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NumPy Beginner's Guide
Second Edition
An action packed guide using real world examples of the
easy to use, high performance, free open source NumPy
mathematical library
Ivan Idris
BIRMINGHAM - MUMBAI
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Numpy Beginner's Guide
Second Edition
Copyright © 2013 Packt Publishing
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First published: November 2011
Second edition: April 2013
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Credits
Author
Ivan Idris
Reviewers
Jaidev Deshpande
Project Coordinator
Abhishek Kori
Proofreader
Mario Cecere
Dr. Alexandre Devert
Mark Livingstone
Miklós Prisznyák
Nikolay Karelin
Acquisition Editor
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Lead Technical Editor
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Technical Editors
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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, Datawarehouse Developer, and QA Analyst. His main professional interests are
Business Intelligence, Big Data, and Cloud Computing. Ivan Idris enjoys writing clean testable
code and interesting technical articles. Ivan Idris is the author of NumPy Beginner's Guide
& Cookbook. You can find more information and a blog with a few NumPy examples at
ivanidris.net.
I would like to take this opportunity to thank the reviewers and the team
at Packt Publishing for making this book possible. Also thanks goes to
my teachers, professors, and colleagues who taught me about science
and programming. Last but not the least, I would like to acknowledge my
parents, family, and friends for their support.
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About the Reviewers
Jaidev Deshpande is an intern at Enthought, Inc, where he works on software for data
analysis and visualization. He is an avid scientific programmer and works on many open
source packages in signal processing, data analysis, and machine learning.
Dr. Alexandre Devert is teaching data-mining and software engineering at the University
of Science and Technology of China. Alexandre also works as a researcher, both as an
academic on optimization problems, and on data-mining problems for a biotechnology
startup. In all those contexts, Alexandre very happily uses Python, Numpy, and Scipy.
Mark Livingstone started his career by working for many years for three international
computer companies (which no longer exist) in engineering/support/programming/training
roles, but got tired of being made redundant. He then graduated from Griffith University on
the Gold Coast, Australia, in 2011 with a Bachelor of Information Technology. He is currently
in his final semester of his B.InfoTech (Hons) degree researching in the area of Proteomics
algorithms with all his research software written in Python on a Mac, and his Supervisor and
research group one by one discovering the joys of Python.
Mark enjoys mentoring first year students with special needs, is the Chair of the IEEE Griffith
University Gold Coast Student Branch, and volunteers as a Qualified Justice of the Peace at
the local District Courthouse, has been a Credit Union Director, and will have completed 100
blood donations by the end of 2013.
In his copious spare time, he co-develops the S2 Salstat Statistics Package available
at http://code.google.com/p/salstat-statistics-package-2/ which is
multiplatform and uses wxPython, NumPy, SciPy, Scikit, Matplotlib, and a number
of other Python modules.
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Miklós Prisznyák is a senior software engineer with a scientific background. He graduated
as a physicist from the Eötvös Lóránd University, the largest and oldest university in Hungary.
He did his MSc thesis on Monte Carlo simulations of non-Abelian lattice quantum field
theories in 1992. Having worked three years in the Central Research Institute for Physics
of Hungary, he joined MultiRáció Kft. in Budapest, a company founded by physicists,
which specialized in mathematical data analysis and forecasting economic data. His main
project was the Small Area Unemployment Statistics System which has been in official
use at the Hungarian Public Employment Service since then. He learned about the Python
programming language here in 2000. He set up his own consulting company in 2002 and
then he worked on various projects for insurance, pharmacy and e-commerce companies,
using Python whenever he could. He also worked in a European Union research institute
in Italy, testing and enhanching a distributed, Python-based Zope/Plone web application.
He moved to Great Britain in 2007 and first he worked at a Scottish start-up, using Twisted
Python, then in the aerospace industry in England using, among others, the PyQt windowing
toolkit, the Enthought application framework, and the NumPy and SciPy libraries. He
returned to Hungary in 2012 and he rejoined MultiRáció where now he is working on a
Python extension module to OpenOffice/EuroOffice, using NumPy and SciPy again, which will
allow users to solve non-linear and stochastic optimization problems. Miklós likes to travel,
read, and he is interested in sciences, linguistics, history, politics, the board game of go, and
in quite a few other topics. Besides he always enjoys a good cup of coffee. However, nothing
beats spending time with his brilliant 10 year old son Zsombor for him.
Nikolay Karelin holds a PhD degree in optics and used various methods of numerical
simulations and analysis for nearly 20 years, first in academia and then in the industry
(simulation of fiber optics communication links). After initial learning curve with Python
and NumPy, these excellent tools became his main choice for almost all numerical analysis
and scripting, since past five years.
I wish to thank my family for understanding and keeping patience during
long evenings when I was working on reviews for the "NumPy Beginner’s
Guide."
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Table of Contents
Preface
Chapter 1: NumPy Quick Start
1
9
Python
Time for action – installing Python on different operating systems
Windows
Time for action – installing NumPy, Matplotlib, SciPy, and IPython
on Windows
Linux
Time for action – installing NumPy, Matplotlib, SciPy, and IPython on Linux
Mac OS X
Time for action – installing NumPy, Matplotlib, and SciPy on Mac OS X
Time for action – installing NumPy, SciPy, Matplotlib, and IPython
with MacPorts or Fink
Building from source
Arrays
Time for action – adding vectors
IPython—an interactive shell
Online resources and help
Summary
9
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Chapter 2: Beginning with NumPy Fundamentals
27
NumPy array object
Time for action – creating a multidimensional array
Selecting elements
NumPy numerical types
Data type objects
Character codes
dtype constructors
dtype attributes
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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
Stacking
Time for action – stacking arrays
Splitting
Time for action – splitting arrays
Array attributes
Time for action – converting arrays
Summary
Chapter 3: Get in Terms with Commonly Used Functions
File I/O
Time for action – reading and writing files
CSV 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 – doing simple statistics
Stock returns
Time for action – analyzing stock returns
Dates
Time for action – dealing with dates
Weekly summary
Time for action – summarizing data
Average true range
Time for action – calculating the 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
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Table of Contents
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
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
Summary
Chapter 5: Working with Matrices and ufuncs
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 function
Universal function methods
Time for action – applying the ufunc methods on add
Arithmetic functions
Time for action – dividing arrays
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
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Table of Contents
Time for action – drawing sawtooth and triangle waves
Bitwise and comparison functions
Time for action – twiddling bits
Summary
Chapter 6: Move Further with NumPy Modules
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
Pseudoinverse
Time for action – computing the pseudo inverse of a matrix
Determinants
Time for action – calculating the determinant of a matrix
Fast Fourier transform
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
Summary
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Chapter 7: Peeking into Special Routines
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Sorting
Time for action – sorting lexically
Complex numbers
Time for action – sorting complex numbers
Searching
Time for action – using searchsorted
Array elements' extraction
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Time for action – extracting elements from an array
Financial functions
Time for action – determining 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
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: Assure Quality with Testing
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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
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Table of Contents
Time for action – checking the array order
Objects 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 tests decorators
Time for action – decorating tests
Docstrings
Time for action – executing doctests
Summary
Chapter 9: Plotting with Matplotlib
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Simple plots
Time for action – plotting a polynomial function
Plot format string
Time for action – plotting a polynomial and its derivative
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 scatter plot
Fill between
Time for action – shading plot regions based on a condition
Legend and annotations
Time for action – using legend and annotations
Three dimensional plots
Time for action – plotting in three dimensions
Contour plots
Time for action – drawing a filled contour plot
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Table of Contents
Animation
Time for action – animating plots
Summary
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Chapter 10: When NumPy is Not Enough – SciPy and Beyond
MATLAB and Octave
Time for action – saving and loading a .mat file
Statistics
Time for action – analyzing random values
Samples’ 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
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
Chapter 11: Playing with Pygame
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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
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Table of Contents
Time for action – drawing the Sierpinski gasket
Simulation game with PyGame
Time for action – simulating life
Summary
Pop Quiz Answers
Index
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Preface
Scientists, engineers, and quantitative data analysts face many challenges nowadays.
Data scientists want to be able to do numerical analysis of large datasets with minimal
programming effort. They want to write readable, efficient, and fast code, which is as close
as possible to the mathematical language package 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 are
amongst others, Matlab, Maple and Mathematica. These products provide powerful scripting
languages, which are still 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 also lack the power of a language such as Python.
Python is a popular general-purpose programming language, widely used in the scientific
community. You can access legacy C, Fortran, or R code easily from Python. It is object-oriented
and considered more high level than C or Fortran. Python allows you to write readable and
clean code with minimal fuss. However, it lacks a Matlab equivalent out of the box. That's
where NumPy comes in. This book is about NumPy and related Python libraries such as SciPy
and Matplotlib.
What is NumPy?
NumPy (from Numerical Python) is an open-source Python library for scientific computing.
NumPy let's you work with arrays and matrices in a natural way. The library contains
a long list of useful mathematical functions including some for linear algebra, Fourier
transformation, and random number generation routines. LAPACK, a linear algebra library,
is used by the NumPy linear algebra module (that is, if you have LAPACK 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 sense, NumPy replaces some
of the functionality of Matlab and Mathematica, allowing rapid interactive prototyping.
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Preface
We will not be discussing NumPy from a developing contributor perspective, but more from
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
a deprecated status now. Neither Numeric nor NumPy made it into the standard Python
library for various reasons. However, you can install NumPy separately as will be explained
in Chapter 1, Numpy Quick Start.
In 2001, a number of people inspired by Numeric created SciPy—an open-source Python
scientific computing 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 alternative to Numeric. Numarray 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 the Numarray features into Numeric. A complete
rewrite took place that culminated in the release of NumPy 1.0 in 2006. At this time, NumPy
has all of the features of Numeric and Numarray and more. Upgrade tools are 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 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 that tries to accomplish the same
task. There are less 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.
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Preface
NumPy's arrays are stored more efficiently than an equivalent data structure in base Python
such as list of lists. Array IO is significantly faster too. The performance improvement scales
with the number of elements of an 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. The drawback of NumPy arrays is that they are more specialized
than plain lists. Outside of the context of numerical computations, NumPy arrays are less
useful. The technical details of NumPy arrays will be discussed in the later chapters.
Large portions of NumPy are written in C. That makes NumPy faster than pure Python code.
A NumPy C API exists as well and it allows further extension of the functionality with the
help of the C language of NumPy. The C API falls outside the scope of this book. Finally,
since NumPy is open-source, you get all of the related advantages. The price is the lowest
possible—free as in "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 to
everyone. This of course is beneficial to the 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, although, they do
implement the same specification. There are some workarounds for this that are discussed in
NumPy Cookbook, Ivan Idris, Packt Publishing.
What this book covers
Chapter 1, NumPy Quick Start will guide you through the steps needed to install NumPy
on your system and create a basic NumPy application.
Chapter 2, Beginning with NumPy Fundamentals introduces you to NumPy arrays
and fundamentals.
Chapter 3, Get to Terms with Commonly Used Functions will teach you about the most
commonly used NumPy functions—the basic mathematical and statistical functions.
Chapter 4, Convenience Functions for Your Convenience will teach 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 will also learn about polynomials,
and manipulating the shape of NumPy objects.
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Preface
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 input and produce a set of scalars as output.
Chapter 6, Move Further with Numpy Modules discusses the number of basic modules
of Universal functions. Universal functions can typically be mapped to mathematical
counterparts such as add, subtract, divide, and multiply.
Chapter 7, Peeking into Special Routines describes some of the more specialized NumPy
functions. As NumPy users, we sometimes find ourselves having special needs. Fortunately,
NumPy provides for most of our needs.
In Chapter 8, Assure Quality with Testing you will learn how to write NumPy unit tests.
Chapter 9, Plotting with Matplotlib covers in-depth Matplotlib, a very useful Python plotting
library. NumPy on its own cannot be used to create graphs and plots. But Matplotlib
integrates nicely with NumPy and has plotting capabilities comparable to Matlab.
Chapter 10, When NumPy is Not Enough – SciPy and Beyond goes into more detail 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. We will learn how to create fun
games with NumPy and Pygame. We also get a taste of artificial intelligence.
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 to have one of the Python versions supported by NumPy as well. Some
code samples make use of the 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 software used to develop and test the code examples:
Python 2.7
NumPy 2.0.0.dev20100915
SciPy 0.9.0.dev20100915
Matplotlib 1.1.1
Pygame 1.9.1
IPython 0.14.dev
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Preface
Needless to say, you don't need to have exactly this software and these versions on your
computer. Python and NumPy is the absolute minimum you will need.
Who this book is for
This book is for you the scientist, engineer, programmer, or analyst, 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.
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 the Next button
moves you to the next screen".
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Preface
Warnings or important notes appear in a box like this.
Tips and tricks appear like this.
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Now that you are the proud owner of a Packt book, we have a number of things to help you
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Downloading the example code
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Preface
Errata
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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. The
IPython interactive shell is introduced briefly. As mentioned in the Preface, 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, we shall:
Install Python, SciPy, Matplotlib, IPython, and NumPy on Windows, Linux,
and Macintosh
Write simple NumPy code
Get to know IPython
Browse online documentation and resources
Python
NumPy is based on Python, so it is required 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 distribution. In this book we will focus on
the standard CPython implementation, which is guaranteed to be compatible with NumPy.
www.it-ebooks.info
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. 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:
1.
Debian and Ubuntu: Python might already be installed on Debian and Ubuntu but
the development headers are usually not. On Debian and Ubuntu install python and
python-dev with the following commands:
sudo apt-get install python
sudo apt-get install python-dev
2.
Windows: The Windows Python installer can be found at www.python.org/
download. On this website, we can also find installers for Mac OS X and source
tarballs for Linux, Unix, and Mac OS X.
3.
Mac: Python comes pre-installed on Mac OS X. We can also get Python through
MacPorts, Fink, or similar projects. We can install, for instance, the Python 2.7
port by running the following command:
sudo port install python27
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 to 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.
Windows
Installing NumPy on Windows is straightforward. You only need to download an installer,
and a wizard will guide you through the installation steps.
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Chapter 1
Time for action – installing NumPy, Matplotlib, SciPy, and IPython
on Windows
Installing NumPy on Windows is necessary but, fortunately, a straightforward task that we
will cover in detail. It is recommended to 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 version. In this example, we chose numpy-1.7.0-win32superpack-python2.7.exe.
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NumPy Quick Start
2.
Open the EXE installer by double clicking on it.
3.
Now, we can see a description of NumPy and its features as shown in the previous
screenshot. Click on the Next button.
4.
If you have Python installed, it should automatically be detected. If it is not
detected, maybe your path settings are wrong. At the end of this chapter,
resources are listed in case you have problems installing NumPy.
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Chapter 1
5.
In this example, Python 2.7 was found. Click on the Next button if Python is found;
otherwise, click on the Cancel button and install Python (NumPy cannot be installed
without Python). Click on the Next button. 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.
6.
Install SciPy and Matplotlib with the Enthought distribution http://www.
enthought.com/products/epd.php. 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.
scipy.org/Wiki/IpythonOnWindows).
What just happened?
We installed NumPy, SciPy, Matplotlib, and IPython on Windows.
Linux
Installing NumPy and related recommended software on Linux depends on the distribution
you have. We will discuss how you would install NumPy from the command line, although,
you could 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 steps
for some of the popular Linux distros:
1.
Run the following instructions from the command line for installing NumPy
and Red Hat:
yum install python-numpy
2.
To install NumPy on Mandriva, run the following command-line instruction:
urpmi python-numpy
3.
To install NumPy on Gentoo run the following command-line instruction:
sudo emerge numpy
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NumPy Quick Start
4.
To install NumPy on Debian or Ubuntu, we need to type the following :
sudo apt-get install python-numpy
The following table gives an overview of the Linux distributions and corresponding package
names for NumPy, SciPy, Matplotlib, and IPython.
Linux
distribution
NumPy
SciPy
Matplotlib
IPython
Arch Linux
pythonnumpy
pythonnumpy
numpy
pythonscipy
pythonscipy
pythonscipy
scipy
pythonmatplotlib
pythonmatplotlib
pythonmatplotlib
matplotlib
ipython
pythonscipy
pythonmatplotlib
ipython
scipy
matplotlib
ipython
Debian
Fedora
Gentoo
OpenSUSE
Slackware
dev-python/
numpy
pythonnumpy,
pythonnumpy-devel
numpy
ipython
ipython
ipython
What just happened?
