(-N/2, N/2, N), amps, label="transformed")
plt.legend(prop={'size':'x-large'})
plt.grid()
plt.tight_layout()
plt.show()
Mathematical optimization
Optimization algorithms try to find the optimal solution for a problem, for instance, finding
the maximum or the minimum of a function. The function can be linear or non-linear. The
solution could also have special constraints. For example, the solution may not be allowed
to have negative values. The scipy.optimize module provides several optimization
algorithms. One of the algorithms is a least squares fitting function, leastsq(). When
calling this function, we provide a residuals (error terms) function. This function minimizes
the sum of the squares of the residuals; it corresponds to our mathematical model for the
solution. It is also necessary to give the algorithm a starting point. This should be a best
guess—as close as possible to the real solution. Otherwise, execution will stop after about
100 * (N+1) iterations, where N is the number of parameters to optimize.
Time for action – fitting to a sine
In the previous Time for action section, we created a simple filter for detrended data. Now,
let's use a more restrictive filter that will leave us only with the main frequency component.
We will fit a sinusoidal pattern to it and plot our results. This model has four parameters—
amplitude, frequency, phase, and vertical offset.
1.
Define a residuals function based on a sine wave model:
def residuals(p, y, x):
A,k,theta,b = p
err = y-A * np.sin(2* np.pi* k * x + theta) + b
return err
2.
Transform the filtered signal back to the original domain:
filtered = -fftpack.irfft(fftpack.ifftshift(amps))
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3.
Guess the values of the parameters of which we are trying to estimate a
transformation from the time domain into the frequency domain:
N = len(qqq)
f = np.linspace(-N/2, N/2, N)
p0 = [filtered.max(), f[amps.argmax()]/(2*N), 0, 0]
print("P0", p0)
The initial values appear as follows:
P0 [2.6679532410065212, 0.00099598469163686377, 0, 0]
4.
Call the leastsq()function:
plsq = optimize.leastsq(residuals, p0, args=(filtered,
dates))
p = plsq[0]
print("P", p)
The final parameter values are as follows:
P [ 2.67678014e+00
-5.01260321e-03]
5.
2.73033206e-03
-8.00007036e+03
Finish the first subplot with detrended data, filtered data, and fit of the filtered data.
Use a date format for the horizontal axis and add a legend:
plt.plot(dates, y, 'o', label="detrended")
plt.plot(dates, filtered, label="filtered")
plt.plot(dates, p[0] * np.sin(2 * np.pi *
dates * p[1] + p[2]) + p[3], '^', label="fit")
fig.autofmt_xdate()
plt.legend(prop={'size':'x-large'})
6.
Add a second subplot with a legend of the main component of the frequency
spectrum:
ax2 = fig.add_subplot(212)
plt.plot(f, amps, label="transformed")
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The following are the resulting charts:
What just happened?
We detrended one year of price data for QQQ. This signal was then filtered until only the
main component of the frequency spectrum was left over. We fitted a sine to the filtered
signal using the scipy.optimize module (see optfit.py):
from __future__ import print_function
from matplotlib.finance import quotes_historical_yahoo
import numpy as np
import matplotlib.pyplot as plt
from scipy import fftpack
from scipy import signal
from matplotlib.dates import DateFormatter
from matplotlib.dates import DayLocator
from matplotlib.dates import MonthLocator
from scipy import optimize
start = (2010, 7, 25)
end = (2011, 7, 25)
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quotes = quotes_historical_yahoo("QQQ", start, end)
quotes = np.array(quotes)
dates = quotes.T[0]
qqq = quotes.T[4]
y = signal.detrend(qqq)
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
fig = plt.figure()
fig.subplots_adjust(hspace=.3)
ax = fig.add_subplot(211)
ax.xaxis.set_minor_locator(alldays)
ax.xaxis.set_major_locator(months)
ax.xaxis.set_major_formatter(month_formatter)
ax.tick_params(axis='both', which='major', labelsize='x-large')
amps = np.abs(fftpack.fftshift(fftpack.rfft(y)))
amps[amps < amps.max()] = 0
def residuals(p, y, x):
A,k,theta,b = p
err = y-A * np.sin(2* np.pi* k * x + theta) + b
return err
filtered = -fftpack.irfft(fftpack.ifftshift(amps))
N = len(qqq)
f = np.linspace(-N/2, N/2, N)
p0 = [filtered.max(), f[amps.argmax()]/(2*N), 0, 0]
print("P0", p0)
plsq = optimize.leastsq(residuals, p0, args=(filtered, dates))
p = plsq[0]
print("P", p)
plt.title('Detrended and filtered signal')
plt.plot(dates, y, 'o', label="detrended")
plt.plot(dates, filtered, label="filtered")
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plt.plot(dates, p[0] * np.sin(2 * np.pi * dates * p[1] + p[2]) + p[3],
'^', label="fit")
fig.autofmt_xdate()
plt.legend(prop={'size':'x-large'})
plt.grid()
ax2 = fig.add_subplot(212)
plt.title('Tranformed signal')
ax2.tick_params(axis='both', which='major', labelsize='x-large')
plt.plot(f, amps, label="transformed")
plt.legend(prop={'size':'x-large'})
plt.grid()
plt.tight_layout()
plt.show()
Numerical integration
SciPy has a numerical integration package, scipy.integrate, which has no equivalent
in NumPy. The quad() function can integrate a one-variable function between two points.
These points can be at infinity. The function uses the simplest numerical integration method:
the trapezoid rule.
Time for action – calculating the Gaussian integral
The Gaussian integral is related to the error() function (also known in mathematics as
erf), but has no finite limits. It evaluates to the square root of pi.
Let's calculate the integral with the quad() function (for the imports check the file in the
code bundle):
print("Gaussian integral", np.sqrt(np.pi),
integrate.quad(lambda x: np.exp(-x**2),
-np.inf, np.inf))
The return value is the outcome and its error would be as follows:
Gaussian integral 1.77245385091 (1.7724538509055159, 1.4202636780944923e08)
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What just happened?
We calculated the Gaussian integral with the quad() function.
Have a go hero – experiment a bit more
Try out other integration functions from the same package. It should just be a matter
of replacing one function call. We should get the same outcome, so you may also want
to read the documentation to learn more.
Interpolation
Interpolation fills in the blanks between known data points in a dataset. The scipy.
interpolate() function interpolates a function based on experimental data. The
interp1d class can create a linear or cubic interpolation function. By default, a linear
interpolation function is constructed, but if the kind parameter is set, a cubic interpolation
function is created instead. The interp2d class works the same way, but in 2D.
Time for action – interpolating in one dimension
We will create data points using a sinc() function and add some random noise to it. After
this, we will do a linear and cubic interpolation and plot the results.
1.
Create the data points and add noise to it:
x = np.linspace(-18, 18, 36)
noise = 0.1 * np.random.random(len(x))
signal = np.sinc(x) + noise
2.
Create a linear interpolation function and apply it to an input array with five times as
many data points:
interpreted = interpolate.interp1d(x, signal)
x2 = np.linspace(-18, 18, 180)
y = interpreted(x2)
3.
Do the same as in the previous step, but with cubic interpolation:
cubic = interpolate.interp1d(x, signal, kind="cubic")
y2 = cubic(x2)
4.
Plot the results with matplotlib:
plt.plot(x, signal, 'o', label="data")
plt.plot(x2, y, '-', label="linear")
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plt.plot(x2, y2, '-', lw=2, label="cubic")
plt.legend()
plt.show()
The following diagram is a plot of the data, linear, and cubic interpolation:
What just happened?
We created a dataset from the sinc() function and added noise to it. We then did linear
and cubic interpolation using the interp1d class of the scipy.interpolate module
(see sincinterp.py):
import numpy as np
from scipy import interpolate
import matplotlib.pyplot as plt
x = np.linspace(-18, 18, 36)
noise = 0.1 * np.random.random(len(x))
signal = np.sinc(x) + noise
interpreted = interpolate.interp1d(x, signal)
x2 = np.linspace(-18, 18, 180)
y = interpreted(x2)
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cubic = interpolate.interp1d(x, signal, kind="cubic")
y2 = cubic(x2)
plt.plot(x, signal, 'o', label="data")
plt.plot(x2, y, '-', label="linear")
plt.plot(x2, y2, '-', lw=2, label="cubic")
plt.title('Interpolated signal')
plt.xlabel('x')
plt.ylabel('y')
plt.grid()
plt.legend(loc='best')
plt.show()
Image processing
With SciPy, we can do image processing using the scipy.ndimage package. The module
contains various image filters and utilities.
Time for action – manipulating Lena
The scipy.misc module is a utility that loads the image of "Lena". This is the image of Lena
Soderberg, traditionally used for image processing examples. We will apply some filters to
this image and rotate it. Perform the following steps to do so:
1.