We installed NumPy, SciPy, Matplotlib, and IPython on various Linux distributions.
Mac OS X
You can install NumPy, Matplotlib, and SciPy on the Mac with a graphical installer or from the
command line with a port manager such as MacPorts or Fink, depending on your preference.
Time for action – installing NumPy, Matplotlib, and SciPy on Mac
OS X
We will install NumPy with a GUI installer using the following steps:
1.
We can get a NumPy installer from the SourceForge website http://
sourceforge.net/projects/numpy/files/. Similar files exist for Matplotlib
and SciPy. Just change numpy in the previous URL to scipy or matplotlib.
IPython didn't have a GUI installer at the time of writing. Download the appropriate
DMG file as shown in the following screenshot, usually the latest one is the best:
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Chapter 1
2.
Open the DMG file as shown in the following screenshot (in this example,
numpy-1.7.0-py2.7-python.org-macosx10.6.dmg):
Double-click on the icon of the opened box, the one having a subscript
that ends with .mpkg. We will be presented with the welcome screen
of the installer.
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NumPy Quick Start
3.
Click on the Continue button to go to the Read Me screen, where we
will be presented with a short description of NumPy as shown in the
following screenshot:
Click on the Continue button to the License the screen.
Read the license, click on the Continue button and then on the Accept button, when
prompted to accept the license. Continue through the next screens and click on the
Finish button at the end.
What just happened?
We installed NumPy on Mac OS X with a GUI installer. The steps to install SciPy and
Matplotlib are similar and can be performed using the URLs mentioned in the first step.
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Chapter 1
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 shown install all these packages. We
only need NumPy for all the tutorials in this book, so please omit the packages you are not
interested in.
1.
For installing with MacPorts, type the following command:
sudo port install py-numpy py-scipy py-matplotlib py-ipython
2.
Fink also has packages for NumPy: scipy-core-py24, scipy-core-py25, and
scipy-core-py26. The SciPy packages are: scipy-py24, scipy-py25, and
scipy-py26. We can install NumPy and the other recommended packages we will
be using in this book for Python 2.6 with the following command:
fink install scipy-core-py26 scipy-py26 matplotlib-py26
What just happened?
We installed NumPy and other 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
Install /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 package.
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 equivalent Python code.
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NumPy Quick Start
Time for action – adding vectors
Imagine that we want to add two vectors called a and b. 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. The
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, or 4. The vector b holds the cubes of integers 0 to n, so if n is equal to 3, then the
vector b is equal to 0, 1, or 8. How would you do that using plain Python? After we come up
with a solution, we will compare it with the NumPy equivalent.
1.
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
2.
The following is a function that achieves the same with NumPy:
def numpysum(n):
a = numpy.arange(n) ** 2
b = numpy.arange(n) ** 3
c = 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.
Now comes the fun part. Remember that it is mentioned in the Preface 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
import sys
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Chapter 1
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.
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
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NumPy Quick Start
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
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
You can download the example code files for all Packt books you have
purchased from your account at http://www.PacktPub.com. 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.
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 are using NumPy or not. However, the result that is
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.
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Chapter 1
Pop quiz Functioning of the arange function
Q1. What does arange(5) do?
1. Creates a Python list of 5 elements with values 1 to 5.
2. Creates a Python list of 5 elements with values 0 to 4.
3. Creates a NumPy array with values 1 to 5.
4. Creates a NumPy array with values 0 to 4.
5. None of the above.
Have a go hero – continue the analysis
The program we used here 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 would be nice to
be able to calculate some statistics such as average times, and so on. 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 resources listed at
the end of this chapter. NumPy has, by the way, 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 experimenting. IPython was created by scientists with
experimentation in mind. The interactive environment that IPython provides is viewed by
many 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 scientific work where IPython
was used. Here is the list of basic IPython features:
Tab completion
History mechanism
Inline editing
Ability to call external Python scripts with %run
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NumPy Quick Start
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 would have to import every package we need, ourselves.
All we need to do is enter the following instruction on the command line:
$ ipython --pylab
Python 2.7.2 (default, Jun 20 2012, 16:23:33)
Type "copyright", "credits" or "license" for more information.
IPython 0.14.dev -- 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.
Welcome to pylab, a matplotlib-based Python environment [backend:
MacOSX].
For more information, type 'help(pylab)'.
In [1]: quit()
The quit() function or Ctrl + D quits the IPython shell. We might 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. We can
run the script as follows:
In [1]: ls
README
vectorsum.py
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Chapter 1
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 and on typing 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
Type 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
ncallstottimepercallcumtimepercallfilename: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}
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NumPy Quick Start
1000 0.000
0.0000.0000.000 {method 'append' of 'list' objects}
1 0.000
0.000
1 0.000
0.0000.0000.000 vectorsum.py:21(numpysum)
0.002
0.002 vectorsum.py:3()
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)
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 into 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!
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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
arange
arccos
arccosh
arcsin
arcsinh
arctan
arctan2
arctanh
argmax
argmin
argsort
argwhere
around
array
array2string
array_equal
array_equiv
array_repr
array_split
array_str
arrow
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 webpage, we can browse the NumPy reference at http://docs.
scipy.org/doc/numpy/reference/ and the user guide as well as several tutorials.
NumPy has a wiki with lots of documentation at http://docs.scipy.org/numpy/
Front%20Page/.
The NumPy and SciPy forum can be found at http://ask.scipy.org/en.
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 and there
is almost no spam to speak of. Most importantly, developers actively involved with NumPy
also answer questions asked on the discussion group. The complete list can be found at
http://www.scipy.org/Mailing_Lists.
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NumPy Quick Start
For IRC users, there is an IRC channel on 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.
Summary
In this chapter, we installed NumPy and other recommended software that we will be using
in some tutorials. We got a vector addition program working and convinced ourselves that
NumPy has superior performance. We were introduced to the IPython interactive shell. In
addition, we explored the available NumPy documentation and online resources.
In the next chapter, we will take a look under the hood and explore some fundamental
concepts including arrays and data types.
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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:
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. However, the
source code delivered alongside the book is regular Python code that uses imports and
print statements.
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Beginning with NumPy Fundamentals
NumPy array object
NumPy has a multi-dimensional 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.
We have already learned, in the previous chapter, how to create an array using the arange
function. Actually, we created a one-dimensional array that contained a set of numbers.
ndarray can have more than one dimension.
The NumPy array is in general homogeneous (there is a special array type that is
heterogeneous)—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 just like in Python, starting from 0. Data types are represented
by special objects. These objects will be discussed comprehensively in this chapter.
We will create an array with the arange function again. Here's how to get the data type
of an array:
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 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.
The example in Chapter 1, NumPy Quick Start, 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. Let's determine the shape of the
vector we created a few minutes ago:
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.
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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 matrix, we would again want to display its shape.
1.
2.
Create a multidimensional array.
Show the array shape:
In: m = array([arange(2), arange(2)])
In: m
Out:
array([[0, 1],
[0, 1]])
In: m.shape
Out: (2, 2)
What just happened?
We created a two-by-two array with the arange function 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.
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 matrix
It shouldn't be too hard now to create a three-by-three matrix. Give it a go and check
whether the array shape is as expected.
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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, let's create a two-by-two matrix again:
In: a = array([[1,2],[3,4]])
In: a
Out:
array([[1, 2],
[3, 4]])
The matrix was created this time by passing the array function a list of lists. We will now
select each item of the matrix one-by-one. 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 as shown in the
following diagram:
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. 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:
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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
complex64
Double precision float: sign bit, 11 bits exponent, 52 bits mantissa
complex128 or
complex
Complex number, represented by two 64-bit floats (real and
imaginary components)
Boolean (True or False) stored as a bit
Complex number, represented by two 32-bit floats (real and
imaginary components)
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
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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. Trying to do that triggers a TypeError:
In [1] : int(42.0+1.j)
TypeError
in 1 int(42.0.+1.j)
TypeError: can’t convert complex to int
The same goes for conversion of a complex number into a float. By the way, the j part is the
imaginary coefficient of the complex number. However, you can convert a float 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. You should instead use dtype objects.
Type
integer
Character code
i
Unsigned integer
u
Single precision float
f
Double precision float
d
bool
b
complex
D
string
S
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Chapter 2
Type
unicode
Character code
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., 6.], dtype=float32)
Likewise this creates an array of complex numbers
In: arange(7, dtype='D')
Out: array([ 0.+0.j, 1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 5.+0.j,
6.+0.j])
dtype constructors
We have a variety of ways to create data types. Take the case of floating point data:
We can use the general Python float:
In: dtype(float)
Out: dtype('float64')
We can specify a single precision float with a character code:
In: dtype('f')
Out: dtype('float32')
We can 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; 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 in sctypeDict.keys():
In: sctypeDict.keys()
Out: [0, …
'i2',
'int0']
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dtype attributes
The dtype class has a number of useful attributes. For example, we can 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 dtype 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, means 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 directly get an element with the flatiter object:
In: b.flat[2]
Out: 2
Or multiple elements:
In: b.flat[[1,3]]
Out: array([1, 3])
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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 selected elements
In: b.flat[[1,3]] = 1
In: b
Out:
array([[7, 1],
[7, 1]])
Time for action – converting arrays
We can 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])
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Chapter 2
We are losing the imaginary part when casting from complex type 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.
Summary
We learned a lot in this chapter about the NumPy fundamentals: data types and arrays.
Arrays have several attributes describing them. We learned that one of these attributes
is the data type, which in NumPy, is represented by a full-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.
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, Get to Terms with Commonly Used Functions. This includes basic
statistical and mathematical functions.
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3
Get in Terms 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 using a historical stock prices example.
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 will not get
far if you are not able to read from and write to files.
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Get in Terms 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.
Perform the following steps to do so:
1.
The identity matrix is a square matrix with ones on the main diagonal and zeroes
for the rest. The identity matrix can be created with the eye function. The only
argument we need to give the eye function is the number of ones. So, for instance,
for a 2 x 2 matrix, write the following code:
i2 = np.eye(2)
print i2
The output is:
[[ 1. 0.]
[ 0. 1.]]
2.
Save the data using the savetxt function. We obviously need to 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. You can check for yourself whether the
contents are as expected. The code for this example can be downloaded from the book
support website http://www.packtpub.com/support (see save.py).
import numpy as np
i2 = np.eye(2)
print i2
np.savetxt("eye.txt", i2)
What just happened?
Reading and writing files is a necessary skill for data analysis. We wrote to a file using
savetxt. We made an identity matrix with the eye function.
CSV files
Files in the comma-separated values (CSV) format are encountered quite frequently. Often,
the CSV file is just a dump from a database file. 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.
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Chapter 3
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 price data for Apple (the company, not the fruit). The data is in the CSV
format. The first column contains a symbol that identifies the stock. In our case, it is AAPL.
Second is the date in the 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 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
would 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 comma-separated value file. The usecols parameter
is set through a tuple to get the seventh and eighth fields, which correspond to the close
price and volume. unpack 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.
What just happened?
CSV files are a special type of file that we have to deal with frequently. We read a CSV file
containing stock quotes with the loadtxt function. We indicated to the loadtxt function
that the delimiter of our file was a comma. We specified which columns we were interested
in, through the usecols argument, and set the unpack parameter to True so that the data
was unpacked for further use.
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. The higher the volume, the more significant a price
move typically is. VWAP is often used in algorithmic trading and is calculated by using volume
values as weights.
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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.
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
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.
The mean function
The mean function is quite friendly and not so mean. This function calculates the arithmetic
mean of an array. Let's see it in action:
print "mean =", np.mean(c)
mean = 351.037666667
Time-weighted average price
In finance, TWAP is another "average" price measure. Now that we are at it, let's compute
the time-weighted average price, 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.
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Chapter 3
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
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 sort of in the middle; 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. After all, how else would we know how much our Apple stocks
would gain or lose?
Time for action – finding highest and lowest values
The min and max functions are the answer to our requirement. Perform the following steps
to find highest and lowest values:
1.
First, we will need to 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)
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These are the values returned:
highest = 364.9
lowest = 333.53
Now, it's trivial to get a midpoint, so it is left as an exercise for the reader 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)
You will see the following:
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.
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. The arithmetic mean and weighted average are
ways to find the center of a distribution of values. However, neither are robust nor sensitive
to outliers. For instance, if we had a close price value of a million dollars, this would have
influenced the outcome of our calculations.
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Chapter 3
Time for action – doing 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. For example, if
we have the values of 1, 2, 3, 4, and 5, the median would be 3, since it is in the middle. The
following are the steps to calculate the median:
1.
Determine the median of the close price. 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 the following, by now:
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. Not because we are paranoid or anything! Obviously, we could 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. We will 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]
It gives us the following output:
middle = 351.99
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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. 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
Success!
5.
Another statistical measure that we are concerned with is variance. Variance tells
us how much a variable varies. 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. With
NumPy, this is just a one liner. See the following code:
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 variance is defined as the mean of the square of deviations from the
mean, divided by the number of elements in the array.
Some books tell us to divide by the number of elements in the array minus one.
print "variance from definition =", np.mean((c - c.mean())**2)
The output is as follows:
variance from definition = 50.1265178889
Just as we expected!
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Chapter 3
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 object 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.
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 the difference
between the log of "a" and the log of "b" is equal to the log of "a divided by b". Log returns,
therefore, also measure 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.
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 built up of the difference between two consecutive array elements. This is sort
of like differentiation in calculus. 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]
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Notice that we don't use the last value in the divisor. Let's 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 is even easier to calculate. We use the log function to get the log of
the close price and then unleash the diff function on the result.
logreturns = np.diff( np.log(c) )
Normally, we would have to check that the input array doesn't have zeroes or
negative numbers. If it did, we would have got 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.
Indices with positive returns (array([ 0, 1, 4, 5,
10, 11, 12, 16, 17, 18, 19, 21, 22, 23, 25, 28]),)
4.
6,
7,
9,
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. See the following code:
annual_volatility = np.std(logreturns)/np.mean(logreturns)
annual_volatility = annual_volatility / np.sqrt(1./252.)
print annual_volatility
5.
Take note 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. Similarly, the monthly volatility is given by:
print "Monthly volatility", annual_volatility * np.sqrt(1./12.)
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Chapter 3
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 the tutorial we
calculated the annual and monthly volatility (see returns.py).
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.)
Dates
Do you sometimes have the Monday blues or the Friday fever? Ever wondered whether
the stock market suffers from said 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 health 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 decision. Please consult a professional statistician first!
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Coders hate dates because they are so complicated! NumPy is very much oriented towards
floating point operations. For that 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 the 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
explicitly tell NumPy 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 so-called converter functions. It is our responsibility to write
the converter function.
Let's 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()
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".
This is, by the way, standard Python and is not related to NumPy itself. 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 we will hook up our date converter function to load the data.
dates, close=np.loadtxt('data.csv', delimiter=',', usecols=(1,6),
converters={1: datestr2num}, unpack=True)
print "Dates =", dates
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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.
The values of the array will be initialized 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, we compute an average for each weekday and store it in the averages
array, as follows:
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
The loop prints the following output:
Day 0 prices [[ 339.32 351.88
351.79
Day 1 prices [[ 345.03 355.2
Average 350.635
Day 2 prices [[ 344.32 358.16
Average 352.136666667
Day 3 prices [[ 343.44 354.54
Average 350.898333333
Day 4 prices [[ 336.1
346.5
351.99]] Average 350.022857143
359.18
353.21
355.36]] Average
359.9
338.61
349.31
355.76]]
363.13
342.62
352.12
352.47]]
358.3
342.88
359.56
346.67]]
356.85
350.56
348.16
360.
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5.
If you want, you can go ahead and find out which day has the highest, and which
the lowest, average. However, it is just as easy to find this out with the max and min
functions, as shown next:
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).
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, 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)
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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.
Weekly summary
The data that we used in the previous Time for action tutorials is end-of-day data.
In essence, it is summarized data compiled from trade data for a certain day. If you are
interested in the cotton market and have decades of data, you might want to summarize
and compress the data even further. Let's do that. 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 from Monday to Friday. During
the period covered by the data, there was one holiday on February 21st, 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, we will just have a look at the first three weeks in the sample—you can
later have a go at improving this.
close = close[:16]
dates = dates[:16]
We will build on the code from the Time for action – dealing with dates tutorial.
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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 a where
function. Then, we will need to extract the first element that has index 0. The result
would be a multidimensional array. Flatten that 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
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
4.
Split the array in pieces of size 5 with the split function.
weeks_indices = np.split(weeks_indices, 5)
print "Weeks indices after split", weeks_indices
It 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])]
5.
7,
In NumPy, 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, that we will define
shortly. Further 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
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6.
Write the summarize function. The summarize function returns, for each week,
a tuple that holds the open, high, low, and close prices for the week, similarly 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 of the week was easily done with the max and
min functions. open for the week is the open for the first day in the week—Monday.
Likewise, 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']]
7.
Store the data in a file with the NumPy savetxt function.
np.savetxt("weeksummary.csv", weeksummary, delimiter=",",
fmt="%s")
As you can see, we specify 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 zeroes, + 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.
Character code
c
Description
d or i
signed decimal integer
e or E
scientific notation with e or E
f
decimal floating point
g or G
use the shorter of e, E, or f
o
signed octal
character
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Character code
s
Description
u
unsigned decimal integer
x or X
unsigned hexadecimal integer
string of characters
View the generated file in your favorite editor or type in the following commands
in 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. Arguments for the
summarize function were neatly passed by apply_along_axis (see weeksummary.py).
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]
# get first Monday
first_monday = np.ravel(np.where(dates == 0))[0]
print "The first Monday index is", first_monday
# get last Friday
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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.
Time for action – calculating the average true range
To calculate the average true range, perform the following steps:
1.
The ATR is based on the low and high price of N days, usually the last 20 days.
N = int(sys.argv[1])
h = h[-N:]
l = l[-N:]
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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:
h – l: The daily range (the difference between high and low price)
h – previousclose: The difference between high price and
previous close
previousclose – l: The difference between the previous close and the
low price
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:
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
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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 this Time for action tutorial,
we saw that the maximum function can do this. After that, we computed a moving average of
the true range values (see atr.py).
import numpy as np
import sys
h, l, c = np.loadtxt('data.csv', delimiter=',', usecols=(4, 5, 6),
unpack=True)
N = int(sys.argv[1])
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)
for i in range(1, N):
atr[i] = (N - 1) * atr[i - 1] + truerange[i]
atr[i] /= N
print "ATR", atr
In the following tutorials, 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 prove your assumptions.
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Simple moving average
The simple moving average 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 simple moving average 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.
Use the following steps to compute the simple moving average:
1.
Use the ones function to create an array of size N and elements initialized to 1;
then, divide the array by N to give us the weights, as follows:
N = int(sys.argv[1])
weights = np.ones(N) / N
print "Weights", weights
For N = 5, this code gives us the following output:
Weights [ 0.2
2.
0.2
0.2
0.2
0.2]
Now call the convolve function with the following 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 be covering 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))
plot(t, c[N-1:], lw=1.0)
plot(t, sma, lw=2.0)
show()
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In the following chart, the smooth thick line is the 5-day simple moving average
and the jagged thin line is the close price:
What just happened?
We computed the simple moving average for the close stock price. Truly, great riches are
within your reach. It turns out that the simple moving average 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 data set with specified weights
(see sma.py).
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
N = int(sys.argv[1])
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))
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plot(t, c[N-1:], lw=1.0)
plot(t, sma, lw=2.0)
show()
Exponential moving average
The exponential moving average is a popular alternative to the simple moving average.
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 and a stop and optionally an array size.
It returns an array of evenly spaced numbers. The following 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.
]
Let's calculate the exponential moving average for our data:
1.
Now, back to the weights—calculate them with exp and linspace.
N = int(sys.argv[1])
weights = np.exp(np.linspace(-1., 0., N))
2.
Normalize the weights. The ndarray object has a sum method that we will use.
weights /= weights.sum()
print "Weights", weights
For N = 5, we get the following weights:
Weights [ 0.11405072
0.31002201]
0.14644403
0.18803785
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3.
After that, it's easy going—we just use the convolve function that we learned
about in the simple moving average tutorial. We will 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))
plot(t, c[N-1:], lw=1.0)
plot(t, ema, lw=2.0)
show()
That gives this nice chart where, again, the close price is the thin jagged line and the
exponential moving average is the smooth thick line:
What just happened?
We calculated the exponential moving average of the close price. First, we computed
exponentially decreasing weights with the exp and linspace functions. linspace 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 that,
we applied the convolve trick that we learned in the simple moving average tutorial
(see ema.py).
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
x = np.arange(5)
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print "Exp", np.exp(x)
print "Linspace", np.linspace(-1, 0, 5)
N = int(sys.argv[1])
weights = np.exp(np.linspace(-1., 0., N))
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))
plot(t, c[N-1:], lw=1.0)
plot(t, ema, lw=2.0)
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, as follows:
A simple moving average
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
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 simple moving average. So, if you need to, please
review the Time for action – computing the simple moving 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
you have to set the values of the array one by one in a loop. Perform the following steps to
envelope with Bollinger bands:
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1.
Starting with an array called sma that contains the moving average values, we will
loop through all the data sets corresponding to those values. After forming the
data set, calculate the standard deviation. Note that it is necessary, at a certain
point, to calculate the difference between each data point and the corresponding
average value. If we did not have NumPy, we would 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 having 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)
upperBB = sma + deviation
lowerBB = sma – deviation
2.
To plot the bands, 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 = numpy.arange(N - 1, C)
plot(t, c_slice, lw=1.0)
plot(t, sma, lw=2.0)
plot(t, upperBB, lw=3.0)
plot(t, lowerBB, lw=4.0)
show()
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The following is a chart of the Bollinger bands for our data. The jagged thin line in the
middle represents the close price and the slightly thicker, smoother line crossing it is the
moving average:
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).
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
N = int(sys.argv[1])
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)
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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]
between_bands = len(np.ravel(between_bands))
print "Ratio between bands", float(between_bands)/len(c_slice)
t = np.arange(N - 1, C)
plot(t, c_slice, lw=1.0)
plot(t, sma, lw=2.0)
plot(t, upperBB, lw=3.0)
plot(t, lowerBB, lw=4.0)
show()
Have a go hero – switching to exponential moving average
It is customary to choose the simple moving average to center the Bollinger band on.
The second most popular choice is the exponential moving average, so try that as an
exercise. You can find a suitable example in this chapter, if you need pointers.
Check that the fill function is faster or is as fast as array.flat = scalar, or set the
value in a loop.
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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.
Time for action – predicting price with a linear model
Keeping an open mind, let's assume that we can express a stock price as a linear combination
of previous values, that is, a sum of those values multiplied by certain coefficients we need
to determine. In linear algebra terms, this boils down to finding a least squares solution.
This recipe goes as follows.
1.
First, form a vector bbx containing N price values.
bbx = c[-N:]
bbx = b[::-1]
print "bbx", x
The result is as follows:
bbx [ 351.99
2.
346.67
352.47
355.76
Second, pre-initialize the matrix A to be N x N and containing zeroes.
A = np.zeros((N, N), float)
print "Zeros N by N", A
Zeros N by N [[ 0. 0. 0. 0.
[ 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0.]]
3.
355.36]
0.]
Third, fill the matrix A with N preceding price values for each value in bbx.
for i in range(N):
A[i, ] = c[-N - 1 - i: - 1 - i]
print "A", A
Now, A looks like this:
A [[ 360.
[ 359.56
[ 352.12
[ 349.31
[ 353.21
355.36 355.76 352.47 346.67]
360.
355.36 355.76 352.47]
359.56 360.
355.36 355.76]
352.12 359.56 360.
355.36]
349.31 352.12 359.56 360. ]]
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4.
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 that.
(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 coefficients xxb that we were after, an array
comprising of 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 numpy.dot(b, x)
The dot product is the linear combination of the coefficients xxb and the prices x.
As a result, we get the following:
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.
What just happened?
We predicted tomorrow's stock price today. If this works in practice, we could 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).
import numpy as np
import sys
N = int(sys.argv[1])
c = np.loadtxt('data.csv', delimiter=',', usecols=(6,), unpack=True)
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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 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
tutorial, we shall resort to a very simple approach that is probably not very useful in real life,
but it should clarify the principle well.
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;
mind you, they are merely estimates. Based on these estimates, it is possible to draw
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support and resistance trend lines. We will define the daily spread to be the difference
between 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 and vstack functions.
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
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, this setup is of no use to us. Make up a condition for points between
the bands and select the where function based on that condition.
condition = (c > support) & (c < resistance)
print "Condition", condition
between_bands = np.where(condition)
The following are the condition values:
Condition [False False True True True True True False False
True False False
False False False True False False False True True True True
False False True True True False 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)
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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:
Tomorrows support 349.389157088
Tomorrows resistance 360.749340996
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 the following:
Number of points between bands 2nd approach 15
5.
Once more, we will plot the results, as follows:
plot(t, c)
plot(t, support)
plot(t, resistance)
show()
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We will get the following plot in which we have the price data and the
corresponding support and resistance lines:
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, 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).
import numpy as np
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
def fit_line(t, y):
A = np.vstack([t, np.ones_like(t)]).T
return np.linalg.lstsq(A, y)[0]
h, l, c = np.loadtxt('data.csv', delimiter=',', usecols=(4, 5, 6),
unpack=True)
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pivots = (h + l + c) / 3
print "Pivots", pivots
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))
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))
plot(t, c)
plot(t, support)
plot(t, resistance)
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 that are a
part of the NumPy library have a counterpart with the same name and functionality in the
ndarray object. 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 var, sum, std,
argmax, argmin, and mean functions that we saw earlier are also ndarray methods.
To clip and compress arrays, look at the following section.
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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)
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 condition a > 2.
Factorial
Many programming books have an example of calculating the factorial. We should not break
with this tradition.
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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 eight. To do that, 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?
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
What just happened?
We used the prod and cumprod functions to calculate factorials
(see ndarraymethods.py).
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()
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Chapter 3
Summary
This chapter informed us about a great number of common NumPy functions. We read a file
with loadtxt and wrote to a file with savetxt. We made an identity matrix with the eye
function. We read a CSV file containing stock quotes with the loadtxt function. The NumPy
average and mean functions allow one to calculate the weighted average and arithmetic
mean of a data set.
A few common statistics functions were also mentioned – first, the min and max functions
that we used to determine the range of the stock prices; second, the median function
that gives the median of a data set; and finally, the std and var functions that return the
standard deviation and variance of a set of numbers.
We calculated the simple stock returns with the diff function that returns back the
differences between sequential elements. The log function computes the natural
logarithms of array elements.
By default, loadtxt tries to convert all data into floats. 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.
We defined a function and passed it as an argument to the apply_along_axis
function. We implemented an algorithm with the requirement to find the maximum
value across arrays.
We learned that the ones function can create an array with ones and the convolve
function calculates the convolution of a data set with the specified weights.
We computed exponentially decreasing weights with the exp and linspace functions.
linspace 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.
We got acquainted with the NumPy fill function. This function fills an array with a scalar
value, the only parameter of the fill function.
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.
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4
Convenience Functions for
Your Convenience
As we have noticed, NumPy has a great number of functions. Many of these
functions are there just for your convenience. Knowing these functions will
greatly increase your productivity. This includes functions that select certain
parts of your arrays (for instance, based on a Boolean condition) or manipulate
polynomials. An example of computing correlation of stock returns is provided
to give you a taste of data analysis in 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 the previous chapter, we had one single data file to play around with. Things have
significantly improved in this chapter—we now have two data files. Let's go ahead and
explore the data with NumPy.
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Correlation
Have you noticed that the stock price of some companies is closely followed by another one,
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 would want to check whether a real
relationship exists. One way is to have a look at the correlation of the stock returns of
both stocks. A high correlation implies a relationship of some sort. It is not proof though,
especially if you don't use sufficient data.
Time for action – trading correlated pairs
For this tutorial, we will use two sample data sets, containing the bare minimum of
end-of-day price data. The first company is BHP Billiton (BHP), which is active in the
mining of petroleum, metals, and diamonds. The second is Vale (VALE), which is also
a metals and mining company. So there is some overlap, albeit not one hundred percent.
For trading 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, there are plenty of examples in the previous chapter.
2.
Covariance tells us how two variables vary together; it is nothing more than
unnormalized correlation. 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]]
3.
View the values on the diagonal with the diagonal function:
print "Covariance diagonal", covariance.diagonal()
The diagonal values of the covariance matrix are as follows:
Covariance diagonal [ 0.00028179
0.00030123]
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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 function:
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:
Try it out:
print covariance/ (bhp_returns.std() * vale_returns.std())
The correlation matrix is as follows:
[[ 1.00173366 0.70264666]
[ 0.70264666 1.0708476 ]]
6.
We will measure the correlation of our pair with the correlation coefficient. The
correlation coefficient takes values between -1 to 1. The correlation of a set of
values with itself is 1 by definition. This would be the ideal value; however, we will
be also 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 probability, 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 the correlation is not that strong.
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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 eventually will
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))
plot(t, bhp_returns, lw=1)
plot(t, vale_returns, lw=2)
show()
The resulting plot:
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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. This was achieved with the corrcoef
function. Further, we saw how the covariance matrix can be computed, from which the
correlation can be derived. As a bonus, a demonstration was given of the diagonal and
trace functions that can give us the diagonal values and the trace of a matrix, respectively
(see correlation.py):
import numpy as np
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
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)
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))
plot(t, bhp_returns, lw=1)
plot(t, vale_returns, lw=2)
show()
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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? Me, I love it! One of the ideas in calculus is Taylor expansion, that is,
representing a differentiable function as an infinite series. In practice, this means that any
differentiable, and therefore, continuous function can be estimated by a polynomial of a
high degree. The terms of the higher degree would then be assumed to be negligibly small.
Time for action – fitting to polynomials
The NumPy polyfit function can fit 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, let's 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, int(sys.argv[1]))
print "Polynomial fit", poly
The polynomial fit (in this example, a cubic polynomial was chosen):
Polynomial fit [ 1.11655581e-03
5.80684638e-01
5.79791202e+01]
2.
-5.28581762e-02
The numbers you see are the coefficients of the polynomial. Extrapolate to the next
value with the polyval function and the polynomial object we got from the fit:
print "Next value", np.polyval(poly, t[-1] + 1)
The next value we predict will be:
Next value 57.9743076081
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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 we 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
0.58068464]
The numbers you see are the coefficients of the derivative polynomial.
5.
Get the roots of the derivative and find the extrema:
print "Extremas", np.roots(der)
The extrema that we get are:
Extremas [ 24.47820054
7.08205278]
Let's double check; compute the values of the fit with polyval:
vals = np.polyval(poly, t)
6.
Now, find the maximum and minimum values with argmax and argmin:
vals = np.polyval(poly, t)
print np.argmax(vals)
print np.argmin(vals)
This gives us the following expected results. 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
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7.
Plot the data and the fit it as follows:
plot(t, bhp - vale)
plot(t, vals)
show()
It results in this plot:
Obviously, the smooth line is the fit and the jagged line is the underlying data. It's not that
good a fit, so 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):
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
bhp=np.loadtxt('BHP.csv', delimiter=',', usecols=(6,),
unpack=True)
vale=np.loadtxt('VALE.csv', delimiter=',', usecols=(6,),
unpack=True)
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t = np.arange(len(bhp))
poly = np.polyfit(t, bhp - vale, int(sys.argv[1]))
print "Polynomial fit", poly
print "Next value", np.polyval(poly, t[-1] + 1)
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)
plot(t, bhp - vale)
plot(t, vals)
show()
Have a go hero – improving the fit
There are a number of things you could do to improve the fit. Try a different power as, in this
tutorial, a cubic polynomial was chosen. Consider smoothing the data before fitting it. One
way you could smooth is with a moving average. Examples of simple and exponential moving
average calculations can be found in the previous chapter.
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. If the close price did
not change then the value of the on-balance volume is zero.