Load the Lena image and display it in a subplot with grayscale colormap:
image = misc.lena().astype(np.float32)
plt.subplot(221)
plt.title("Original Image")
img = plt.imshow(image, cmap=plt.cm.gray)
Note that we are dealing with a float32 array.
2.
The median filter scans the image and replaces each item by the median of
neighboring data points. Apply a median filter to the image and display it in
a second subplot:
plt.subplot(222)
plt.title("Median Filter")
filtered = ndimage.median_filter(image, size=(42,42))
plt.imshow(filtered, cmap=plt.cm.gray)
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3.
Rotate the image and display it in the third subplot:
plt.subplot(223)
plt.title("Rotated")
rotated = ndimage.rotate(image, 90)
plt.imshow(rotated, cmap=plt.cm.gray)
4.
The Prewitt filter is based on computing the gradient of image intensity. Apply a
Prewitt filter to the image and display it in the fourth subplot:
plt.subplot(224)
plt.title("Prewitt Filter")
filtered = ndimage.prewitt(image)
plt.imshow(filtered, cmap=plt.cm.gray)
plt.show()
The following are the resulting images:
What just happened?
We manipulated the image of Lena in several ways using the scipy.ndimage module (see
images.py):
from scipy import misc
import numpy as np
import matplotlib.pyplot as plt
from scipy import ndimage
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image = misc.lena().astype(np.float32)
plt.subplot(221)
plt.title("Original Image")
img = plt.imshow(image, cmap=plt.cm.gray)
plt.axis("off")
plt.subplot(222)
plt.title("Median Filter")
filtered = ndimage.median_filter(image, size=(42,42))
plt.imshow(filtered, cmap=plt.cm.gray)
plt.axis("off")
plt.subplot(223)
plt.title("Rotated")
rotated = ndimage.rotate(image, 90)
plt.imshow(rotated, cmap=plt.cm.gray)
plt.axis("off")
plt.subplot(224)
plt.title("Prewitt Filter")
filtered = ndimage.prewitt(image)
plt.imshow(filtered, cmap=plt.cm.gray)
plt.axis("off")
plt.show()
Audio processing
Now that we have done some image processing, you will probably not be surprised that we
can do exciting things with WAV files too. Let's download a WAV file and replay it a couple of
times. We will skip the explanation of the download part, which is just regular Python.
Time for action – replaying audio clips
We will download a WAV file of Austin Powers exclaiming "Smashing baby". This file can
be converted to a NumPy array with the read() function from the scipy.io.wavfile
module. The write() function from the same package will be used to create a new WAV file
at the end of this section. We will further use the tile() function to replay the audio clip
several times.
1.
Read the file with the read() function:
sample_rate, data = wavfile.read(WAV_FILE)
This gives us two items – sample rate and audio data. For this section we are only
interested in the audio data.
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2.
Apply the tile() function:
repeated = np.tile(data, 4)
3.
Write a new file with the write() function:
wavfile.write("repeated_yababy.wav",
sample_rate, repeated)
The original audio data and the audio clip repeated four times appear in the
following plot:
What just happened?
We read an audio clip, repeated it four times, and then created a new WAV file with the new
array (see repeat_audio.py):
from __future__ import print_function
from scipy.io import wavfile
import matplotlib.pyplot as plt
import urllib.request
import numpy as np
response = urllib.request.urlopen('http://www.thesoundarchive.com/
austinpowers/smashingbaby.wav')
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print(response.info())
WAV_FILE = 'smashingbaby.wav'
filehandle = open(WAV_FILE, 'wb')
filehandle.write(response.read())
filehandle.close()
sample_rate, data = wavfile.read(WAV_FILE)
print("Data type", data.dtype, "Shape", data.shape)
plt.subplot(2, 1, 1)
plt.title("Original audio signal")
plt.plot(data)
plt.grid()
plt.subplot(2, 1, 2)
# Repeat the audio fragment
repeated = np.tile(data, 4)
# Plot the audio data
plt.title("Repeated 4 times")
plt.plot(repeated)
wavfile.write("repeated_yababy.wav",
sample_rate, repeated)
plt.grid()
plt.tight_layout()
plt.show()
Summary
In this chapter, we only scratched the surface of what is possible with SciPy and SciKits.
Still, we learned a bit about file I/O, statistics, signal processing, optimization, interpolation,
audio, and image processing.
In the next chapter, we will create some simple, yet fun, games with Pygame—the open
source Python game library. In the process, we will learn about NumPy integration with
Pygame, a machine learning Scikits module, and more.
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11
Playing with Pygame
This chapter is for developers who want to create games quickly and easily
with NumPy and Pygame. Basic game development experience would help,
but it isn't necessary.
The things you will learn are as follows:
Pygame basics
matplotlib integration
Surface pixel arrays
Artificial intelligence
Animation
OpenGL
Pygame
Pygame is a Python framework, originally written by Pete Shinners, which, as its name
suggests, can be used to create video games. Pygame is free, open source since 2004 and
licensed under the GPL license, which means that you are allowed to basically make any type
of game. Pygame is built on top of the Simple DirectMedia Layer (SDL). SDL is a C framework
that gives access to graphics, sound, keyboard, and other input devices on various operating
systems including Linux, Mac OS X, and Windows.
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Time for action – installing Pygame
We will install Pygame in this section. Pygame should be compatible with all Python versions.
At the time of writing, there were some incompatibility issues with Python 3, but in all
probability, these will be fixed soon.
Installing on Debian and Ubuntu: Pygame can be found in the Debian archives at
https://packages.qa.debian.org/p/pygame.html.
Installing on Windows: From the Pygame website (http://www.pygame.org/
download.shtml), download the appropriate binary installer for the Python
version you are using.
Installing Pygame on the Mac: Binary Pygame packages for Mac OS X 10.3 and up
can be found at http://www.pygame.org/download.shtml.
Installing from source: Pygame is using the distutils system for compiling
and installing. To start installing Pygame with the default options, simply run the
following command:
$ python setup.py
If you need more information about the available options, type the following:
$ python setup.py help
To compile the code, you need a compiler for your operating system. Setting this
up is beyond the scope of this book. More information about compiling Pygame on
Windows can be found at http://pygame.org/wiki/CompileWindows. For
more information about compiling Pygame on Mac OS X, refer to http://pygame.
org/wiki/MacCompile.
Hello World
We will create a simple game that we will improve on further in this chapter. As is traditional
in programming books, we start with a Hello World! example.
Time for action – creating a simple game
It's important to notice the so-called main game loop, where all the action happens, and
the usage of the Font module to render text. In this program, we will manipulate a Pygame
Surface object that is used for drawing, and we will handle a quit event.
1.
First, import the required Pygame modules. If Pygame is installed properly, we
should get no errors, otherwise please return to the installation Time for action:
import pygame, sys
from pygame.locals import *
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2.
Initialize Pygame, create a display of 400 by 300 pixels, and set the window title
to Hello world!:
pygame.init()
screen = pygame.display.set_mode((400, 300))
pygame.display.set_caption('Hello World!')
3.
Games usually have a game loop, which runs forever until, for instance, a quit event
occurs. In this example, only set a label with the text Hello world! at coordinates
(100, 100). The text has font size 19 and a red color:
while True:
sysFont = pygame.font.SysFont("None", 19)
rendered = sysFont.render('Hello World', 0, (255, 100, 100))
screen.blit(rendered, (100, 100))
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
pygame.display.update()
We get the following screenshot as an end result:
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Following is the complete code for the Hello World! example:
import pygame, sys
from pygame.locals import *
pygame.init()
screen = pygame.display.set_mode((400, 300))
pygame.display.set_caption('Hello World!')
while True:
sysFont = pygame.font.SysFont("None", 19)
rendered = sysFont.render('Hello World', 0, (255, 100, 100))
screen.blit(rendered, (100, 100))
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
pygame.display.update()
What just happened?
It may not seem like much, but we learned a lot in this section. The functions that passed the
review are summarized in the following table:
Function
pygame.init()
Description
pygame.display.set_
mode((400, 300))
This function creates a so-called Surface object to
draw on. We give this function a tuple representing
the dimensions of the surface.
pygame.display.set_
caption('Hello World!')
This function sets the window title to a specified
string value.
pygame.font.SysFont("None",
19)
This function creates a system font from a commaseparated list of fonts (in this case none) and an
integer font size parameter.
sysFont.render('Hello
World', 0, (255, 100, 100))
This function draws text on a Surface. The last
parameter is a tuple representing the RGB values of
a color.
screen.blit(rendered, (100,
100))
This function draws on a Surface.
This function performs initialization and you must call
it before calling other Pygame functions.