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Time for action – balancing volume
In other words we need to multiply the sign of the close price with the volume. In this
tutorial, we will go over 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 close 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 0.63 -1.54 -0.28 0.25 -0.6
2.15
0.69 -1.33 1.16
1.59 -0.26 -1.29 -0.13 -2.12 -3.91 1.28 -0.57 -2.07 -2.07
2.5
1.18
-0.88 1.31 1.24 -0.59]
2.
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
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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. 2650200. -4667300. -5359800.
7768400.
-4799100. 3448300. 4719800. -3898900. 3727700. 3379400.
-2463900.
-3590900. -3805000. -3271700. -5507800. 2996800. -3434800.
-5008300.
-7809799. 3947100. 3809700. 3098200. -3500200. 4285600.
3918800.
-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):
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
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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. 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 that
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. We will let it calculate the profit of a single trading day:
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
that we still have to write:
func = np.vectorize(calc_profit)
3.
We can now apply func as if it is a function. Apply the func 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, we will return 0. Otherwise, we
sell at the close price and the profit is just the difference between the buy price and
the close price. Actually, it is more interesting to have a look at the relative profit:
def calc_profit((open, high, low, close):
#buy just below the open
buy = open * float(sys.argv[1])
# daily range
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if low < buy < high:
return (close - buy)/buy
else:
return 0
print "Profits", profits
5.
There are two days with zero profits: there was either no net gain, or a loss.
Select the days with trades and calculate 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 are 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 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 are:
Number of winning trades 16 53.33 %
Average profit % 0.72
7.
As pessimists, we are interested in losing trades with profit less than zero. Select the
days with losing trades and calculate 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 are:
Number of losing trades 12 40.0 %
Average loss % -0.92
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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 summary of the losing and winning trades (see simulation.py):
import numpy as np
import sys
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 * float(sys.argv[1])
# 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)
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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 that much.
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 windowing function formed by a weighted cosine. There are
other window functions that will be covered in greater detail 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 N length window
(in this example, N is 8):
N = int(sys.argv[1])
weights = np.hanning(N)
print "Weights", weights
The weights are as follows:
Weights [ 0.
0.1882551
0.95048443 0.61126047
0.1882551
0.
]
2.
0.61126047
0.95048443
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]
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3.
Plotting with Matplotlib:
t = np.arange(N - 1, len(bhp_returns))
plot(t, bhp_returns[N-1:], lw=1.0)
plot(t, smooth_bhp, lw=2.0)
plot(t, vale_returns[N-1:], lw=1.0)
plot(t, smooth_vale, lw=2.0)
show()
The chart would appear as follows:
The thin lines on the 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:
K = int(sys.argv[1])
t = np.arange(N - 1, len(bhp_returns))
poly_bhp = np.polyfit(t, smooth_bhp, K)
poly_vale = np.polyfit(t, smooth_vale, K)
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5.
Now, we need to compute for a situation where the polynomials we found in
the previous step are 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
27.51284094+0.j
24.32064343+0.j
18.86423973+0.j
12.43797190+1.73218179j 12.437971901.73218179j
6.34613053+0.62519463j
6.34613053-0.62519463j]
6.
The numbers we get are complex; 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
The real intersection points are as follows:
Real intersection points [ 27.73321597
24.32064343 18.86423973
0.
7.
27.51284094
0.
0. 0.]
We managed to pick up some zeroes. The trim_zeros function strips the
leading and trailing zeros from a one-dimensional array. Get rid of the zeroes
with trim_zeros:
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
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18.86423973]
Convenience Functions for Your Convenience
What just happened?
We applied the hanning function to smooth arrays containing stock returns. We subtracted
two polynomials with the polysub function. We 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 strip_zeroes function (see smoothing.py):
import numpy as np
import sys
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
N = int(sys.argv[1])
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,), un
pack=True)
vale_returns = np.diff(vale) / vale[ : -1]
smooth_vale = np.convolve(weights/weights.sum(), vale_returns)[N-1:
-N+1]
K = int(sys.argv[1])
t = np.arange(N - 1, len(bhp_returns))
poly_bhp = np.polyfit(t, smooth_bhp, K)
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
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print "Sans 0s", np.trim_zeros(xpoints)
plot(t, bhp_returns[N-1:], lw=1.0)
plot(t, smooth_bhp, lw=2.0)
plot(t, vale_returns[N-1:], lw=1.0)
plot(t, smooth_vale, lw=2.0)
show()
Have a go hero – smoothing variations
Experiment with the other smoothing functions—hamming, blackman, bartlett,
and kaiser. They work more or less in the same way as hanning.
Summary
We calculated the correlation of the stock returns of two stocks with the corrcoef function.
As a bonus, a demonstration of the diagonal and trace functions was given, 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 that gives back the derivative of
a polynomial.
Hopefully, we increased our productivity so that we can continue in the next chapter
with matrices and universal functions (ufuncs).
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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 mathematical counterparts, such as
add, subtract, divide, multiply, and likewise. We will also be introduced to
trigonometric, bitwise, and comparison universal functions.
In this chapter, we shall cover the following topics:
Matrix creation
Matrix operations
Basic ufuncs
Trigonometric functions
Bitwise functions
Comparison functions
Matrices
Matrices in NumPy are subclasses of ndarray. Matrices can be created using a special string
format. They are, just like in mathematics, two-dimensional. 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.
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Working with Matrices and ufuncs
Time for action – creating matrices
Matrices can be created with the mat function. This 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, 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:
print "Inverse A", A.I
The inverse matrix is printed as follows (be warned that this is a O(n3) operation):
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]]
4.
Instead of using a string to create a matrix, let's 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]]
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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):
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 two-by-two identity matrix:
A = np.eye(2)
print "A", A
The identity matrix looks like this:
A [[ 1. 0.]
[ 0. 1.]]
Create another matrix like A and multiply by 2:
B = 2 * A
print "B", B
The second matrix is as follows:
B [[ 2. 0.]
[ 0. 2.]]
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2.
Create the compound matrix from a string. The string uses the same format as the
mat function; only, you can 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):
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
Universal functions
Ufuncs expect a set of scalars as input and produce a set of scalars as output. Universal
functions can typically be mapped to mathematical counterparts, such as, add, subtract,
divide, multiply, and likewise.
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Chapter 5
Time for action – creating universal function
We can create a universal function from a Python function with the NumPy 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; 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 universal function with frompyfunc; specify 1 as as 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]
We can do the same for a two-dimensional array by using 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]
[42 42]]
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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):
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 objects representing functions. Universal functions have four methods. They only make
sense for functions such as add. That is, they have two input parameters and return one
output parameter. If the signature of an ufunc does not match this condition, this will result
in a ValueError, so call this method only for binary universal functions. The four methods
are listed as follows:
reduce
accumulate
reduceat
outer
Time for action – applying the ufunc methods on add
Let's call the four methods on add function.
1.
The input array is reduced by applying the universal function 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)
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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:
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]
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
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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
1
2
3
2
3
4
3
4
5
4
5
6
5
6
7
6
7
8
7 8]
8 9]
9 10]]
What just happened?
We applied the four methods, reduce, accumulate, reduceat, and outer, of universal
functions to the add function (see ufuncmethods.py):
import numpy as np
a = np.arange(9)
print
print
print
print
print
print
print
print
"Reduce", np.add.reduce(a)
"Accumulate", np.add.accumulate(a)
"Reduceat", np.add.reduceat(a, [0, 5, 2, 7])
"Reduceat step I", np.add.reduce(a[0:5])
"Reduceat step II", a[5]
"Reduceat step III", np.add.reduce(a[2:7])
"Reduceat step IV", np.add.reduce(a[7:])
"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. This means that when you use one of those operators
on a NumPy array, the corresponding universal function will get called. Division involves a
slightly more complex process. There are three universal functions that have to do with array
division: divide, true_divide, and floor_division. Two operators correspond to
division: / and //.
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Time for action – dividing arrays
Let's see the array division in action:
1.
The divide function does truncate 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)
The floor_divide function results in:
Floor Divide [2 3 1] [0 0 0]
Floor Divide 2 [ 3. 3. 3.] [ 0.
4.
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 would appear as follows:
print "/ operator", a/b, b/a
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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?
We found that there are three different NumPy division functions. 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 would get by calling the
divide and floor functions consecutively (see dividing.py):
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)
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.
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Chapter 5
Modulo operation
The modulo or remainder can be calculated using the NumPy mod, remainder, and fmod
functions. Also, one 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]
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]
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What just happened?
We demonstrated the NumPy mod, remainder, and fmod functions, which compute the
modulo, or remainder (see modulo.py):
import numpy as np
a = np.arange(-4, 4)
print
print
print
print
"Remainder", np.remainder(a, 2)
"Mod", np.mod(a, 2)
"% operator", a % 2
"Fmod", np.fmod(a, 2)
Fibonacci numbers
The Fibonacci numbers are based on a recurrence relation. It is difficult to express this
relation directly with NumPy code. However, we can express this relation in a matrix form
or use the golden ratio formula. 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 integer.
Time for action – computing Fibonacci numbers
The Fibonacci recurrence relation can be represented by a matrix. Calculation of Fibonacci
numbers can be expressed as 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 eighth 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:
8th Fibonacci 21
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Chapter 5
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 Fibonacci numbers are:
Fibonacci [
1.
1.
2.
3.
5.
8.
13.
21.]
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):
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 time it. Create a universal
Fibonacci function with frompyfunc and time it too.
Lissajous curves
All the standard trigonometric functions, such as, sin, cos, tan and likewise are represented
by universal functions in NumPy. Lissajous curves are a fun way of using trigonometry.
I remember producing Lissajous figures on an oscilloscope in the physics lab. Two
parametric equations can describe the figures:
x = A sin(at + π/2)
y = B sin(bt)
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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 = float(sys.argv[1])
b = float(sys.argv[2])
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.
Matplotlib will be covered later in Chapter 9, Plotting with Matplotlib. Plot as
shown here:
plot(x, y)
show()
The result for a = 9 and b = 8:
What just happened?
We plotted the Lissajous curve with the previously mentioned 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):
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import numpy as np
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
import sys
a = float(sys.argv[1])
b = float(sys.argv[2])
t = np.linspace(-np.pi, np.pi, 201)
x = np.sin(a * t + np.pi/2)
y = np.sin(b * t)
plot(x, y)
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.
The formula of this particular series representing the square wave is as follows:
Time for action – drawing a square wave
We will initialize t just like in the previous tutorial. 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 initial values for the function to 0:
t
k
k
f
=
=
=
=
np.linspace(-np.pi, np.pi, 201)
np.arange(1, float(sys.argv[1]))
2 * k - 1
np.zeros_like(t)
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2.
This step should be a straightforward application of the sin and sum functions:
for i in range(len(t)):
f[i] = np.sum(np.sin(k * t[i])/k)
f = (4 / np.pi) * f
3.
The code to plot is almost identical to the one in the previous tutorial:
plot(t, f)
show()
The resulting square wave generated with k = 99 is as follows:
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 linspace and the k values with the arange function
(see squarewave.py):
import numpy as np
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
import sys
t
k
k
f
=
=
=
=
np.linspace(-np.pi, np.pi, 201)
np.arange(1, float(sys.argv[1]))
2 * k - 1
np.zeros_like(t)
for i in range(len(t)):
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Chapter 5
f[i] = np.sum(np.sin(k * t[i])/k)
f = (4 / np.pi) * f
plot(t, f)
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 like 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:
Time for action – drawing sawtooth and triangle waves
We will initialize t just like in the previous tutorial. 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(-ny.pi, np.pi, 201)
k = np.arange(1, float(sys.argv[1]))
f = np.zeros_like(t)
2.
This computation of function values should again be a straightforward application
for the sin and sum functions:
for i in range(len(t)):
f[i] = np.sum(np.sin(2 * np.pi * k * t[i])/k)
f = (-2 / np.pi) * f
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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 here:
plot(t, f, lw=1.0)
plot(t, np.abs(f), lw=2.0)
show()
In the following figure, the triangle wave is the one with the thicker line:
What just happened?
We drew a sawtooth wave using the sin function. The input values were assembled with
linspace 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
from matplotlib.pyplot import plot
from matplotlib.pyplot import show
import sys
t = np.linspace(-np.pi, np.pi, 201)
k = np.arange(1, float(sys.argv[1]))
f = np.zeros_like(t)
for i in range(len(t)):
f[i] = np.sum(np.sin(2 * np.pi * k * t[i])/k)
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Chapter 5
f = (-2 / np.pi) * f
plot(t, f, lw=1.0)
plot(t, np.abs(f), lw=2.0)
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 double.
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 likewise. These operators allow
you to do some clever tricks, which should be good for performance; however, they could
make your code quite unreadable, so use them with care.
Time for action – twiddling bits
We will go over three tricks—checking whether the signs of integers are different, checking
whether a number is a power of two, and calculating the modulus of a number that is a
power of two. 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. ^ corresponds to the bitwise_xor
function. < 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 True True
True False True True
True True True True True True]
Sign different? [ True True True True
True False True True
True True True True True True]
As expected, all the signs differ, except for zero.
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True
True
True
True
True
True
True
True
Working with Matrices and ufuncs
2.
A power of two 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 two
would 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 the AND operator a power
of two, and the integer that is one less than that, then we should get 0. The NumPy
counterpart of & is bitwise_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
[False False
True
False True
Power of 2?
[-9 -8 -7 -6
[False False
True
False True
3.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8]
False False False False False False False True
False False False
True]
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8]
False False False False False False False True
False False False
True
True
True]
The trick of computing the modulus of four actually works when taking the modulus
of integers that are a power of two, such as, 4, 8, 16, and likewise. A bitwise left shift
leads to doubling of values. We saw in the previous step that subtracting one from a
power of two 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 two. The NumPy equivalent of << is the
left_shift universal function.
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)
The result is shown as follows:
Modulus 4
[-9 -8 -7 -6
[3 0 1 2 3 0
Modulus 4
[-9 -8 -7 -6
[3 0 1 2 3 0
-5 -4 -3 -2 -1 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0]
4
5
6
7
8]
-5 -4 -3 -2 -1 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0]
4
5
6
7
8]
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Chapter 5
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 two, and calculating the modulus of a number that
is a power of two. We saw the NumPy counterparts of the operators ^, &, <<, and < (see
bittwidling.py):
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)
Summary
We learned, in this chapter, about matrices and universal functions. We covered how to
create matrices and how universal functions work. We had a brief introduction to arithmetic,
trigonometric, bitwise, and comparison universal functions.
In the next chapter, we shall cover the NumPy modules.
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6
Move Further with NumPy Modules
NumPy has a number of modules that have been inherited from its predecessor,
Numeric. Some of these packages have a SciPy counterpart, which may have
fuller functionality. This will be discussed in a later chapter. The numpy.dual
package contains functions that are defined both in NumPy and SciPy. The
packages discussed in this chapter are also part of the numpy.dual package.
In this chapter, we shall 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.
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 A* A-1 = I.
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Move Further with NumPy Modules
The inv function in the numpy.linalg package can do this for us. Let's invert an example
matrix. To invert matrices, perform the following steps:
1.
We will create the example matrix with the mat function that 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 is printed as follows:
A
[[ 0 1
[ 1 0
[ 4 -3
2.
2]
3]
8]]
Now, we can see the inv function in action, using which we will invert the matrix.
inverse = np.linalg.inv(A)
print "inverse of A\n", inverse
The inverse matrix is shown 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 exception is raised.
If you want, you can check the result manually. This is left as an exercise for
the reader.
3.
Let's check what we get when we multiply the original matrix with the result of the
inv function:
print "Check\n", A * inverse
The result is the identity matrix, as expected.
Check
[[ 1.
[ 0.
[ 0.
0.
1.
0.
0.]
0.]
1.]]
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Chapter 6
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).
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.
2.
3.
4.
array
create_matrix
mat
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.
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; here A is a matrix, b can be
1D or 2D 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.
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Move Further with NumPy Modules
Time for action – solving a linear system
Let's solve an example of a linear system. To solve a linear system, perform the
following steps:
1.
Let's create the matrices 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
The matrices A and b are shown as follows:
2.
Solve this linear system by calling the solve function.
x = np.linalg.solve(A, b)
print "Solution", x
The following is the solution of the linear system:
Solution [ 29.
3.
16.
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 (see solution.py).
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])
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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. The eigvals function in the numpy.linalg package calculates eigenvalues.
The eig function returns a tuple containing eigenvalues and eigenvectors.
Time for action – determining eigenvalues and eigenvectors
Let's calculate the eigenvalues of a matrix. Perform the following steps to do so:
1.
Create a matrix as follows:
A = np.mat("3 -2;1 0")
print "A\n", A
The matrix we created looks like the following:
A
[[ 3 -2]
[ 1 0]]
2.
Calculate eigenvalues by calling the eig 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
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The eigenvalues and eigenvectors will be shown as follows:
First tuple of eig [ 2. 1.]
Second tuple of eig
[[ 0.89442719 0.70710678]
[ 0.4472136
0.70710678]]
4.
Check the result with the dot function by calculating the right- and left-hand sides
of the eigenvalues equation Ax = ax.
for i in range(len(eigenvalues)):
print "Left", np.dot(A, eigenvectors[:,i])
print "Right", eigenvalues[i] * eigenvectors[:,i]
print
The output is as follows:
Left [[ 1.78885438]
[ 0.89442719]]
Right [[ 1.78885438]
[ 0.89442719]]
Left [[ 0.70710678]
[ 0.70710678]]
Right [[ 0.70710678]
[ 0.70710678]]
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).
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 in range(len(eigenvalues)):
print "Left", np.dot(A, eigenvectors[:,i])
print "Right", eigenvalues[i] * eigenvectors[:,i]
print
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Chapter 6
Singular value decomposition
Singular value decomposition is a type of factorization that decomposes a matrix into
a product of three matrices. The singular value decomposition is a generalization of the
previously discussed eigenvalue decomposition. The svd function in the numpy.linalg
package can perform this decomposition. This function returns three matrices – U, Sigma,
and V – such that U and V are orthogonal and Sigma contains the singular values of the
input matrix.
The asterisk denotes the Hermitian conjugate or the conjugate transpose.
Time for action – decomposing a matrix
It's time to decompose a matrix with the singular value decomposition. In order to
decompose a matrix, perform the following steps:
1.
First, create a matrix as follows:
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 7 -2]]
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
The result is a tuple containing the two orthogonal matrices U and V on the
left- and right-hand sides 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 0.33333333 -0.66666667]]
3.
We do not actually have the middle matrix—we only have the diagonal values.
The other values are all 0. We can form the middle matrix with the diag function.
Multiply the three matrices. This is shown, as follows:
print "Product\n", U * np.diag(Sigma) * V
The product of the three matrices looks like the following:
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).
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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Chapter 6
Pseudoinverse
The Moore-Penrose pseudoinverse of a matrix can be computed with the pinv
function of the numpy.linalg module (visit http://en.wikipedia.org/wiki/
Moore%E2%80%93Penrose_pseudoinverse). The pseudoinverse is calculated using the
singular value decomposition. The inv function only accepts square matrices; the pinv
function does not have this restriction.
Time for action – computing the pseudo inverse of a matrix
Let's compute the pseudo inverse of a matrix. Perform the following steps to do so:
1.
First, create a matrix as follows:
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 7 -2]]
2.
Calculate the pseudoinverse matrix with the pinv function, as follows:
pseudoinv = np.linalg.pinv(A)
print "Pseudo inverse\n", pseudoinv
The following is the pseudoinverse:
Pseudo inverse
[[-0.00555556 0.07222222]
[ 0.02222222 0.04444444]
[ 0.05555556 -0.05555556]]
3.
Multiply the original and pseudoinverse matrices.
print "Check", A * pseudoinv
What we get is not an identity matrix, but it comes close to it, as follows:
Check [[ 1.00000000e+00
0.00000000e+00]
[ 8.32667268e-17
1.00000000e+00]]
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What just happened?
We computed the pseudoinverse 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).
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 visit http://en.wikipedia.org/wiki/
Determinant. For an n x n real value matrix the determinant corresponds to the scaling an
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, perform the following steps:
1.
Create the matrix as follows:
A = np.mat("3 4;5 6")
print "A\n", A
The matrix we created is shown as follows:
A
[[ 3.
[ 5.
2.
4.]
6.]]
Compute the determinant with the det function.
print "Determinant", np.linalg.det(A)
The determinant is shown as follows:
Determinant -2.0
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What just happened?
We calculated the determinant of a matrix with the det function from the numpy.linalg
module (see determinant.py).
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). FFT improves on more naïve algorithms and is of order O(NlogN). DFT has
applications in signal processing, image processing, solving partial differential equations,
and more. NumPy has a module called fft that offers fast Fourier transform functionality.
A lot of the functions in this module are paired; this means that, for many functions, there is
a function that does the inverse operation. For instance, the fft and ifft functions form
such a pair.
Time for action – calculating the Fourier transform
First, we will create a signal to transform. In order to calculate the Fourier transform,
perform 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)
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.
print np.all(np.abs(np.fft.ifft(transformed) - wave) < 10 ** -9)
The result is shown as follows:
True
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4.
Plot the transformed signal with Matplotlib.
plot(transformed)
show()
The resulting screenshot shows the fast Fourier transform:
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).
import numpy as np
from matplotlib.pyplot import plot, show
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)
plot(transformed)
show()
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Chapter 6
Shifting
The fftshift function of the numpy.linalg module shifts zero-frequency components to
the center of a spectrum. The ifftshift function reverses this operation.
Time for action – shifting frequencies
We will create a signal, transform it, and then shift the signal. In order to shift the
frequencies, perform 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)
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.
print np.all((np.fft.ifftshift(shifted) - transformed) < 10 ** -9)
The result is shown as follows:
True
5.
Plot the signal and transform it with Matplotlib.
plot(transformed, lw=2)
plot(shifted, lw=3)
show()
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The following screenshot shows the shift in the fast Fourier transform:
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
from matplotlib.pyplot import plot, show
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)
plot(transformed, lw=2)
plot(shifted, lw=3)
show()
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Chapter 6
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.
Pseudo random numbers are random enough for most intents and purposes, except for
some very special cases. The functions related to random numbers can be found in the
NumPy random module. The core random number generator is based on the Mersenne
Twister algorithm. Random numbers can be generated 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.
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.
Imagine a 17th-century gambling house where you can bet on flipping of 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 1000 coins in our possession. We will use the
binomial function from the random module for that purpose.
In order to understand the binomial function, go through the following steps:
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 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.
0 9
3.
Plot the cash array with Matplotlib.
plot(np.arange(len(cash)), cash)
show()
As you can see in the following screenshot, our cash balance performs
a random walk:
What just happened?
We did a random walk experiment using the binomial function from the NumPy random
module (see headortail.py).
import numpy as np
from matplotlib.pyplot import plot, show
cash = np.zeros(10000)
cash[0] = 1000
outcome = np.random.binomial(9, 0.5, size=len(cash))
for i in range(1, len(cash)):
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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()
plot(np.arange(len(cash)), cash)
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. 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, there is
one ball in there that is bad. Every time it is pulled out, the contestants lose six points. If
however, they manage to get out three of the 25 normal balls, they get one point. So, what
is going to happen if we have 100 questions in total? In order to get a solution for this, go
through the following steps:
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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3.
Plot the points array with Matplotlib.
plot(np.arange(len(points)), points)
show()
The following screenshot 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 are pulled
out of a jar in each session (see urn.py).
import numpy as np
from matplotlib.pyplot import plot, show
points = np.zeros(100)
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:
print outcomes[i]
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plot(np.arange(len(points)), points)
show()
Continuous distributions
Continuous distributions are modeled by the probability density functions (pdf).
The probability for a certain interval is determined by integration of the probability
density function. The NumPy random module has a number of 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
Random numbers can be generated from a normal distribution and their distribution may be
visualized with a histogram. To draw a normal distribution, perform 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: Draw the histogram and theoretical pdf
with a center value of 0 and standard deviation of 1. We will use Matplotlib for
this purpose.
dummy, bins, dummy = 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 screenshot, 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
normal_values = np.random.normal(size=N)
dummy, bins, dummy = 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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Chapter 6
Lognormal distribution
A lognormal distribution is a distribution of a 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 probability density function with
a histogram. Perform the following steps:
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: Draw the histogram and theoretical pdf
with a center value of 0 and standard deviation of 1. We will use Matplotlib for
this purpose.
dummy, bins, dummy = 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()
The fit of the histogram and theoretical pdf is excellent, as you can see in the
following screenshot:
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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 probability density
function and a histogram of randomly generated values (see lognormaldist.py).
import numpy as np
import matplotlib.pyplot as plt
N=10000
lognormal_values = np.random.lognormal(size=N)
dummy, bins, dummy = 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(-(np.log(x) - mu)**2 / (2 * sigma**2))/ (x * sigma *
np.sqrt(2 * np.pi))
plt.plot(x, pdf,lw=3)
plt.show()
Summary
We 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 shall cover specialized routines. These are functions that you
probably would not use often, but are very useful when you do need them.
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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, as follows:
The sort function returns a sorted array
The lexsort function performs sorting with a list of keys
The argsort function returns the indices that would sort an array
The ndarray class has a sort method that performs 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
From this list argsort and sort are available as methods on NumPy arrays as well.
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Peeking into Special Routines
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. Perform the following steps:
1.
Now for something completely different, let's go back to Chapter 3, Get to Terms
with Commonly Used Functions. In that chapter we used stock price data of AAPL.
This is by now pretty old data. We will load the close prices and the always complex
dates. In fact, we will need 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 we will now 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:
['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']
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).
import numpy as np
import datetime
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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, 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.
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.
Time for action – sorting complex numbers
We will create an array of complex numbers and sort it. Perform the following steps to do so:
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 shown as follows:
Sorted
[ 0.39342751+0.34955771j
0.41516850+0.26221878j
0.86631422+0.74612422j
0.40597665+0.77477433j
0.92293095+0.81335691j]
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What just happened?
We generated random complex numbers and sorted them using the sort_complex
function (see sortcomplex.py).
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
Searching
NumPy has several functions that can search through arrays, as follows:
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.
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Chapter 7
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 could be inserted to maintain the sort order. It uses 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 allows us to get the index of a value in a sorted array, where
it could be inserted so that the array remains sorted. An example should make this clear.
Perform the following steps:
1.
To demonstrate we will need an array that is sorted. Create an array with arange,
which of course is sorted.
a = np.arange(5)
2.
It's time to call the searchsorted function.
indices = np.searchsorted(a, [-2, 7])
print "Indices", indices
The following are the indices which should maintain the sort order:
Indices [0 5]
3.
Let's 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]
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What just happened?
The searchsorted function gave us indices 5 and 0 for 7 and -2. With these indices,
we would make the array [-2, 0, 1, 2, 3, 4, 7]—so the array remains sorted
(see sortedsearch.py).
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, Get to Terms
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 from an array. Perform the following steps to do so:
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
3.
Extract the even elements based on our condition with the extract function.
print "Even numbers", np.extract(condition, a)
This gives us the even numbers, as required:
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, as follows:
Non zero (array([1, 2, 3, 4, 5, 6]),)
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Chapter 7
What just happened?
We extracted the even elements from an array based on a Boolean condition with the
NumPy extract function (see extracted.py).
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, as follows:
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. 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.
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 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. In this tutorial, let's take an interest
rate of three percent, quarterly payments of 10 for 5 years and present value of 1,000.
Call the fv function with the appropriate values to calculate the future value.
print "Future value", np.fv(0.03/4, 5 * 4, -10, -1000)
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The future value is as follows:
Future value 1376.09633204
This corresponds with saving for 10 years, with quarterly additional savings of 10 at an
interest rate of three percent. If we vary the number of years and if we save and keep the
other parameters constant, we will get following plot:
What just happened?
We calculated the future value using the NumPy fv function starting with a present value of
1,000; interest rate of three percent; and quarterly payments of 10 for 5 years. We plotted
the future value for various saving periods (see futurevalue.py).
import numpy as np
from matplotlib.pyplot import plot, show
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))
plot(fvals, 'bo')
show()
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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.
Time for action – getting the present value
Let's reverse—compute the present value with numbers from the previous tutorial.
Plug in the figures from the Time for action – determining future value tutorial to calculate
the present value.
print "Present value", np.pv(0.03/4, 5 * 4, -10, 1376.09633204)
This gives us 1,000 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 previous Time for action tutorial 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.
Time for action – calculating the net present value
We will calculate the net present value for a randomly generated cash flow series. Perform
the following steps to do so:
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
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The cash flows would be shown 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 three percent.
print "Net present value", np.npv(0.03, cashflows)
The net present value would be shown as follows:
Net present value 107.435682443
What just happened?
We computed the net present value from a randomly generated cash flow series with the
NumPy npv function (see netpresentvalue.py).
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 interest rate, which does not take into
account inflation. The NumPy irr function returns the internal rate of return for
a given cash flow series.
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 tutorial.
Call the irr function with the cash flow series from the previous Time for action tutorial.
print "Internal rate of return", np.irr([-100, 38, 48, 90,
17, 36])
The internal rate of return would be shown as follows:
Internal rate of return 0.373420226888
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Chapter 7
What just happened?
We calculated the internal rate of return from the cash flow series of the previous Time for
action tutorial. 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 1 million with interest rate of 10 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 values mentioned previously.
print "Payment", np.pmt(0.01/12, 12 * 30, 10000000)
The monthly payment would be shown as follows:
Payment -32163.9520447
What just happened?
We calculated the monthly payment for a loan of 1 million at an annual rate of 10 percent.
Given that we have 30 years to repay the loan, the pmt function tells us that we need to pay
32,163.9520447 per month.
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 9,000 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)
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The number of payments would be shown as follows:
Number of payments 167.047511801
What just happened?
We determined the number of payments needed to pay off a loan of 9,000 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, and the present value and 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 tutorial and reverse compute the interest rate from the other parameters.
Fill in the numbers from the previous Time for action tutorial.
print "Interest rate", 12 * np.rate(167, -100, 9000, 0)
The interest rate is approximately 10 percent, as expected.
Interest rate 0.0999756420664
What just happened?
We used the NumPy rate function and the values from the previous Time for action tutorial
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 such as bartlett,
blackman, hamming, hanning, and kaiser. An example of the hanning function can be
found in Chapter 4, Convenience Functions for Your Convenience and Chapter 3, Get to Terms
with Commonly Used Functions.
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Time for action – plotting the Bartlett window
The Bartlett window is a triangular smoothing window. Perform the following steps to plot
the Bartlett window:
1.
Call the NumPy bartlett function to calculate the Bartlett window.
window = np.bartlett(42)
2.
Plot the Bartlett window with Matplotlib, which is very easy.
plot(window)
show()
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 formed by summing the first three terms of cosines, as follows:
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The NumPy blackman function returns the Blackman window. The only parameter is the
number of points in the output window. If this number is 0 or less than 0, an empty array
is returned.
Time for action – smoothing stock prices with the Blackman
window
Let's smooth the close prices from the small AAPL stock prices data file. Perform the
following steps to do so:
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 = int(sys.argv[1])
window = np.blackman(N)
smoothed = np.convolve(window/window.sum(),
closes, mode='same')
2.
Plot the smoothed prices with Matplotlib. We will omit the first five and the last
five data points in this example. The reason for this is that there is a strong
boundary effect.
plot(smoothed[N:-N], lw=2, label="smoothed")
plot(closes[N:-N], label="closes")
legend(loc='best')
show()
The closing prices of AAPL smoothed with the Blackman window should appear,
as follows:
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Chapter 7
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
from matplotlib.pyplot import plot, show, legend
from matplotlib.dates import datestr2num
import sys
closes=np.loadtxt('AAPL.csv', delimiter=',', usecols=(6,),
converters={1:datestr2num}, unpack=True)
N = int(sys.argv[1])
window = np.blackman(N)
smoothed = np.convolve(window/window.sum(), closes, mode='same')
plot(smoothed[N:-N], lw=2, label="smoothed")
plot(closes[N:-N], label="closes")
legend(loc='best')
show()
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Hamming window
The Hamming window is formed by a weighted cosine. The formula is as follows:
The NumPy hamming function returns the Hamming window. The only parameter is the
number of points 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. Perform the following steps to do so:
1.