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Function
pygame.event.get()
Description
pygame.quit()
This function cleans up the resources used by
Pygame. Call this function before exiting the game.
pygame.display.update()
This function refreshes the surface.
This function gets a list of Event objects. Events
represent a special occurrence in the system, such as
a user quitting the game.
Animation
Most games, even the most static ones, have some level of animation. From a programmer's
standpoint, animation is nothing more than displaying an object at a different place at a
different time, thus simulating movement.
Pygame offers a Clock object, which manages how many frames are drawn per second. This
ensures that the animation is independent of how fast the user's CPU is.
Time for action – animating objects with NumPy and Pygame
We will load an image and use NumPy again to define a clockwise path around the screen.
1.
Create a Pygame clock as follows:
clock = pygame.time.Clock()
2.
As part of the source code accompanying this book, there should be a picture of a
head. Load this image and move it around on the screen:
img = pygame.image.load('head.jpg')
3.
Define some arrays to hold the coordinates of the positions, where we would like to
put the image during the animation. Since we will move the object, there are four
logical sections of the path: right, down, left, and up. Each of these sections will
have 40 equidistant steps. Initialize all the values in the sections to 0:
steps = np.linspace(20, 360, 40).astype(int)
right = np.zeros((2, len(steps)))
down = np.zeros((2, len(steps)))
left = np.zeros((2, len(steps)))
up = np.zeros((2, len(steps)))
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Playing with Pygame
4.
It's straight-forward to set the coordinates of the positions of the image. However,
there is one tricky bit to notice—the [::-1] notation leads to reversing the order
of the array elements:
right[0] = steps
right[1] = 20
down[0] = 360
down[1] = steps
left[0] = steps[::-1]
left[1] = 360
up[0] = 20
up[1] = steps[::-1]
5.
We can join the path sections, but before doing this, transpose the arrays with the T
operator because they are not aligned properly for concatenation:
pos = np.concatenate((right.T, down.T, left.T, up.T))
6.
In the main event loop, let the clock tick at a rate of 30 frames per second:
clock.tick(30)
A screenshot of the moving head is as follows:
You should be able to watch a movie of this animation at https://www.youtube.
com/watch?v=m2TagGiq1fs, and it is also part of the code bundle
(animation.mp4).
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The code of this example uses almost everything we have learnt so far, but should
still be simple enough to understand:
import pygame, sys
from pygame.locals import *
import numpy as np
pygame.init()
clock = pygame.time.Clock()
screen = pygame.display.set_mode((400, 400))
pygame.display.set_caption('Animating Objects')
img = pygame.image.load('head.jpg')
steps = np.linspace(20, 360, 40).astype(int)
right = np.zeros((2, len(steps)))
down = np.zeros((2, len(steps)))
left = np.zeros((2, len(steps)))
up = np.zeros((2, len(steps)))
right[0] = steps
right[1] = 20
down[0] = 360
down[1] = steps
left[0] = steps[::-1]
left[1] = 360
up[0] = 20
up[1] = steps[::-1]
pos = np.concatenate((right.T, down.T, left.T, up.T))
i = 0
while True:
# Erase screen
screen.fill((255, 255, 255))
if i >= len(pos):
i = 0
screen.blit(img, pos[i])
i += 1
for event in pygame.event.get():
if event.type == QUIT:
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pygame.quit()
sys.exit()
pygame.display.update()
clock.tick(30)
What just happened?
We learned a bit about animation in this section. The most important concept we learned
about is the clock. The following table describes the new functions we used:
Function
pygame.time.Clock()
Description
clock.tick(30)
This function executes a tick of the game clock. Here, 30 is
the number of frames per second.
This creates a game clock.
matplotlib
matplotlib is an open source library for easy plotting, which we learned about in Chapter
9, Plotting with matplotlib. We can integrate matplotlib into a Pygame game and create
various plots.
Time for Action – using matplotlib in Pygame
In this recipe, we take the position coordinates of the previous section and make a graph
of them.
1.
To integrate matplotlib with Pygame, we need to use a non-interactive backend;
otherwise matplotlib will present us with a GUI window by default. We will
import the main matplotlib module and call the use() function. Call this function
immediately after importing the main matplotlib module and before importing
other matplotlib modules:
import matplotlib as mpl
mpl.use("Agg")
2.
We can draw non-interactive plots on a matplotlib canvas. Creating this canvas
requires imports, creating a figure and a subplot. Specify the figure to be 3 by 3
inches large. More details can be found at the end of this recipe:
import matplotlib.pyplot as plt
import matplotlib.backends.backend_agg as agg
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fig = plt.figure(figsize=[3, 3])
ax = fig.add_subplot(111)
canvas = agg.FigureCanvasAgg(fig)
3.
In non-interactive mode, plotting data is a bit more complicated than in the default
mode. Since we need to plot repeatedly, it makes sense to organize the plotting
code in a function. Pygame eventually draws the plot on the canvas. The canvas
adds a bit of complexity to our setup. At the end of this example, you can find
more detailed explanation of the functions:
def plot(data):
ax.plot(data)
canvas.draw()
renderer = canvas.get_renderer()
raw_data = renderer.tostring_rgb()
size = canvas.get_width_height()
return pygame.image.fromstring(raw_data, size, "RGB")
The following screenshot shows the animation in action. You can also view
a screencast in the code bundle (matplotlib.mp4) and on YouTube at:
https://www.youtube.com/watch?v=t6qTeXxtnl4.
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Playing with Pygame
We get the following code after the changes:
import pygame, sys
from pygame.locals import *
import numpy as np
import matplotlib as mpl
mpl.use("Agg")
import matplotlib.pyplot as plt
import matplotlib.backends.backend_agg as agg
fig = plt.figure(figsize=[3, 3])
ax = fig.add_subplot(111)
canvas = agg.FigureCanvasAgg(fig)
def plot(data):
ax.plot(data)
canvas.draw()
renderer = canvas.get_renderer()
raw_data = renderer.tostring_rgb()
size = canvas.get_width_height()
return pygame.image.fromstring(raw_data, size, "RGB")
pygame.init()
clock = pygame.time.Clock()
screen = pygame.display.set_mode((400, 400))
pygame.display.set_caption('Animating Objects')
img = pygame.image.load('head.jpg')
steps = np.linspace(20, 360, 40).astype(int)
right = np.zeros((2, len(steps)))
down = np.zeros((2, len(steps)))
left = np.zeros((2, len(steps)))
up = np.zeros((2, len(steps)))
right[0] = steps
right[1] = 20
down[0] = 360
down[1] = steps
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left[0] = steps[::-1]
left[1] = 360
up[0] = 20
up[1] = steps[::-1]
pos = np.concatenate((right.T, down.T, left.T, up.T))
i = 0
history = np.array([])
surf = plot(history)
while True:
# Erase screen
screen.fill((255, 255, 255))
if i >= len(pos):
i = 0
surf = plot(history)
screen.blit(img, pos[i])
history = np.append(history, pos[i])
screen.blit(surf, (100, 100))
i += 1
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
pygame.display.update()
clock.tick(30)
What just happened?
The following table explains the plotting related functions:
Function
mpl.use("Agg")
Description
plt.figure(figsize=[3, 3])
This function creates a figure of 3 by 3 inches
agg.FigureCanvasAgg(fig)
This function creates a canvas in non-interactive mode
This function specifies to use the non-interactive backend
canvas.draw()
This function draws on the canvas
canvas.get_renderer()
This function gets a renderer for the canvas
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Surface pixels
The Pygame surfarray module handles the conversion between Pygame Surface
objects and NumPy arrays. As you may recall, NumPy can manipulate big arrays in a
fast and efficient manner.
Time for Action – accessing surface pixel data with NumPy
In this section, we will tile a small image to fill the game screen.
1.
The array2d() function copies pixels into a two-dimensional array (and there is
a similar function for three-dimensional arrays). Copy the pixels from the avatar
image into an array:
pixels = pygame.surfarray.array2d(img)
2.
Create the game screen from the shape of the pixels array using the shape attribute
of the array. Make the screen seven times larger in both directions:
X = pixels.shape[0] * 7
Y = pixels.shape[1] * 7
screen = pygame.display.set_mode((X, Y))
3.
Tiling the image is easy with the NumPy the tile() function. The data needs to be
converted into integer values, because Pygame defines colors as integers:
new_pixels = np.tile(pixels, (7, 7)).astype(int)
4.
The surfarray module has a special function blit_array() to display the array
on the screen:
pygame.surfarray.blit_array(screen, new_pixels)
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The following code performs the tiling of the image:
import pygame, sys
from pygame.locals import *
import numpy as np
pygame.init()
img = pygame.image.load('head.jpg')
pixels = pygame.surfarray.array2d(img)
X = pixels.shape[0] * 7
Y = pixels.shape[1] * 7
screen = pygame.display.set_mode((X, Y))
pygame.display.set_caption('Surfarray Demo')
new_pixels = np.tile(pixels, (7, 7)).astype(int)
while True:
screen.fill((255, 255, 255))
pygame.surfarray.blit_array(screen, new_pixels)
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
pygame.display.update()
What just happened?