Call the NumPy hamming function to calculate the Hamming window.
window = np.hamming(42)
2.
Plot the window with Matplotlib.
plot(window)
show()
The Hamming window plot is shown as follows:
What just happened?
We plotted the Hamming window with the NumPy hamming function.
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Kaiser window
The Kaiser window is formed by the Bessel function. The formula is as follows:
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, an empty array is returned. The second parameter is the beta.
Time for action – plotting the Kaiser window
Let's plot the Kaiser window. Perform the following steps to do so:
1.
Call the NumPy kaiser function to calculate the Kaiser window.
window = np.kaiser(42, 14)
2.
Plot the window with Matplotlib.
plot(window)
show()
The Kaiser window would appear as follows:
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What just happened?
We plotted the Hamming window with the NumPy kaiser function.
Special mathematical functions
We will end this chapter with some special mathematical functions. Bessel functions are
solutions of the Bessel differential equations (visit http://en.wikipedia.org/wiki/
Bessel_function). 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 visit http://en.wikipedia.org/wiki/Sinc_function.
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 to calculate the function values.
vals = np.i0(x)
3.
Plot the modified Bessel function with Matplotlib.
plot(x, vals)
show()
The modified Bessel function would have the following output:
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Chapter 7
What just happened?
We plotted the modified Bessel function of the first kind 0th order with the
NumPy i0 function.
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. Perform the following steps to do so:
1.
Compute evenly spaced values with the NumPy linspace function.
x = np.linspace(0, 4, 100)
2.
Call the NumPy sinc function to compute the function values.
vals = np.sinc(x)
3.
Plot the sinc function with Matplotlib.
plot(x, vals)
show()
The sinc function would have the following output:
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The sinc2d function requires a two-dimensional array. We can create it with the
outer function resulting in the following plot:
What just happened?
We plotted the well-known sinc function with the NumPy sinc function
(see plot_sinc.py).
import numpy as np
from matplotlib.pyplot import plot, show
x = np.linspace(0, 4, 100)
vals = np.sinc(x)
plot(x, vals)
show()
We did the same for two dimensions (see sinc2d.py).
import numpy as np
from matplotlib.pyplot import imshow, show
x = np.linspace(0, 4, 100)
xx = np.outer(x, x)
vals = np.sinc(xx)
imshow(vals)
show()
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Chapter 7
Summary
This was a special chapter covering some of the more special NumPy topics. We covered
sorting and searching, special functions, financial utilities, and window functions.
The next chapter will be about the very important subject of testing.
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8
Assure Quality with Testing
Some programmers test only in production. If you are not one of them 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. Only a small unit of
code is tested by each test. The benefits are increased confidence in the quality
of the code, reproducible tests, and as a side effect, more clear 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 as much as possible of the code. The next time the code is changed we can
run the tests and catch potential regressions. In other words functionality already present
will still work.
This chapter's topics include:
Unit testing
Asserts
Floating point precision
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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
floating-point 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:
Function
assert_almost_equal
Description
assert_approx_equal
Raises an exception if two numbers are not equal up to a
certain significance
assert_array_almost_equal
Raises an exception if two arrays are not equal up to a
specified precision
assert_array_equal
Raises an exception if two arrays are not equal
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
assert_equal
Raises an exception if two objects are not equal
assert_raises
Fails if a specified exception is not raised by a callable
invoked with defined arguments
assert_warns
Fails if a specified warning is not thrown
assert_string_equal
Asserts that two strings are equal
assert_allclose
Raise an assertion if two objects are not equal up to
desired tolerance
Raises an exception if two numbers are not equal up to a
specified precision
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 seven 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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2.
Call the function with higher precision (up to eight decimal places):
print "Decimal 7", np.testing.assert_almost_equal(0.123456789,
0.123456780, decimal=8)
The result is:
Decimal 7
Traceback (most recent call last):
…
raiseAssertionError(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 precision.
Pop quiz – specifying decimal precision
Q1. Which parameter of the assert_almost_equal function specifies the
decimal precision?
1.
2.
3.
4.
decimal
precision
tolerance
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 result is an exception that is triggered
by the condition:
abs(actual - expected) >= 10**-(significant - 1)
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Time for action – asserting approximately equal
Let's take the numbers from the previous Time for action tutorial 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:
Significance 8 None
2.
Call the function with high significance:
print "Significance 9",
np.testing.assert_approx_equal
(0.123456789, 0.123456780, significant=9)
An exception is thrown:
Significance 9
Traceback (most recent call last):
...
raiseAssertionError(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 precision.
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:
|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 tutorial by adding a 0 to
each array:
1.
Calling 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:
Decimal 8 None
2.
Calling the function with higher precision:
print "Decimal 9", np.testing.assert_array_almost_equal([0,
0.123456789], [0, 0.123456780], decimal=9)
An exception is thrown:
Decimal 9
Traceback (most recent call last):
…
assert_array_compare
raiseAssertionError(msg)
AssertionError:
Arrays are not almost equal
(mismatch 50.0%)
x: array([ 0.
y: array([ 0.
,
0.12345679])
,
0.12345678])
What just happened?
We compared two arrays with the NumPy array_almost_equal function
Have a go hero – comparing array with different shapes
Use the NumPy array_almost_equal function to compare two arrays with
different shapes.
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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 atol (absolute tolerance) and rtol (relative
tolerance). For two arrays a and b, these parameters satisfy the 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 tutorial 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:
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])
An exception is thrown:
Fail
Traceback (most recent call last):
…
assert_array_compare
raiseAssertionError(msg)
AssertionError:
Arrays are not equal
(mismatch 50.0%)
x: array([ 0.
y: array([ 0.
,
0.12345679,
,
0.12345678,
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nan])
nan])
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:
Pass None
2.
Call the assert_array_less function on failing the test:
print "Fail", np.testing.assert_array_less([0, 0.123456789,
np.nan], [0, 0.123456780, np.nan])
An exception is thrown:
Fail
Traceback (most recent call last):
...
raiseAssertionError(msg)
AssertionError:
Arrays are not less-ordered
(mismatch 100.0%)
x: array([ 0.
y: array([ 0.
,
0.12345679,
,
0.12345678,
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nan])
nan])
Assure Quality with Testing
What just happened?
We checked the ordering of two arrays with the assert_array_less function.
Objects 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:
1.
Call the assert_equal function:
print "Equal?", np.testing.assert_equal((1, 2), (1, 3))
An exception is thrown:
Equal?
Traceback (most recent call last):
...
raiseAssertionError(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 an
exception is thrown and the difference between the strings is shown. The case of the string
characters matters.
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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")
An exception is thrown:
Fail
Traceback (most recent call last):
…
raiseAssertionError(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.
ULP stands for Unit of Least Precision 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 millimetres, but beyond
that you can only estimate half millimetres.
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Machine epsilon is the largest relative rounding error in floating point arithmetic. Machine
epsilon is equal to ULP relative to one. 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:
EPS 2.22044604925e-16
2.
Compare two almost equal floats: Compare 1.0 with 1 + epsilon (eps) using the
assert_almost_equal_nulp function. Do the same for 1 + 2 * epsilon (eps):
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:
1 None
2
Traceback (most recent call last):
…
assert_array_almost_equal_nulp
raiseAssertionError(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 (eps) 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 tutorial, 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:
EPS 2.22044604925e-16
2.
Do the comparisons as done in the previous Time for action tutorial, 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:
1 1.0
2 2.0
What just happened?
We compared the same values as the previous Time for action tutorial, 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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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 for abnormal conditions.
1.
We start by writing the factorial function
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 is using the arange and cumprod functions we have already seen 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. We test for calling the factorial function with:
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))
We rigged one of the tests to fail as you can see in the following output:
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Chapter 8
$ 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
raiseValueError, "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 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):
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#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))
if __name__ == '__main__':
unittest.main()
Nose tests 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. The numpy.testing
module has a number of decorators:
Decorator
numpy.testing.decorators.
deprecated
Description
numpy.testing.decorators.
knownfailureif
Raises KnownFailureTest exception
based on a condition.
numpy.testing.decorators.
setastest
Marks a function as being a test or not being
a test.
numpy.testing.decorators.skipif
Raises SkipTest exception based on a
condition.
numpy.testing.decorators.slow
Labels test functions or methods as slow.
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 we will install nose in case you don't have it yet.
1.
Install nose with setuptools
easy_install nose
Or pip:
pip install nose
2.
We will 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.
We can 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.
We will 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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6.
Let's disable the second test method from the previous step.
decorate_methods(TestClass2, setastest(False), 'test_false2')
7.
We can 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
raiseKnownFailureTest(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
from
from
from
numpy.testing.decorators import setastest
numpy.testing.decorators import skipif
numpy.testing.decorators import knownfailureif
numpy.testing import decorate_methods
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@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
Docstrings are strings embedded in Python code that resemble interactive sessions.
These strings can be used to test certain assumptions, or just provide examples.
The numpy.testing module has a function to run these tests.
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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 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).
We will 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.
We will write the following line of NumPy code to compute the factorial:
return np.arange(1, n+1).cumprod()[-1]
We want this code to fail from time to time for demonstration purposes.
3.
We can 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
raiseAssertionError("Some doctests failed:\n%s" % "\n".join(msg))
AssertionError: Some doctests failed:
******************************************************************
****
File "docstringtest.py", line 10, in docstringtest.factorial
Failed example:
factorial(0)
Exception raised:
Traceback (most recent call last):
File "…/doctest.py", line 1254, in __run
compileflags, 1) in test.globs
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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 run
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
We 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.sourceforge.net/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:
Simple plots
Subplots
Histograms
Plot customization
Three-dimensional plots
Contour plots
Animation
Logplots
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Plotting with Matplotlib
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 will be updated
interactively without waiting for the show function. This is comparable to the way text
output is printed on the fly.
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 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 that 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 xlabel function.
plt.xlabel('x’)
6.
Add a label to the y axis with ylabel function.
plt.ylabel('y(x)’)
7.
Call the show function to display the graph.
plt.show()
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Here is a plot with polynomial coefficients 1, 2, 3, and 4:
What just happened?
We displayed a graph of a polynomial on our screen. We added labels to the x and y axis
(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 1 and 2.
4. It does neither 1, 2, or 3.
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Plotting with Matplotlib
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 derivative
Let’s plot a polynomial and its first order derivative using the derive function with m as 1.
We already did the first part in the previous Time for action tutorial. We want to have two
different line styles to be able 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)
x = np.linspace(-10, 10, 30)
y = func(x)
y1 = func1(x)
2.
Plot the polynomial and its derivative in two different styles: red circles and green
dashes. You cannot see the colors in a print copy of this book so you will have to try
it out for yourself.
plt.plot(x, y, 'ro’, x, y1, 'g--’)
plt.xlabel('x’)
plt.ylabel('y’)
plt.show()
The graph again with polynomial coefficients 1, 2, 3, and 4:
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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))
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. Still, you would like to have
everything grouped together. We can achieve this with the subplot function.
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)
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Plotting with Matplotlib
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, you can combine the three parameters into a single number such as
311. The subplots will be organized in 3 rows and 1 column. Give the subplot the
title "Polynomial". Make a solid red line.
plt.subplot(311)
plt.plot(x, y, 'r-’)
plt.title("Polynomial")
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()
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The three subplots with polynomial coefficients 1, 2, 3, and 4:
What just happened?
We plotted a polynomial and its first and second derivative using three different line styles
and three subplots in 3 rows and 1 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-’)
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Plotting with Matplotlib
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 us monitor our stock investments. The matplotlib.finance
package has utilities with which we can download stock quotes from Yahoo Finance
(http://finance.yahoo.com/). The data can then be plotted 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 will require a connection to Yahoo Finance, which will be the data source.
1.
Determine the start date by subtracting 1 year from today.
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)
2.
We need to create so-called locators. These objects from the matplotlib.dates
package are needed to locate months and days on the x-axis.
alldays = DayLocator()
months = MonthLocator()
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3.
Create a date formatter to format the dates on the x axis. This formatter will create 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)
11.
Format the labels on the x axis as dates. This should rotate the labels on the x axis,
so that they fit better.
fig.autofmt_xdate()
plt.show()
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Plotting with Matplotlib
The candlestick chart for DISH (Dish Network Corp.) would appear 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)
alldays = DayLocator()
months = MonthLocator()
month_formatter = DateFormatter("%b %Y")
symbol = 'DISH’
if len(sys.argv) == 2:
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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 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 1 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]
3.
Draw the histogram with a reasonable number of bars.
plt.hist(close, np.sqrt(len(close)))
plt.show()
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Plotting with Matplotlib
The histogram for DISH would appear as follows:
What just happened?
We charted the stock price distribution of DISH as 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]
quotes = quotes_historical_yahoo(symbol, start, today)
quotes = np.array(quotes)
close = quotes.T[4]
plt.hist(close, np.sqrt(len(close)))
plt.show()
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Have a go hero – drawing a bell curve
Overlay a bell curve (related to 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
axis 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, create the figure, and add to it a subplot. We already went through these steps in
the previous Time for action tutorial, so we will skip them here.
1.
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 tutorial as well. The stock volume using a
logarithmic scale for DISH would appear as follows:
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Plotting with Matplotlib
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]
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
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Scatter plots
A scatter plot displays values for two numerical variables in the same data set.
The Matplotlib scatter function creates a scatter plot. Optionally, we can specify
the color and size of the data points in the plot as well as alpha transparency.
Time for action – plotting price and volume returns with 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 = pyplot.figure()
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)
pyplot.show()
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Plotting with Matplotlib
The scatter plot for DISH will appear 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
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]
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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 region of a plot with a specified color. We can also
choose an 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 the 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 1 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)
3.
Plot the closing price.
ax.plot(dates, close)
4.
Shade the regions of the plot below the closing price using different colors
depending 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=close"))
4.
Add a legend and let Matplotlib decide where to put it.
leg = ax.legend(loc=’best’, fancybox=True)
5.
Make the legend transparent by setting the alpha channel value
leg.get_frame().set_alpha(0.5)
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The stock price and moving averages with legend and annotations would appear
as follows:
What just happened?
We plotted the close price of a stock and three of its exponential moving averages.
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)
symbol = 'DISH’
if len(sys.argv) == 2:
symbol = sys.argv[1]
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Plotting with Matplotlib
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)
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 3D plots, we need an Axes3D object associated with a 3d projection.
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Time for action – plotting in three dimensions
We will plot in three dimensions a simple three-dimensional function:
1.
We need to 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, we will use the meshgrid function.
This will be used 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" on the surface. The choice for
colormap is a matter of taste.
ax.plot_surface(x, y, z,
cmap=cm.YlGnBu_r)
rstride=4, cstride=4,
The result is the following 3D plot:
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Plotting with Matplotlib
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()
Contour plots
Matplotlib contour 3D plots come in two flavors—filled and unfilled. We can create normal
contour plots with the contour function. For the filled contour plots we can 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. 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 we need this
line of code:
ax.contourf(x, y, z)
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Chapter 9
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
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()
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Plotting with Matplotlib
Animation
Matplotlib offers fancy animation capabilities. Matplotlib has 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.
We will plot 3 random datasets as circles, dots and triangles in different colors.
circles, triangles, dots = ax.plot(x, 'ro’, y, 'g^’, z, 'b.’)
2.
This function will get called to update the screen regularly. We will 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.
We will generate random data with NumPy.
def generate():
while True: yield np.random.rand(2, N)
Here is a snapshot of the animation in action:
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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
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, 3D plots, contour plots, and logplots. We also saw
a few examples of displaying stock charts. Obviously, we only scratched the surface and 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/johnhunter/). The memorial fund set up by the NumFocus Foundation
is an opportunity for us, as fans of John Hunter’s work, to "give back" so to say. Again, for
more details, check out the previous link to the NumFocus website.
The next chapter is about SciPy—a scientific Python framework that is built on top of NumPy.
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10
When NumPy is Not
Enough – SciPy and Beyond
SciPy is the world famous Python open-source scientific computing library
built on top of NumPy. It adds functionality 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.
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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 the said arrays within a MATLAB or Octave
environment, the easiest thing to do is create a .mat file. We then can load the file within
MATLAB or Octave. Let’s go through the necessary steps:
1.