Following is a brief description of the new functions and attributes we used:
Function
pygame.surfarray.array2d(img)
pygame.surfarray.blit_
array(screen, new_pixels)
Description
This function copies pixel data into a twodimensional array
This function displays array values on the screen
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Artificial Intelligence
Often we need to mimic intelligent behavior within a game. The scikit-learn project
aims to provide an API for machine learning, and what I like most about it is its amazing
documentation. We can install scikit-learn with the package manager of our operating
system, though this option may or may not be available, depending on your operating system,
but should be the most convenient route. Windows users can just download an installer
from the project website. On Debian and Ubuntu, the project is called python-sklearn.
On MacPorts, the ports are called py26-scikits-learn and py27-scikits-learn. We
can also install from source or using easy_install. There are third-party distributions from
Python(x,y), Enthought, and NetBSD.
We can install scikit-learn by typing at command line:
$ [sudo] pip install -U scikit-learn
We can also type the following instead of the preceding line:
$ [sudo] easy_install -U scikit-learn
This may not work because of permissions, so you might need to put sudo in front of the
commands or log in as admin.
Time for Action – clustering points
We will generate some random points and cluster them, which means that the points that
are close to each other are put into the same cluster. This is just one of the many techniques
that you can apply with scikit-learn. Clustering is a type of machine learning algorithm,
which aims to group items based on similarities. Next, we will calculate a square affinity
matrix. An affinity matrix is a matrix containing affinity values: for instance, the distances
between points. Finally, we will cluster the points with the AffinityPropagation class
from scikit-learn.
1.
Generate 30 random point positions within a square of 400 by 400 pixels:
positions = np.random.randint(0, 400, size=(30, 2))
2.
Calculate the affinity matrix using the Euclidean distance to the origin as the
affinity metric:
positions_norms = np.sum(positions ** 2, axis=1)
S = - positions_norms[:, np.newaxis] positions_norms[np.newaxis, :] + 2 *
np.dot(positions, positions.T)
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3.
Give the AffinityPropagation class the result from the previous step. This class
labels the points with the appropriate cluster number:
aff_pro = sklearn.cluster.AffinityPropagation().fit(S)
labels = aff_pro.labels_
4.
Draw polygons for each cluster. The function involved requires a list of points, a
color (let's paint it red), and a surface:
pygame.draw.polygon(screen, (255, 0, 0), polygon_points[i])
The result is a bunch of polygons for each cluster, as shown in the following picture:
The clustering example code is as follows:
import numpy as np
import sklearn.cluster
import pygame, sys
from pygame.locals import *
np.random.seed(42)
positions = np.random.randint(0, 400, size=(30, 2))
positions_norms = np.sum(positions ** 2, axis=1)
S = - positions_norms[:, np.newaxis] - positions_norms[np.newaxis,
:] + 2 * np.dot(positions,
positions.T)
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aff_pro = sklearn.cluster.AffinityPropagation().fit(S)
labels = aff_pro.labels_
polygon_points = []
for i in xrange(max(labels) + 1):
polygon_points.append([])
# Sorting points by cluster
for label, position in zip(labels, positions):
polygon_points[labels[i]].append(positions[i])
pygame.init()
screen = pygame.display.set_mode((400, 400))
while True:
for point in polygon_points:
pygame.draw.polygon(screen, (255, 0, 0), point)
for event in pygame.event.get():
if event.type == QUIT:
pygame.quit()
sys.exit()
pygame.display.update()
What just happened?
The most important lines in the artificial intelligence example are described in more detail in
the following table:
Function
sklearn.cluster.
AffinityPropagation().fit(S)
Description
pygame.draw.polygon(screen, (255,
0, 0), point)
This function draws a polygon given a
surface, a color (red in this case), and a list
of points
This function creates an
AffinityPropagation object and
performs a fit using an affinity matrix
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OpenGL and Pygame
OpenGL specifies an API for two-dimensional and three-dimensional computer graphics.
The API consists of functions and constants. We will be concentrating on the Python
implementation called PyOpenGL. Install PyOpenGL with the following command:
$ [sudo] pip install PyOpenGL PyOpenGL_accelerate
You might need to have root access to execute this command. The corresponding
easy_install command is as follows:
$ [sudo] easy_install PyOpenGL PyOpenGL_accelerate
Time for Action – drawing the Sierpinski gasket
For the purpose of demonstration, we will draw a Sierpinski gasket, also known as Sierpinski
triangle or Sierpinski Sieve with OpenGL. This is a fractal pattern in the shape of a triangle
created by the mathematician Waclaw Sierpinski. The triangle is obtained via a recursive
and, in principle infinite procedure.
1.
First, start out by initializing some of the OpenGL related primitives. This includes
setting the display mode and background color. A line-by-line explanation is given
at the end of this section:
def display_openGL(w, h):
pygame.display.set_mode((w,h),
pygame.OPENGL|pygame.DOUBLEBUF)
glClearColor(0.0, 0.0, 0.0, 1.0)
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
gluOrtho2D(0, w, 0, h)
2.
The algorithm requires us to display points, the more the better. First, we set the
drawing color to red. Second, we define the vertices (I call them points myself) of
a triangle. Then, we define random indices, which are to be used to choose one of
the three triangle vertices. We pick a random point somewhere in the middle—it
doesn't really matter where. After this, draw points halfway between the previous
point and one of the vertices picked at random. Finally, flush the result:
glColor3f(1.0, 0, 0)
vertices = np.array([[0, 0], [DIM/2, DIM], [DIM, 0]])
NPOINTS = 9000
indices = np.random.random_integers(0, 2, NPOINTS)
point = [175.0, 150.0]
[ 287 ]
Playing with Pygame
for i in xrange(NPOINTS):
glBegin(GL_POINTS)
point = (point + vertices[indices[i]])/2.0
glVertex2fv(point)
glEnd()
glFlush()
The Sierpinski triangle looks like the following:
The full Sierpinski gasket demo code with all the imports is as follows:
import pygame
from pygame.locals import *
import numpy as np
from OpenGL.GL import *
from OpenGL.GLU import *
def display_openGL(w, h):
pygame.display.set_mode((w,h), pygame.OPENGL|pygame.DOUBLEBUF)
glClearColor(0.0, 0.0, 0.0, 1.0)
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
[ 288 ]
Chapter 11
gluOrtho2D(0, w, 0, h)
def main():
pygame.init()
pygame.display.set_caption('OpenGL Demo')
DIM = 400
display_openGL(DIM, DIM)
glColor3f(1.0, 0, 0)
vertices = np.array([[0, 0], [DIM/2, DIM], [DIM, 0]])
NPOINTS = 9000
indices = np.random.random_integers(0, 2, NPOINTS)
point = [175.0, 150.0]
for i in xrange(NPOINTS):
glBegin(GL_POINTS)
point = (point + vertices[indices[i]])/2.0
glVertex2fv(point)
glEnd()
glFlush()
pygame.display.flip()
while True:
for event in pygame.event.get():
if event.type == QUIT:
return
if __name__ == '__main__':
main()
What just happened?
As promised, the following is a line-by-line explanation of the most important parts of
the example:
Function
pygame.display.set_mode((w,h),
pygame.OPENGL|pygame.DOUBLEBUF)
Description
glClear(GL_COLOR_BUFFER_BIT|GL_
DEPTH_BUFFER_BIT)
This function clears the buffers using a mask. Here
we clear the color buffer and depth buffer bits
gluOrtho2D(0, w, 0, h)
This function defines a two-dimensional
orthographic projection matrix with the coordinates
of the left, right, top, and bottom clipping planes
This function sets the display mode to the required
width, height, and OpenGL display
[ 289 ]
Playing with Pygame
Function
glColor3f(1.0, 0, 0)
Description
glBegin(GL_POINTS)
This function delimits the vertices of primitives or a
group of primitives. Here the primitives are points
glVertex2fv(point)
This function renders a point given a vertex
glEnd()
This function closes a section of code started with
glBegin()
glFlush()
This function forces the execution of GL commands
This function defines the current drawing color
using three float values for RGB (red, green, blue).
In this case, we will be painting in red
Simulation game with Pygame
As a last example, we will simulate life with Conway's Game of Life. The original game
of life is based on a few basic rules. We start out with a random configuration on a twodimensional square grid. Each cell in the grid can be either dead or alive. This state depends
on the neighbors of the cell. You can read more about the rules at http://en.wikipedia.
org/wiki/Conway%27s_Game_of_Life#Rules At each step in time, the following
transitions occur:
1. Live cells with less than two live neighbors die.
2. Live cells with two or three live neighbors live on to the next generation.
3. Live cells with more than three live neighbors die.
4. Dead cells with exactly three live neighbors become a live cell.
Convolution can be used to evaluate the basic rules of the game. We need the SciPy package
for the convolution process.
Time for Action – simulating life
The following code is an implementation of the Game of Life, with some modifications:
Clicking once with the mouse draws a cross until we click again
The r key resets the grid to a random state
Pressing b creates blocks based on the mouse position
g creates gliders
[ 290 ]
Chapter 11
The most important data structure in the code is a two-dimensional array, holding the color
values of the pixels on the game screen. This array is initialized with random values and then
recalculated for each iteration of the game loop. Find more information about the involved
functions in the next section.
1.
To evaluate the rules, use the convolution as follows:
def get_pixar(arr, weights):
states = ndimage.convolve(arr, weights, mode='wrap')
bools = (states == 13) | (states == 12 ) | (states == 3)
return bools.astype(int)
2.
Draw a cross using the basic indexing tricks that we learned in Chapter 2, Beginning
with NumPy Fundamentals:
def draw_cross(pixar):
(posx, posy) = pygame.mouse.get_pos()
pixar[posx, :] = 1
pixar[:, posy] = 1
3.
Initialize the grid with random values:
def random_init(n):
return np.random.random_integers(0, 1, (n, n))
The following is the code in its entirety:
from __future__ import print_function
import os, pygame
from pygame.locals import *
import numpy as np
from scipy import ndimage
def get_pixar(arr, weights):
states = ndimage.convolve(arr, weights, mode='wrap')
bools = (states == 13) | (states == 12 ) | (states == 3)
return bools.astype(int)
def draw_cross(pixar):
(posx, posy) = pygame.mouse.get_pos()
pixar[posx, :] = 1
pixar[:, posy] = 1
[ 291 ]
Playing with Pygame
def random_init(n):
return np.random.random_integers(0, 1, (n, n))
def draw_pattern(pixar, pattern):
print(pattern)
if pattern == 'glider':
coords = [(0,1), (1,2), (2,0), (2,1), (2,2)]
elif pattern == 'block':
coords = [(3,3), (3,2), (2,3), (2,2)]
elif pattern == 'exploder':
coords = [(0,1), (1,2), (2,0), (2,1), (2,2), (3,3)]
elif pattern == 'fpentomino':
coords = [(2,3),(3,2),(4,2),(3,3),(3,4)]
pos = pygame.mouse.get_pos()
xs = np.arange(0, pos[0], 10)
ys = np.arange(0, pos[1], 10)
for x in xs:
for y in ys:
for i, j in coords:
pixar[x + i, y + j] = 1
def main():
pygame.init ()
N = 400
pygame.display.set_mode((N, N))
pygame.display.set_caption("Life Demo")
screen = pygame.display.get_surface()
pixar = random_init(N)
weights = np.array([[1,1,1], [1,10,1], [1,1,1]])
cross_on = False
while True:
pixar = get_pixar(pixar, weights)
if cross_on:
draw_cross(pixar)
pygame.surfarray.blit_array(screen, pixar * 255 ** 3)
pygame.display.flip()
[ 292 ]
Chapter 11
for event in pygame.event.get():
if event.type == QUIT:
return
if event.type == MOUSEBUTTONDOWN:
cross_on = not cross_on
if event.type == KEYDOWN:
if event.key == ord('r'):
pixar = random_init(N)
print("Random init")
if event.key == ord('g'):
draw_pattern(pixar, 'glider')
if event.key == ord('b'):
draw_pattern(pixar, 'block')
if event.key == ord('e'):
draw_pattern(pixar, 'exploder')
if event.key == ord('f'):
draw_pattern(pixar, 'fpentomino')
if __name__ == '__main__':
main()
You should be able to view a screencast from the code bundle (life.mp4) or on
YouTube (https://www.youtube.com/watch?v=NNsU-yWTkXM). A screenshot
of the game in action is as follows:
[ 293 ]
Playing with Pygame
What just happened?
We used some NumPy and SciPy functions that need explaining:
Function
ndimage.convolve(arr,
weights, mode='wrap')
Description
bools.astype(int)
This function converts the array of Booleans to integers.
np.arange(0, pos[0], 10)
This function creates an array from 0 to pos[0] in
steps of 10. So, if pos[0] is equal to 1000, we will get
0, 10, 20, … 990.
This function applies the convolve operation on the
given array, using weights in the wrap mode. The mode
has to do with the array borders.
Summary
You might find the mention of Pygame in this book a bit odd. However, after reading this
chapter, I hope you realized that NumPy and Pygame go well together. Games after all involve
lots of computation for which NumPy and SciPy are ideal choices, and they also require
artificial intelligence capabilities as found in scikit-learn. In any event, making games is
fun and we hope this last chapter was the equivalent of a nice dessert or coffee after a tencourse meal! If you are still hungry for more, please check out NumPy Cookbook, Second
Edition, Ivan Idris, Packt Publishing, which builds further on this book with minimum overlap.
[ 294 ]
A
Pop Quiz Answers
Chapter 1, NumPy Quick Start
Pop quiz – functioning of the arange() function
What does arange(5) do?
It creates a NumPy array with values 0-4
The created NumPy array has values 0, 1, 2, 3, and 4
Chapter 2, Beginning with NumPy Fundamentals
Pop quiz – the shape of ndarray
How is the shape of an ndarray
stored?
It is stored in a tuple
[ 295 ]
Pop Quiz Answers
Chapter 3, Getting Familiar with Commonly
Used Functions
Pop quiz – computing the weighted average
Which function returns the
weighted average of an array?
average
Chapter 4, Convenience Functions for Your Convenience
Pop quiz – calculating covariance
Which function returns the
covariance of two arrays?
cov
Chapter 5, Working with Matrices and ufuncs
Pop quiz – defining a matrix with a string
What is the row delimiter in a string
accepted by the mat and bmat
functions?
Semicolon
Chapter 6, Move Further with NumPy Modules
Pop quiz – creating a matrix
Which function can create matrices?
mat
[ 296 ]
Appendix A
Chapter 7, Peeking into Special Routines
Pop quiz – generating random numbers
Which NumPy module deals with
random numbers?
random
Chapter 8, Assuring Quality with Testing
Pop quiz – specifying decimal precision
Which parameter of the assert_
almost_equal function specifies
the decimal precision?
decimal
Chapter 9, Plotting with matplotlib
Pop quiz – the plot() function
What does the plot function do?
It does neither 1, 2, or 3
Chapter 10, When NumPy Is Not Enough –Scipy
and Beyond
Pop quiz – loading .mat files
Which function loads .mat files?
loadmat
[ 297 ]
B
Additional Online Resources
This appendix contains links to the relevant websites.
Python
Learn Python the Hard Way (for Python 2) at http://learnpythonthehardway.
org/
Dive Into Python 3 (for Python 3) at http://www.diveintopython3.net/
Beginner's Guide to Python at https://wiki.python.org/moin/
BeginnersGuide
Non-programmers Tutorial for Python 3 can be found at http://en.wikibooks.
org/wiki/Non-Programmer%27s_Tutorial_for_Python_3
A Byte of Python is available at http://www.swaroopch.com/notes/python/
An Introduction to Interactive Programming in Python can be found at
https://www.coursera.org/course/interactivepython1
Learn Python online by Code Mentor at https://www.codementor.io/learnpython-online
Learn Python by visualizing code execution at http://pythontutor.com/
Find Codecademy Python exercises at http://www.codecademy.com/tracks/
python
Google's Python class is available at https://developers.google.com/edu/
python/
A Python style guide from Google can be found at https://googlestyleguide.googlecode.com/svn/trunk/pyguide.html
The IPython website can be found at http://ipython.org/
matplotlib a Python plotting library at http://matplotlib.org/
[ 299 ]
Additional Online Resources
NumPy and SciPy documentation can be accessed at http://docs.scipy.org/
doc/
NumPy and SciPy mailing lists can be found at http://www.scipy.org/
scipylib/mailing-lists.html
Mathematics and statistics
Linear algebra tutorials are available from Khan Academy at https://www.
khanacademy.org/math/linear-algebra
Pre-calculus tutorials from Khan Academy are available at https://www.
khanacademy.org/math/precalculus
Probability and statistics tutorials from Khan Academy can be found at
https://www.khanacademy.org/math/probability
Trigonometry tutorials from Khan Academy can be found at
https://www.khanacademy.org/math/trigonometry
Access Alcumus by Art of Problem Solving(AoPS) at http://www.