Create a NumPy array and call savemat to create a .mat file. This function has two
parameters – a filename 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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Pop quiz – loading .mat files
Q1. Which function loads .mat files?
1. Loadmatlab
2. loadmat
3. loadoct
4. 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 can be found that can perform a great number of statistical tests.
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. Perform the following steps
to do so:
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 us the mean
and standard deviation of the data set.
print “Mean”, “Std”, stats.norm.fit(generated)
The mean and standard deviation would be shown as follows:
Mean Std (0.0071293257063200707, 0.95537708218972528)
3.
Skewness tells us how skewed (asymmetric) a probability distribution is. Perform
a skewness test. This test returns two values. The second value is the p-value; the
probability that the skewness of the data set corresponds to a normal distribution.
The pvalue instances range from 0 to 1.
print “Skewtest”, “pvalue”, stats.skewtest(generated)
The result of the skewness test would be shown as follows:
Skewtest pvalue (-0.62120640688766893, 0.5344638245033837)
So there is a 53 percent chance that we are dealing with a normal distribution.
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4.
Kurtosis tells us how “curved” a probability distribution is. Perform a kurtosis
test. This test is set up in a similar way as the skewness test, but of course,
applies to kurtosis.
print “Kurtosistest”, “pvalue”,
stats.kurtosistest(generated)
The result of the kurtosis test would be shown as follows:
Kurtosistest pvalue (1.3065381019536981, 0.19136963054975586)
5.
A normality test tells us how likely it is that a data set complies to the normal
distribution. Perform a normality test. This test also returns two values,
of which the second is the p-value
print “Normaltest”, “pvalue”, stats.normaltest(generated)
The result of the normality test would be shown as follows:
Normaltest pvalue (2.09293921181506, 0.35117535059841687)
6.
We can easily find the value at a certain percentile with SciPy.
print “95 percentile”,
stats.scoreatpercentile(generated, 95)
The value at the 95th percentile would be shown 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 would be shown as follows:
Percentile at 1 85.5555555556
8.
Plot the generated values in a histogram with Matplotlib. More information about
Matplotlib can be found in the previous chapter.
plt.hist(generated)
plt.show()
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The following is the histogram of the generated random values:
What just happened?
We created a data set from a normal distribution and analyzed it with the scipy.stats
module (see statistics.py).
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.hist(generated)
plt.show()
Have a go hero – improving the data generation
Judging from the histogram in the Time for action – analyzing random values section, there
is still room for improvement when it comes to generating the data. Try using NumPy or
different parameters of the scipy.stats.norm.rvs function.
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Samples’ comparison and SciKits
Often we will have two data samples, maybe from different experiments, that are somehow
related. Statistical tests exist that can compare the samples. Some of these have been
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 the following command:
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, we can retrieve quotes from Yahoo! Finance. We will
compare the log returns of the close price of DIA and SPY. Also we will perform the JarqueBera test on the difference of the log returns. Perform the following steps to do so:
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. The log returns are calculated by taking the
natural logarithm of the close price and then taking the difference of consecutive
values.
spy =
dia =
3.
np.diff(np.log(get_close(“SPY”)))
np.diff(np.log(get_close(“DIA”)))
The means comparison test checks whether two different samples could have the
same mean value. Two values are returned, of which the second is a p-value from
0 to 1.
print “Means comparison”, stats.ttest_ind(spy, dia)
The result of the means comparison test would be shown as follows:
Means comparison (-0.017995865641886155, 0.98564930169871368)
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So there is about a 98 percent chance that the two samples have the same mean
log return.
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 would be shown 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()
The histograms of the log returns and difference are shown in the
following screenshot:
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What just happened?
We compared samples of log returns for DIA and SPY. We also performed the Jarque-Bera
test on the difference of the log returns (see pair.py).
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”)))
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.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()
Signal processing
The scipy.signal module contains filter functions and B-spline interpolation algorithms.
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Spline interpolation uses a polynomial called a spline for 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. Still we can
get the trend back easily after detrending. Let’s do that for 1 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]
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, ‘-’)
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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 screenshot shows the QQQ prices with a trend line:
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)
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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.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.show()
Fourier analysis
Signals in the real world often have a periodic nature. A commonly used tool to deal with
these signals is the Fourier transform. The 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.
The 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 names as their MATLAB counterparts and similar functions as their
MATLAB equivalents.
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Time for action – filtering a detrended signal
We learned in how to detrend a signal in the Time for action – detecting a trend in QQQ
section. 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 tutorial, such as
downloading the data and setting up Matplotlib objects. These steps are omitted here.
1.
Apply Fourier transforms, which will give 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’})
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()
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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
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]
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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.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’})
ax2 = fig.add_subplot(212)
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.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. Several optimization algorithms are provided by the scipy.optimize
module. One of the algorithms is a least squares fitting function, leastsq. When calling this
function, we are required to provide a residuals (error terms) function. This function is used
to minimize the sum of the squares of the residuals. It corresponds to our mathematical
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model for the solution. Also, it is 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 800 iterations.
Time for action – fitting to a sine
In the Time for action – filtering a detrended signal 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. Perform the following
steps to fit to a sine:
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))
3.
Guess the values of the parameters for 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 would be shown 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 following are the final parameter values:
P [ 2.67678014e+00
-5.01260321e-03]
2.73033206e-03
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5.
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”)
The following shows the resulting charts:
What just happened?
We detrended 1 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 matplotlib.finance import quotes_historical_yahoo
import numpy as np
import matplotlib.pyplot as plt
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from
from
from
from
from
from
scipy import fftpack
scipy import signal
matplotlib.dates import DateFormatter
matplotlib.dates import DayLocator
matplotlib.dates import MonthLocator
scipy import optimize
start = (2010, 7, 25)
end = (2011, 7, 25)
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)
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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.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’})
ax2 = fig.add_subplot(212)
ax2.tick_params(axis=’both’, which=’major’, labelsize=’x-large’)
plt.plot(f, amps, label=”transformed”)
plt.legend(prop={‘size’:’x-large’})
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 as erf in mathematics),
but has no finite limits. It evaluates to the square root of pi. Let’s calculate the integral with
the quad function.
Calculate the Gaussian integral with the quad function.
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 shown 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.
Interpolation
Interpolation “fills in the blanks” between known data points in a data set. 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 them. After
that, we will do a linear and cubic interpolation, and plot the results. Perform the following
steps to do so:
1.
Create the data points and add noise to them.
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”)
plt.plot(x2, y2, ‘-’, lw=2, label=”cubic”)
plt.legend()
plt.show()
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The following screenshot is a plot of the data, linear, and cubic interpolation:
What just happened?
We created a data set 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)
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.legend()
plt.show()
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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
In the scipy.misc module, there 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 on 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 signal 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)
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()
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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
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”)
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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 be not 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 tutorial. We will further use the tile function to replay the audio clip several times.
Perform the following steps to do so:
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 tutorial we are only
interested in the audio data.
2.
Apply the tile function.
repeated = np.tile(data, int(sys.argv[1]))
3.
Write a new file with the write function.
wavfile.write(“repeated_yababy.wav”,
sample_rate, repeated)
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The original audio data and the audio clip repeated four times are shown 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 scipy.io import wavfile
import matplotlib.pyplot as plt
import urllib2
import numpy as np
import sys
response = urllib2.urlopen(‘http://www.thesoundarchive.com/
austinpowers/smashingbaby.wav’)
print response.info()
WAV_FILE = ‘smashingbaby.wav’
filehandle = open(WAV_FILE, ‘w’)
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”)
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plt.plot(data)
plt.subplot(2, 1, 2)
# Repeat the audio fragment
repeated = np.tile(data, int(sys.argv[1]))
# Plot the audio data
plt.title(“Repeated”)
plt.plot(repeated)
wavfile.write(“repeated_yababy.wav”,
sample_rate, repeated)
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, and
audio and image processing.
In the next chapter we will create some simple, yet fun, games with Pygame – the
open-source Python game library. During this 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 with NumPy and
Pygame quickly and easily. Basic game development experience would help but
isn't necessary.
In this chapter we will cover the following topics:
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 General Public 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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Playing with Pygame
Time for action – installing Pygame
We will install Pygame in this tutorial. 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. Perform the following steps to install Pygame:
1.
Depending on the operating system, you have the following options with which you
install Pygame:
Debian and Ubuntu: Pygame can be found in the Debian archives
at http://packages.qa.debian.org/p/pygame.html.
Windows: From the Pygame website (http://www.pygame.org/
download.shtml) we can download the appropriate binary installer
for the Python version we are using.
2.
Mac: Binary Pygame packages for Mac OS X 10.3 and up can be found
at http://www.pygame.org/download.shtml.
Pygame uses 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:
python setup.py help
3.
In order to compile the code, you need to have 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. More information about compiling Pygame on Mac OS X can be
found at http://pygame.org/wiki/MacCompile.
Hello World
We will create a simple game that we will further improve later in this chapter. As is
traditional in books about programming, we will 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. Perform the
following steps to create a simple game:
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1.
First import the required Pygame modules. If Pygame is installed properly, we should
get no errors, otherwise please return to the installation recipe.
import pygame, sys
from pygame.locals import *
2.
We will initialize Pygame, create a display of 400 x 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 we will only set a label with the text Hello World at
coordinates (100, 100). The text has font size of 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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The 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 might not seem like much, but we learned a lot in this tutorial. The functions that passed
the review are summarized in the following table:
Function
pygame.init()
Description
pygame.display.set_mode((400,
300))
This 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 sets the window title to a specified string
value.
pygame.font.SysFont("None", 19)
This creates a system font from a commaseparated list of fonts (in this case. none) and a
font size parameter.
sysFont.render('Hello World',
0, (255, 100, 100))
This draws text on a Surface object. The
last parameter is a tuple representing the RGB
values of a color.
This function does initialization and needs to be
called before other Pygame functions are called.
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Function
screen.blit(rendered, (100,
100))
pygame.event.get()
Description
This draws on a Surface object.
This gets a list of Event objects. The Event
objects represent some special occurrence in
the system, such as a user quitting the game.
pygame.quit()
This cleans up resources used by Pygame. Call
this function before exiting the game.
pygame.display.update()
This refreshes the surface.
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 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.
Perform the following steps to do so:
1.
We can 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. We will load this image and move it around on the screen.
img = pygame.image.load('head.jpg')
3.
We will define some arrays to hold the coordinates of the positions where we would
like to put the image during the animation. Since the object will be moved, there are
four logical sections of the path – right, down, left, and up. Each of these sections
will have 40 equidistant steps. We will initialize all the values in these 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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4.
It's trivial to set the coordinates of the positions of the image. However, there is one
tricky bit to be aware of – 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.
The path sections can be joined, but before we can do this, the arrays have to
be transposed 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 we will set the clock tick at a rate of 30 frames per second:
clock.tick(30)
The following is a screenshot of the moving head:
You should be able to watch a movie of this animation at https://www.youtube.
com/watch?v=m2TagGiq1fs.
The code of this example uses almost everything we learned so far, but should still
be simple enough to understand:
import pygame, sys
from pygame.locals import *
import numpy as np
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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:
pygame.quit()
sys.exit()
pygame.display.update()
clock.tick(30)
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What just happened?
We learned a bit about animation in this tutorial. The most important concept we learned
about, is about the clock. The new functions that we used are described in the following table:
Function
pygame.time.Clock()
Description
clock.tick(30)
This 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 that 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 will take the position coordinates of the previous tutorial and make a graph
from them. Perform the following steps to do so:
1.
Using a noninteractive backend: In order to integrate Matplotlib with Pygame we
need to use a noninteractive backend, otherwise Matplotlib will present us with
a GUI window by default. We will import the main Matplotlib module and call the
use function. This function has to be called immediately after importing the main
Matplotlib module and before other Matplotlib modules are imported.
import matplotlib as mpl
mpl.use("Agg")
2.
Noninteractive plots can be drawn on a Matplotlib canvas. Creating this canvas
requires imports, creating a figure and a subplot. We will specify the figure to be 3 x
3 inches large. More details can be found at the end of this section.
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)
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3.
In noninteractive 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. The plot is eventually drawn on the canvas. The canvas adds a bit
of complexity to our setup. At the end of this example you can find a 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 on YouTube at https://www.youtube.com/watch?v=t6qTeXxtnl4.
4.
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")
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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
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))
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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 plotting-related functions are explained in the following table:
Function
mpl.use("Agg")
Description
plt.figure(figsize=[3, 3])
This creates a figure of 3 x 3 inches.
agg.FigureCanvasAgg(fig)
This creates a canvas in noninteractive mode.
canvas.draw()
This draws on the canvas.
canvas.get_renderer()
This gets a renderer for the canvas.
This specifies the use of the noninteractive backend.
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.
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Time for action – accessing surface pixel data with NumPy
In this tutorial we will tile a small image to fill the game screen. Perform the following steps
to do so:
1.
The array2d function copies pixels into a two-dimensional array. There is a similar
function for three-dimensional arrays. We will copy the pixels from the avatar image
into an array:
pixels = pygame.surfarray.array2d(img)
2.
Let's create the game screen from the shape of the pixels array using the shape
attribute of the array. The screen will be 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 tile function. The data needs to be
converted to integer values, since colors are defined 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)
This produces the following screenshot:
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The following code does 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?
The following is a brief description of the new functions and attributes we used:
Function
pygame.surfarray.array2d(img)
Description
pygame.surfarray.blit_
array(screen, new_pixels)
This displays array values on the screen.
This copies pixel data into a 2D array.
Artificial intelligence
Often we need to mimic intelligent behavior within a game. The scikit-learn project
aims to provide an API for machine learning. What I like the most about it is the amazing
documentation. We can install scikit-learn with the package manager of our operating
system. This option may or may not be available depending on the operating system, but
should be the most convenient route. Windows users can just download an installer from the
project website.
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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 in the following command at the command line:
pip install -U scikit-learn
Or you can also do it with the following command:
easy_install -U scikit-learn
This might not work because of permissions, so you may need to put sudo in front of the
commands or log in as an admin.
Time for action – clustering points
We will generate some random points and cluster them, which means that points that are
close to each other are put in the same cluster. This is only 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. Second, we will calculate a square affinity
matrix. An affinity matrix is a matrix containing affinity values; for instance, distances
between points. Finally, we will cluster the points with the AffinityPropagation class
from scikit-learn. Perform the following steps to cluster points:
1.
We will generate 30 random point positions within a square of 400 x 400 pixels:
positions = np.random.randint(0, 400, size=(30, 2))
2.
We will use the Euclidean distance to the origin as affinity matrix.
positions_norms = np.sum(positions ** 2, axis=1)
S = - positions_norms[:, np.newaxis] - positions_norms[np.newaxis,
:] + 2 * np.dot(positions, positions.T)
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.
We will 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])
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Chapter 11
The result is a bunch of polygons for each cluster, as shown in the
following screenshot:
The clustering example code is as follows:
import numpy as np
import sklearn.cluster
import pygame, sys
from pygame.locals import *
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)
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 i in xrange(len(labels)):
polygon_points[labels[i]].append(positions[i])
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Playing with Pygame
pygame.init()
screen = pygame.display.set_mode((400, 400))
while True:
for i in xrange(len(polygon_points)):
pygame.draw.polygon(screen, (255, 0, 0), polygon_points[i])
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), polygon_points[i])
This draws a polygon given a surface,
a color (red in this case), and a list of
points.
This creates an
AffinityPropagation object and
performs a fit using an affinity matrix.
OpenGL and Pygame
OpenGL specifies an API for 2D and 3D 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:
pip install PyOpenGL PyOpenGL_accelerate
You might need to have root access to execute this command. The following is the
corresponding easy_install command:
easy_install PyOpenGL PyOpenGL_accelerate
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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. Perform the following steps to draw the Sierpinski gasket:
1.