artofproblemsolving.com/alcumus
Find the Pre-Calculus Coursera course at https://www.coursera.org/course/
precalculus
The Coursera course on linear algebra, which uses Python, can be found at
https://www.coursera.org/course/matrix
An introduction to probability by Harvard University can be accessed at
https://itunes.apple.com/us/course/statistics-110-probability/
id502492375
The statistics wikibook is available at https://en.wikibooks.org/wiki/
Statistics
The Electronic Statistics Textbook. Tulsa, OK: StatSoft. WEB can be found at:
http://www.statsoft.com/Textbook
[ 300 ]
C
NumPy Functions' References
This appendix contains a list of useful NumPy functions and their descriptions.
numpy.apply_along_axis(func1d, axis, arr, *args): Applies the function
func1d along an axis on 1D slices of arr.
numpy.arange([start,] stop[, step,], dtype=None): Creates a NumPy
array with evenly spaced values within a specified range.
numpy.argsort(a, axis=-1, kind='quicksort', order=None): Returns
the indices that would sort the input array.
numpy.argmax(a, axis=None): Returns the indices of the maximum values
along an axis.
numpy.argmin(a, axis=None): Returns the indices of the minimum values
along an axis.
numpy.argwhere(a): Finds the indices of non-zero elements.
numpy.array(object, dtype=None, copy=True, order=None,
subok=False, ndmin=0): Creates a NumPy array from an array-like sequence,
such as a Python list.
numpy.testing.assert_allclose((actual, desired, rtol=1e-07,
atol=0, err_msg='', verbose=True): Raises an error if two objects are
unequal up to a specified precision.
numpy.testing.assert_almost_equal(): Raises an exception if two numbers
are not equal up to a specified precision.
numpy.testing.assert_approx_equal(): Raises an exception if two numbers
are not equal up to a certain significance.
numpy.testing.assert_array_almost_equal(): Raises an exception if two
arrays are not equal up to a specified precision.
[ 301 ]
NumPy Functions' References
numpy.testing.assert_array_almost_equal_nulp(x, y, nulp=1):
Compares arrays to their unit of least precision (ULP).
numpy.testing.assert_array_equal(): Raises an exception if two arrays are
not equal.
numpy.testing.assert_array_less(): Raises an exception if two arrays do
not have the same shape, and the elements of the first array are strictly less than
the elements of the second array.
numpy.testing.assert_array_max_ulp(a, b, maxulp=1, dtype=None):
Determines whether the array elements differ by, at most, a specified number
of ULP.
numpy.testing.assert_equal(): Tests whether two NumPy arrays are equal.
numpy.testing.assert_raises(): Fails if a specified exception is not raised by
a callable invoked with defined arguments.
numpy.testing.assert_string_equal(): Asserts that two strings are equal.
numpy.testing.assert_warns(): Fails if a specified warning is not thrown.
numpy.bartlett(M): Returns the Bartlett window with M points. This window is
similar to a triangular window.
numpy.random.binomial(n, p, size=None): Draws random samples from
the binomial distribution.
numpy.bitwise_and(x1, x2[, out]): Calculates the bit-wise AND of arrays.
numpy.bitwise_xor(x1, x2[, out]): Calculates the bit-wise XOR of arrays.
numpy.blackman(M): Returns a Blackman window with M points, which is close
to optimal and a little bit worse than a Kaiser window.
numpy.column_stack(tup): Stacks 1D arrays provided as a tuple column wise.
numpy.concatenate ((a1, a2, ...), axis=0): Concatenates a sequence
of arrays.
numpy.convolve(a, v, mode='full'): Computes the linear convolution
of 1D arrays.
numpy.dot(a, b, out=None): Calculates the dot product of two arrays.
numpy.diff(a, n=1, axis=-1): Computes the nth difference for a given axis.
numpy.dsplit(ary, indices_or_sections): Splits an array into subarrays
along the third axis.
numpy.dstack(tup): Stacks arrays given as a tuple along the third axis.
numpy.eye(N, M=None, k=0, dtype=): Returns the
identity matrix.
[ 302 ]
Appendix C
numpy.extract(condition, arr): Selects elements of an array using
a condition.
numpy.fft.fftshift(x, axes=None): Shifts the zero-frequency component
of a signal to the center of the spectrum.
numpy.hamming(M): Returns the Hamming window with M points.
numpy.hanning(M): Returns the Hanning window with M points.
numpy.hstack(tup): Stacks arrays given as a tuple horizontally.
numpy.isreal(x): Returns a Boolean array, where True corresponds to an
element of the input array, which is a real number (as opposed to a complex
number).
numpy.kaiser(M, beta): Returns a Kaiser window with M points for a given
beta parameter.
numpy.load(file, mmap_mode=None): Loads NumPy arrays or pickled objects
from .npy, .npz or pickles. A memory-mapped array is stored in the filesystem
and doesn't have to be completely loaded in memory. This is especially useful for
large arrays.
numpy.loadtxt(fname, dtype=, comments='#',
delimiter=None, converters=None, skiprows=0, usecols=None,
unpack=False, ndmin=0): Loads data from a text file into a NumPy array.
numpy.lexsort (keys, axis=-1): Sorts using multiple keys.
numpy.linspace(start, stop, num=50, endpoint=True,
retstep=False, dtype=None): Returns evenly spaced numbers over an interval.
numpy.max(a, axis=None, out=None, keepdims=False): Returns the
maximum of an array along an axis.
numpy.mean(a, axis=None, dtype=None, out=None, keepdims=False):
Calculates the arithmetic mean along the given axis.
numpy.median(a, axis=None, out=None, overwrite_input=False):
Calculates the median along the given axis.
numpy.meshgrid(*xi, **kwargs): Returns coordinate matrices for coordinate
vectors. For instance:
In: numpy.meshgrid([1, 2], [3, 4])
Out:
[array([[1, 2],
[1, 2]]), array([[3, 3],
[4, 4]])]
[ 303 ]
NumPy Functions' References
numpy.min(a, axis=None, out=None, keepdims=False): Returns the
minimum of an array along an axis.
numpy.msort(a): Returns a copy of an array sorted along the first axis.
numpy.nanargmax(a, axis=None): Returns the indices of the maximums given
an axis ignoring NaNs.
numpy.nanargmin(a, axis=None): Returns the indices of the minimums given
an axis ignoring NaNs.
numpy.nonzero(a): Returns indices of non-zero array elements.
numpy.ones(shape, dtype=None, order='C'): Creates a NumPy array of
specified shape and data type, containing 1s.
numpy.piecewise(x, condlist, funclist, *args, **kw): Evaluates a
function piecewise.
numpy.polyder(p, m=1): Differentiates a polynomial to a given order.
numpy.polyfit(x, y, deg, rcond=None, full=False, w=None,
cov=False): Performs a least squares polynomial fit.
numpy.polysub(a1, a2): Subtracts polynomials.
numpy.polyval(p, x): Evaluates a polynomial at specified values.
numpy.prod(a, axis=None, dtype=None, out=None, keepdims=False):
Returns the product of array elements over a specified axis.
numpy.ravel(a, order='C'): Flattens an array or returns a copy if necessary.
numpy.reshape(a, newshape, order='C'): Changes the shape of a NumPy
array.
numpy.row_stack(tup): Stacks arrays row wise.
numpy.save(file, arr): Saves a NumPy array to a file in the NumPy .npy
format.
numpy.savetxt(fname, X, fmt='%.18e', delimiter=' ',
newline='\n', header='', footer='', comments='# '): Saves a NumPy
array to a text file.
numpy.sinc(a): Computes the sinc function.
numpy.sort_complex(a): Sorts array elements with the real part first, then
followed by the imaginary part.
numpy.split(a, indices_or_sections, axis=0): Splits an array into
subarrays.
[ 304 ]
Appendix C
numpy.std(a, axis=None, dtype=None, out=None, ddof=0,
keepdims=False): Returns the standard deviation along the given axis.
numpy.take(a, indices, axis=None, out=None, mode='raise'):
Selects elements from an array using specified indices.
numpy.vsplit(a, indices_or_sections): Splits an array into subarrays
vertically.
numpy.vstack(tup): Stacks arrays vertically.
numpy.where(condition, [x, y]): Selects array elements from input arrays
based on a Boolean condition.
numpy.zeros(shape, dtype=float, order='C'): Creates a NumPy array of
specified shape and data type, containing zeros.
[ 305 ]
Index
Symbols
.mat file
loading 246
saving 246
== operator 44
A
absolute tolerance (atol) 202
add() function
ufuncs methods, applying 127, 128
affinity matrix 284
animation
about 241
clock.tick(30) function 278
in Pygame 275
pygame.time.Clock() function 278
URL 276
YouTube, URL 279
annotate() function 234
annotations
about 234
using 235, 236
argmax() function 178
argmin() function 178
argwhere() function 178
arithmetic functions 129
arithmetic mean
about 57
reference link 57
array2d() function 282
array initialization 118
arrays
about 17
clipping 94, 95
comparing 202, 203
compressing 94, 95
dividing 129, 130
extracting, from element 179, 180
ndarray methods, using 94
ordering 203
vectors, adding 17-20
artificial intelligence
about 284
points, clustering 284-286
pygame.draw.polygon () function 286
sklearn.cluster.AffinityPropagation().fit(S)
function 286
assert_almost_equal() function
using 198
assert_approx_equal() function
using 199
assert_array_almost_equal() function
using 200
assert_array_almost_equal_nulp() function
using 206
assert_array_less() function 203
assert_array_max_ulp() function
using 207
assert_equal() function
using 204
assert functions
about 198
assert_allclose() 198
[ 307 ]
assert_almost_equal() 198
assert_approx_equal() 198
assert_array_almost_equal() 198
assert_array_equal() 198
assert_array_less() 198
assert_equal() 198
assert_raises() 198
assert_string_equal() 198
assert_warns() 198
assert_string_equal() function
using 204, 205
at() method
using, for fancy indexing 144
attributes, one-dimensional NumPy arrays
flat attribute 49, 50
imag attribute 49
itemsize attribute 48
ndim attribute 48
real attribute 49
size attribute 48
T attribute 48
audio clips
replaying 268, 269
audio processing
about 268
audio clips, replaying 268, 269
Average True Range (ATR)
about 74
calculating 75, 76
minimum() function, using 77
axes 71
B
bartlett() function 187
Bartlett window
plotting 187, 188
basic arithmetic, Python 4
Bessel function
about 191
URL 191
Binet's formula 133
binomial distribution models 161
binomial() function
using 161, 162
bits
twiddling 141, 142
bitwise functions 140
blackman() function 188
Blackman window
about 188
used, for smoothing stock prices 189, 190
bmat() function 123
Bollinger Bands
about 82
components 82, 83
enveloping 83-85
Exponential Moving Average, switching 86
bootstrapping 169
B-spline interpolation algorithms 253
C
calc_profit() function 111
calculus 104
character codes 33
clustering 284
comma-separated value (CSV) files
about 55
loading 55
complex conjugate 152
complex numbers
about 176
sorting 177
concatenate() function 43
conjugate transpose 152
constructors 34
continuous distributions
about 165
normal distribution, drawing 165, 166
contour plots
about 240
drawing 240
convolution
about 77
references 77
corrcoef() function 101-103
correlation
about 100
correlated pairs, trading 100-104
URL 100
[ 308 ]
covariance
URL 100
Cowboy's Game of Life
about 290
implementation 290-293
transitions 290
URL 290
D
data type objects 33
dates
about 65
datetime64 data type, using 69, 70
dealing with 65-67
TWAP, calculating 68
VWAP, calculating 68
datetime64 data type
about 69
URL 69
using 69, 70
datetime object
reference link 66
deriv() function 219
determinant
about 155
of matrix, calculating 155, 156
URL 155
detrended signal
filtering 256, 257
diagonal() method 101
diff() function 109
Discrete Fourier transform
about 156
URL 256
Dish Network Corp (DISH) 225
distribution (distro) 15
divide() function 129, 130
docstring 213
doctests
executing 214, 215
dot() function 149
dtype attribute
about 35
record data type, creating 35, 36
E
eigenvalues
about 149
determining 150
URL 149
eigenvectors
about 149
determining 150
eig() function 149
eigvals() function 149
element
array, extracting from 179, 180
Enthought
about 284
URL 14
error() function 263
Exponential Moving Average (EMA)
about 80
calculating 80-82
extract() function
using, for array element extraction 179, 180
extrema 106
F
factorial
about 95
calculating 95, 96
fancy indexing
about 143
using, with at() method 144
Fast Fourier transform (FFT)
about 156
calculating 156-158
fft() function 157
fftshift() function 158
Fibonacci numbers
about 132, 133
calculations, timing 134
computing 133, 134
URL 132
file handle
about 54
reference link 54
[ 309 ]
file I/O
about 53
files, reading 54, 55
files, writing 54, 55
fill_between() function 232, 233
financial functions
fv() function 180
irr() function 181
mirr() function 181
nper() function 181
npv() function 180
pmt() function 180
pv() function 180
rate() function 181
used, for determining future value 181, 182
Fink
used, for installing Numpy 16
flatten() function 41
floating-point numbers
comparing 205
comparing, with assert_array_almost_
equal_nulp function 206
floats
comparing 207
comparing, with maxulp of 2 207
floor_divide() function 129
floor() function 129
fmod() function 132
for loop
about 9
implementing 9, 10
format string
plotting 219
polynomial derivatives, plotting 219, 220
Fourier analysis
about 256
detrended signal, filtering 256, 257
Fourier series
about 136, 156
URL 136
frequencies
shifting 158-160
frompyfunc() function 125
full() function
used, for creating value initialized
arrays 119, 120
full_like() function
used, for creating value initialized
arrays 119, 120
functional programming
about 73
reference link 73
functions, Python
defining 11
future value
about 180
determining, with financial functions 181, 182
URL 181
fv() function 180
G
Gaussian integral
calculating 263, 264
golden ratio formula 132, 133
H
hamming() function 190
Hamming window
about 190
plotting 190, 191
hanning() function
used, for smoothing 114
Hello World game
about 272
creating 272-274
pygame.display.set_caption () function 274
pygame.display.set_mode() function 274
pygame.display.update() function 275
pygame.event.get() function 275
pygame.font.SysFont () function 274
pygame.init() function 274
pygame.quit() function 275
screen.blit() function 274
sysFont.render() function 274
Hermitian conjugate 152
hist() function 226
histograms
about 226
bell curve, drawing 228
stock price distributions, charting 226-228
[ 310 ]
hypergeometric distribution
about 163
game show, simulating 163, 164
hypergeometric() function 164
I
identity matrix
URL 54
ifft() function 157
if statement
about 8
implementing 8
image processing
about 266
Lena, manipulating 266, 267
interest rate
about 186
figuring 186, 187
internal rate of return
about 181-184
determining 185
interpolation
about 264
in one direction 264, 265
inv() function 146
IPython
about 21-24
features 21
installing, on Linux 15
installing, on Windows 13, 14
URL 21
IRC channel
URL 25
irr() function 181
isreal() function 117
J
Jackknife resampling
about 96
Not a Number (NaN), handling 97
URL 96
Jarque-Bera normality test 250, 251
K
kaiser() function 191
Kaiser window
about 191
plotting 192
Kolmogorov-Smirnov 251
kurtosis 248
L
leastsq() function 259
least-squares method
about 87
reference link 87
legend() function 234
legends
about 234
using 235, 236
Lena Soderberg
manipulating 266, 267
lexsort() function
sorting lexically 174, 175
linear algebra
about 145
matrices, inverting 146, 147
URL 145
Linear Algebra PACKage (LAPACK) 2
linear model
about 86
price, predicting 86-89
linear systems
solving 148
linspace() function 138, 218
Linux
IPython, installing 15
matplotlib, installing 15
NumPy, installing 15
SciPy, installing 15
Lissajous curves
about 134
drawing 135, 136
loadmat() function 245
locators 224
[ 311 ]
logarithmic plots
about 228
stock volume, plotting 228, 229
lognormal distribution
about 167
drawing 167, 168
lognormal() function 167
M
MacPorts
used, for installing Numpy 16
Maple 21
mat() function 122, 123, 146
Mathematica 21
mathematical optimization
about 259
sinusoidal pattern, fitting 259-261
MATLAB 21, 245
matplotlib
about 217
installing, on Linux 15
installing, on Windows 13, 14
URL 217
using, in Pygame 278-281
matplotlib.finance package
about 223
used, for plotting year's worth of stock
quotes 223-225
matrices
about 40, 121, 122
creating 122, 123
matrix, creating from 123, 124
transposing, URL 40
URL 122
matrix
creating, from other matrices 123, 124
decomposing, with SVD 152
determinant, calculating 155, 156
inverting, in linear algebra 146, 147
inverting, URL 122
pseudo inverse, computing 154, 155
matrix() function 133
median
about 59
reference link 59
Mersenne Twister algorithm
URL 160
methods 34
mirr() function 181
missing values 96
mod() function 131
modified Bessel function
plotting 193
modified internal rate of return 181
modules, Python
about 12
importing 12
modulo
calculating 131
computing 131, 132
msvcp71.dll file
URL 14
multidimensional arrays
indexing 36-39
slicing 36-39
N
nanargmax() function 178
nanargmin() function 178
NetBSD 284
net present value
about 180-183
calculating 184
URL 183
normal distribution
drawing 165, 166
URL 165
normality test 248
nose tests, decorators
about 210
numpy.testing.decorators.deprecated 210
numpy.testing.decorators.knownfailureif 210
numpy.testing.decorators.setastest 210
numpy.testing.decorators.skipif 210
numpy.testing.decorators.slow 210
Not a Number (NaN)
handling, with nanmean() function 97
handling, with nanstd() function 97
handling, with nanvar() function 97
npv() function 180, 183
[ 312 ]
number of periodic payments
about 181-186
determining 186
numerical integration
about 263
Gaussian integral, calculating 263, 264
NumPy
about 1
array object 28
financial functions 180
functions 301-305
installing, on Debian and Ubuntu 15
installing, on Gentoo 15
installing, on Linux 15
installing, on Mac OS X 16
installing, on Mandriva 15
installing, on Red Hat 15
installing, on Windows 13, 14
installing, with Fink 16
installing, with MacPorts 16
matrices 122
numerical types 31
search functions 178
sorting routines 173
URL 2
used, for accessing surface pixels 282, 283
used, for animating objects 275, 276
NumPy 1.8 143
NumPy and SciPy functions
bools.astype() function 294
ndimage.convolve() function 294
np.arange() function 294
NumPy array object
about 28, 29
character codes 33
data type objects 33
dtype attribute 35
dtype constructor 34
elements, selecting 30-32
multidimensional array, creating 29, 30
numerical types 31
three-by-three array, creating 30
numpy.random.choice() function
used, for sampling 169, 170
numpy.testing.assert_array_almost_equal_
nulp() function 302
O
objects
comparing 204
Octave matrices 245
on-balance volume indicator 108
one-dimensional NumPy arrays
attributes 48-50
column_stack() function 45
column stacking 43, 44
concatenate() function 45
converting 51
depth stacking 43
depth-wise splitting 47
dstack() function 45
horizontal splitting 46
horizontal stacking 42
hstack() function 45
row_stack() function 45
row stacking 44
shapes, manipulating 39-41
slicing 36
splitting 45-48
stacking 41
vertical splitting 46
vertical stacking 42
vstack() function 45
online resources, Python 299
OpenGL
and Pygame 287
outer() method 128
P
partial sorting
partition() function, using 175, 176
URL 175
partition() function
about 176
used, for partial sorting 175, 176
payment against loan 180
periodic payments
about 185
calculating 185
piecewise() function 109
pinv() function 154
[ 313 ]
plot() function 218, 219
plot regions, based on condition
shading 232, 233
plots
animating 241, 242
pmt() function 180
points
clustering 284
poly1d() function 218
polyfit() function 105, 107
polynomials
about 104
fitting to 105-108
polysub() function 117
present value
about 183
obtaining 183
URL 183
print() function
about 6
used, for printing 6
pseudo inverse, of matrix
computing 154, 155
URL 154
pseudo-random numbers
about 160
URL 144
p-value 247
pv() function 180
Pygame
about 271
agg.FigureCanvasAgg() function 281
and OpenGL 287
canvas.draw() function 281
canvas.get_renderer() function 281
installing 272
installing, on Debian and Ubuntu 272
matplotlib, using 278-281
mpl.use () function 281
plt.figure() function 281
Sierpinski gasket, drawing 287
used, for animating objects 275-277
Pygame installation
from source 272
on Debian and Ubuntu, URL 272
on Mac OS X, URL 272
on Mac, URL 272
on Windows, URL 272
Python
about 1
basic arithmetic 4
classes, URL 34
comparison operators, URL 44
decorators, URL 210
functions 11
help system 3
installer, URL 2
installing, on Debian and Ubuntu 2
installing, on different operating systems 2
installing, on Mac 2
installing, on Windows 2
mathematics and statistics, URLs 300
modules 12
online resources 299
URLs 299, 300
using, as calculator 4
values, assigning to variables 5
Python shell 3
Q
QQQ
trend, detecting 253-255
quad() function 263
R
random numbers
about 160
binomial() function 161
rate() function 181, 186
rate of interest 181
ravel() function 39, 41
read() function 268
regression line
URL 100
relative tolerance (rtol) 202
remainder() function 131
reshape() function 41
resistance levels 90
resize() function 41
rint() function 133, 134
row_stack() function 44
[ 314 ]
S
sample variance 61
savemat() function 245, 246
sawtooth
about 138
drawing 139, 140
scatter() function 230
scatter plot
about 230
used, for plotting price 230, 231
used, for plotting volume returns 230, 231
SciKits
about 250
URL 250
SciPy
installing, on Linux 15
installing, on Windows 13, 14
URL 25
scipy.interpolate() function 264
scipy.stats module 247
ScipySuperpack
URL 16
search functions
argmax() function 178
argmin() function 178
argwhere() function 178
extract() function 178
nanargmax() function 178
nanargmin() function 178
searchsorted() function
about 178
using 178, 179
select() function 116
semilogx() function 228
semilogy() function 228
shapes, one-dimensional NumPy arrays
flatten 40
ravel 39
resize() method 41
setting, up with tuple 40
transpose 40
show() function 217, 218
Sierpinski gasket
drawing 287-290
glBegin() function 290
glColor3f() function 290
glEnd() function 290
glFlush() function 290
gluOrtho2D() function 289
glVertex2fv() function 290
pygame.display.set_mode((w,h) function 289
Sierpinski triangle 287
signal processing
about 253
trend, detecting in QQQ 253-255
sign() function 109
Simple DirectMedia Layer (SDL) 271
Simple Moving Average (SMA)
about 77
computing 77-79
simple plots
about 217
polynomial function, plotting 218, 219
simulation
about 111
loops, avoiding with vectorize()
function 111-114
sinc() function
about 192, 264
plotting 194, 195
URL 192
sin() function 138
singular value decomposition. See SVD
skewness
about 247
URL 247
smoothing
about 114
hanning() function, using 114-117
variations 118
solve() function 148
sorting routines
about 173
argsort() function 173
lexsort() function 173
msort() function 173
ndarray class 173
sort_complex() function 173
sort() function 173
source code, for Numpy
building 16
URL 16
[ 315 ]
special mathematical functions
about 192
modified Bessel function, plotting 193
sinc() function 192
spline interpolation
URL 253
square waves
about 136
drawing 137, 138
Stack Overflow
URL 25
statistics
about 59, 247
bootstrapping 169
data generation, improving 250
numpy.random.choice() function,
using 169
performing 59-62
random values, analyzing 247-249
stock log returns
comparing 250-252
DIA 250
SPY 250
stock price distributions
charting 226, 227
stock returns
about 62
analyzing 63, 64
stock volume
plotting 228, 229
strings
comparing 204, 205
subplot() function 221
subplots
about 221
First Derivative 222
Polynomial 221
polynomial derivatives, plotting 221-223
Second Derivative 222
support levels 90
surface pixels
accessing, with NumPy 282, 283
pygame.surfarray.array2d(img) function 283
pygame.surfarray.blit_array(screen,
new_pixels) function 283
SVD
about 151
matrix, decomposing 152, 153
svd() function 151
T
Taylor series
URL 105
test-driven development (TDD) 197
three-dimensional function
plotting 238, 239
three-dimensional plots 238
tile() function 268, 282
time-weighted average price (TWAP)
about 57
averages, calculating 58
weighted average, computing 57
trace() method 101
trend line
about 89
drawing 90-93
triangle waves
about 138
drawing 139, 140
trim_zeros() function 117
true_divide() function 129
U
ufuncs
about 125
creating 125, 126
fancy indexing, using with at() method 144
methods 126
ufuncs methods
about 126
applying, to add() function 127, 128
Unit of Least Precision (ULP) 205
unit tests
about 207
writing 208, 209
universal functions. See ufuncs
[ 316 ]
V
value initialized arrays
creating, with full() functions 119
creating, with full_like() function 119
value range
about 58
highest value, searching 58, 59
lowest value, searching 58, 59
variable assignment, Python 4
variance
about 61
reference link 61
vectorize() function
used, for avoiding loops 111
vectors
adding, with NumPy 18, 20
adding, with Python 17
volume
about 108
balancing 109, 110
Volume Weighted Average Price (VWAP)
about 56
calculating 56
mean() function 56, 57
reference link 56
vstack() function 42
W
weekly summary
about 70
code, modifying 74
data, summarizing 70-73
window functions
about 187
bartlett() function 187
Bartlett window, plotting 187, 188
blackman() function 188
hamming() function 190
kaiser() function 191
Windows
IPython, installing 13, 14
matplotlib, installing 13, 14
Numpy, installing 13, 14
SciPy, installing 13, 14
Windows IPython installer
URL 14
write() function 268, 269
X
xlabel() function 218
XOR operation
URL 141
Y
year's worth of stock quotes
plotting 223-225
ylabel() function 218
Z
zeros_like() function 126
[ 317 ]
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