First, we will 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 that we draw points halfway between the previous point
and one of the vertices picked at random. Finally, we "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]
for i in xrange(NPOINTS):
glBegin(GL_POINTS)
point = (point + vertices
[indices[i]])/2.0
glVertex2fv(point)
glEnd()
glFlush()
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The Sierpinski triangle looks like the following screenshot:
The following is the full Sierpinski gasket demo code with all the imports:
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)
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):
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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 clears the buffers using a mask. Here
we clear the color buffer and depth buffer
bits.
gluOrtho2D(0, w, 0, h)
This defines a 2D orthographic projection
matrix with the coordinates of the left,
right, top, and bottom clipping planes.
glColor3f(1.0, 0, 0)
This defines the current drawing color using
three float values for RGB. In this case we
will be painting in red.
glBegin(GL_POINTS)
This delimits the vertices of primitives or a
group of primitives. Here the primitives are
points.
glVertex2fv(point)
This renders a point given a vertex.
glEnd()
This closes a section of code started with
glBegin.
glFlush()
This forces execution of GL commands.
This sets the display mode to the required
width, height, and OpenGL display.
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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
two-dimensional square grid. Each cell in the grid can be either dead or alive. This
state depends on the eight neighbors of the cell. Convolution can be used to evaluate
the basic rules of the game. We will need the SciPy package for the convolution process.
Time for action – simulating life
The following code is an implementation of Game of Life with some modifications,
as follows:
Clicking once with the mouse draws a cross until we click again
Pressing the r key resets the grid to a random state
Pressing b creates blocks based on the mouse position
Pressing g creates gliders
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. More information about the involved
functions can be found in the next section.
1.
To evaluate the rules, we will use 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.
We can draw a cross using 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))
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The following is the code in its entirety:
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
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
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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()
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()
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Chapter 11
You should able to view a screencast on YouTube at https://www.youtube.com/
watch?v=NNsU-yWTkXM. The following is a screenshot of the game in action:
What just happened?
We used some NumPy and SciPy functions that need an explanation, as follows:
Function
ndimage.convolve(arr, weights,
mode='wrap')
Description
bools.astype(int)
This converts the array of Booleans to integers.
np.arange(0, pos[0], 10)
This 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 applies the convolve operation on the
given array, using weights in wrap mode. The
mode has to do it with the array borders.
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Playing with Pygame
Summary
You might have found the mention of Pygame in this book a bit odd. 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. They also
require artificial intelligence capabilities as found in scikit-learn. Anyway, making
games is fun and we hope this last chapter was the equivalent of a nice dessert or coffee
after a ten-course meal. If you are still hungry for more, please check out NumPy Cookbook,
Ivan Idris, Packt Publishing; it builds further on this book with minimum overlap.
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Pop Quiz Answers
Chapter 1, NumPy Quick Start
What does arrange(5) do?
It creates a NumPy array with values 0 to 4.
The created NumPy array has values 0, 1, 2, 3, 4.
Chapter 2, Beginning with NumPy Fundamentals
How is the shape of an ndarray stored?
It is stored in a tuple.
Chapter 3, Get into Terms with Commonly Used Functions
Which function returns the weighted average of an array?
average
Chapter 4, Convenience functions for your convenience
Which function returns the covariance of two arrays?
cov
Chapter 5, Working with Matrices and ufuncs
What is the row delimiter in a string accepted by the mat
and bmat functions?
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Semicolon
Pop Quiz Answers
Chapter 6, Move further with NumPy modules
mat
Which function can create matrices?
Chapter 7, Peeking into special routines
random
Which NumPy module deals with random numbers?
Chapter 8, Assure Quality with Testing
decimal
Which parameter of the assert_almost_equal
function specifies the decimal precision?
Chapter 9, Plotting with Matplotlib
What does the plot function do?
It does neither 1, 2, or 3.
Chapter 10, When NumPy is not enough Scipy and beyond
loadmat
Which function loads .mat files?
[ 276 ]
www.it-ebooks.info
Symbols
.mat file
loading 226, 227
saving 226
% operator 121
A
accumulate method
applying, on add function 117
AffinityPropagation class 264
agg.FigureCanvasAgg() function 261
AI
about 263
points, clustering 264, 266
almost equal arrays
asserting 178
AND operator 130
annotate function 215
apply_along_axis function 66
approximately equal arrays
asserting 180
arange function 28, 29, 97, 160
argmax function 158
argmin function 64, 158
argsort function 155
argwhere function 159
arithmetic functions
about 118
array division 119, 120
array attributes
about 45
dtype 45
flat 47
imag 47
itemsize 46
ndim 45
real 47
shape 45
size 46
T attribute 46
arrays
comparing 182
converting 48
ordering 183
arrays almost equal
asserting 181
array shapes
manipulating 38
array shapes, manipulating
flatten function 38
ravel function 38
reshape function 39
resize method 39
transpose matrices 39
arrays, NumPy
about 17
splitting 43
stacking 39
arrays spiltting
about 43
depth-wise splitting 44
horizontal splitting 43
vertical splitting 44
arrays stacking
column stacking 42
depth stacking 41
horizontal stacking 40
row stacking 42
www.it-ebooks.info
Index
vertical stacking 41
assert_allclose function 178
assert_almost_equal function
about 178
using 178
assert_approx_equal function
about 178
using 179
assert_array_almost_equal function
about 178
using 180
assert_array_almost_equal_nulp function
using 186
assert_array_equal function
about 178
using 182
assert_array_less function
about 178
using 183
assert_array_max_ulp function
about 187
using 187
assert_equal function
about 178
using 184
assert functions
about 178
assert_allclose 178
assert_almost_equal 178
assert_approx_equal 178
assert_array_almost_equal 178
assert_array_equal 178
assert_array_less 178
assert_equal 178
assert_raises 178
assert_string_equal 178
assert_warns 178
assert_raises function 178
assert_string_equal function
about 178
using 184, 185
assert_warns function 178
astype function 48
audio clips
replaying 247, 248
audio processing
about 247
audio clips, replaying 247, 248
average true range (ATR)
about 69
calculating 69-71
B
bartlett function 109, 167
Bartlett window
about 167
plotting 167
binomial distribution models 147
binomial function
using 147, 148
bits
twiddling 129, 130
bitwise_and function 130
Bitwise-ANDing 130
bitwise functions 129
bitwise_xor function 129
blackman function 109, 168
Blackman window
about 167
plotting 168, 169
Bollinger bands
about 76
enveloping with 76-78
bools.astype() function 273
C
calc_profit function 102
canvas.draw() function 261
canvas.get_renderer() function 261
character codes 32
clip method 87
clock object, Pygame
animating 255, 256
column_stack function 42
column stacking 42
comma-separated values. See CSV files
comparison functions 129
complex numbers
about 157
sorting 157, 158
compress method 87
concatenate function 40
[ 278 ]
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consecutive wins and losses
analyzing 105
continuous distributions 151
contour function 220
contour plots
about 220
filled contour plot, drawing 220
convolution 72
convolve function 73
correlation
about 92
correlated pairs, trading 92-95
CPython 9
CSV files
about 52
dealing with 53
loading from 53
cumprod method 88
detrend function 233
diff function 59, 100
discrete Fourier transform (DFT) 143
DISH (Dish Network Corp.) 206
divide function 119
docstrings
about 193
doctests, executing 194
doctests
executing 194
documentation website, NumPy and SciPy
URL 25
dsplit function 44
dstack function 41
dtype attribute 34, 45
dtype constructors 33
E
D
data
summarizing weekly 65-68
data sorting routines
AAPL stock prices, sorting lexically 156
argsort function 155
lexsort function 155
msort function 155
sort_complex function 155
sort function 155
sort method 155
data type objects 32
dates
dealing with 61-64
Debian and Ubuntu
NumPy, installing 14
Python, installing 10
decorate_methods function
calling 190
depth stacking 41
depth-wise splitting 44
determinant, of matrix
about 142
calculating 142
detrended signal
filtering 236, 237
easy_install command 266
Eigenvalues
about 137
determining 137, 138
Eigenvectors
about 137
determining 137, 138
elements
extracting, from array 160, 161
error function 242
exponential moving average
calculating 74, 75
extract function 159, 160
F
factorial
calculating 88
fast Fourier transform (FFT)
about 143
calculating 143, 144
fftshift function 145
Fibonacci numbers
about 122
computing 122, 123
file I/O
files, reading and writing 52
[ 279 ]
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fill_between function
about 213
using 213
financial functions 161
future value, determining 161
fv 161
irr 161
mirr 161
nper 161
npv 161
pmt 161
pv 161
rate 161
flat attribute 47
floating-point comparisons
about 185
assert_array_almost_equal_nulp function,
using 185
floats
comapring, maxulp of 2 used 187
floor_divide function 119
fmod function 121
Fourier analysis
about 235
detrended signal, filtering 236, 237
frequencies
shifting 145, 146
fv function 161
G
Game of Life
implementing 270, 273
Gaussian integral
calculating 242
Gentoo
NumPy, installing 13
glBegin() function 269
glClear() function 269
glColor3f() function 269
glEnd() function 269
glFlush() function 269
gluOrtho2D() function 269
glVertex2fv() function 269
H
hamming function 109, 170
Hamming window
about 170
plotting 170
hanning function 105
Hello World example 252
hist function 207
histograms
about 207
stock price distributions, charting 207, 208
horizontal splitting 43
horizontal stacking 40
hstack function 40
hypergeometric distribution
about 149
game show, simulating 149, 150
I
imag attribute 47
image processing
about 245
image processingLena, manipulating 245, 249
Lena image, manipulating 245, 246
installation, Python
on Debian and Ubuntu 10
on Mac 10
on Windows 10
interest rate
calculating 166
internal rate of return
about 164
determining 164
interp1d class 243, 244
interp2d class 243
interpolation
about 243
in one dimension 243, 244
IPython
about 21
features 21
installation instructions 21
installing, on Linux 13
[ 280 ]
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installing, on Mac OS X 14
installing, on Windows 13
installing, with MacPorts or Fink 17
online resources 25
packages, importing 22-24
pylab mode 25
Pylab switch 22
IRC channel 26
irr function 161
isreal function 108
itemsize attribute 46
K
kaiser function 109, 171
Kaiser window
about 171
plotting 171
L
leastsq function 239
left_shift universal function 130
legend function 215
legends and annotations
about 215
using 215, 217
Lena image
manipulating 245, 246
lexsort function
about 155
using 156
linear algebra
about 133
matrices, inverting 133-135
linear model
price, predicting with 80, 81
linear systems
solving 135, 136
linspace function 124
about 74
Linux
IPython, installing 13
Matplotlib, installing 13
NumPy, installing 13
SciPy, installing 13
Lissajous curves
about 123
drawing 124
loadmat function 225
loadtxt function 53, 62
logarithmic plots
about 209
stock volume, plotting 209
log function 60
lognormal distribution
about 153
drawing 153
lstsq function 81
M
Mac
Python, installing 10
Mac OS X
IPython, installing 14
Matplotlib, installing 14
NumPy, installing 14-16
SciPy, installing 14
Mandriva
NumPy, installing 13
mathematical optimization
about 238
sine, fitting to filtered signal 239, 240
Matlab
Matlababout 225
MATLAB 225
Matplotlib
about 197, 258
contour plots 220
fill_between function 213
finance 204
histograms 207
installing, on Linux 13
installing, on Mac OS X 14
installing, on Windows 13
installing, with MacPorts or Fink 17
legend and annotations 215
logarithmic plots 209
plot format string 200
plots, animating 222
scatter plots 211
simple plots 198
[ 281 ]
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subplots 201
three dimensional plots 218
using, in Pygame 258, 259
matplotlib.pyplot package 198
matrices
about 111
creating 112, 113
matrix
creating, from matrices 113, 114
matrix function 122
max function 56
mean function 54, 58
median function 58
Mersenne Twister algorithm 147
meshgrid function 219
min function 56
mirr function 161
mod function 121
modified Bessel function
about 172
plotting 172, 173
modulo operation
about 121
computing 121
Moore-Penrose pseudoinverse 141
mpl.use() function 261
msort function 57, 155
multidimensional arrays
indexing 36, 37
slicing 35
multidimensional NumPy array
creating 29
N
nanargmax function 158
nanargmin function 158
ndarray 28
ndarray methods
about 86
clip method 87
compress method 87
ndimage.convolve() function 273
ndim attribute 45
net present value
about 163
calculating 163
nonzero function 160
normal distribution
drawing 151, 152
nose tests decorators
about 190
numpy.testing.decorators.deprecated 190
numpy.testing.decorators.knownfailureif 190
numpy.testing.decorators.setastest 190
numpy.testing.decorators.skipif 190
numpy.testing.decorators.slow 190
np.arange() function 273
nper function 161
npv function 161
number of periodic payments
determining 165
numerical integration
about 242
Gaussian integral, calculating 242
NumPy
about 9
approximately equal arrays, asserting 180
arithmetic functions 118
array order, checking 183
arrays 17
arrays almost equal, asserting 181
assert functions 178
ATR calculation 69
bitwise functions 129
Blackman window 167
Bollinger bands 76
character codes 32
comparison functions 129
complex numbers, sorting 157
continuous distributions 151
correlation 92
CSV files 52
data sorting routines 155
data, summarizing weekly 65
data type objects 32
dates, dealing with 61
determinants, calculating 142
docstrings 193
dtype attributes 34
dtype constructors 33
Eigenvalues, finding 137
Eigenvectors, finding 137
elements, extracting from array 160
[ 282 ]
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elements, selecting 30
equal arrays, asserting 182
exponential moving average, calculating 74
factorial, calculating 87
Fast Fourier transform, calculating 143
file I/O 51
floating point comparisons 185
floats, comparing with ULPs 187
frequencies, shifting 145
Hamming window 170
hypergeometric distribution 149
installing, on Debian or Ubuntu 14
installing, on Gentoo 13
installing, on Linux 13
installing, on Mac OS X 14, 15
installing, on Mandriva 13
installing, on Windows 10-12
installing, with MacPorts or Fink 17
interest rate, calculating 166
internal rate of return, determining 164
Kaiser window 171
linear algebra 133
linear model 80
linear systems, solving 136
Lissajous curves 123
lognormal distribution 153
matrices 111
modulo operation 121
ndarray methods 86
net present value 163
nose tests decorators 190
number of periodic payments, determining 165
numerical types 30
objects, comparing 184
on-balance volume 99
one-dimensional slicing 35
periodic payments, calculating 165
polynomials 96
present value 163
pseudoinverse, calculating 141
random numbers 147
searching 158
simple moving average, computing 72
simulation 102
sinc function 173
singular value decomposition 139
smoothing 105
source code, retrieving 17
special mathematical functions 172
square waves 125
statistics, performing 56
stock returns, analyzing 59
strings, comparing 185
trend line 82
unit tests 187
universal functions 114
value range, finding 55
vectors, adding 18, 20
VWAP, calculating 53
window functions 166
NumPy and SciPy forum
URL 25
NumPy array object
about 28
multi-dimensional array object 28
NumPy division functions
divide function 119
floor_divide function 120
true_divide function 119
numpy.linalg package 133
NumPy numerical types
about 30, 32
bool 31
complex64 31
complex128 31
float16 31
float32 31
float64 31
int8 31
int16 31
int32 31
int64 31
inti 31
uint8 31
uint16 31
uint32 31
uint64 31
NumPy reference
URL 25
NumPy wiki documentation
URL 25
[ 283 ]
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O
objects
comparing 184
Octave 225
on-balance volume
computing 99
one-dimensional slicing 35
optimization
about 238
sine, fitting to 239, 240
outer method
applying, on add function 118
P
periodic payments
calculating 165
piecewise function 100
plot format string
about 200
polynomial and derivative, plotting 200, 201
plot regions
shading, based on condition 213
plots
animating 222, 223
plt.figure() function 261
pmt function 161
polyder function 97
polyfit function 96, 98
polynomial function
plotting 198, 199
polynomials
about 96
fitting to 96-98
polysub function 108
polyval function 96
present value
about 163
computing 163
probability density functions (pdf) 151
prod function 88
pseudoinverse 141
pseudoinverse, of matrix
computing 141
Pseudo random numbers 147
pv function 161
Pygame
about 251
AI 263
animation 255
clock object 255
for Debian and Ubuntu 252
for Mac 252
for Windows 252
game, simulating 270
Hello World example 252
installing 252
Matplotlib, using 258
surface pixel data, accessing 261
pygame.display.set_caption() function 254
pygame.display.set_mode() function 254
pygame.display.set_mode((w,h) function 269
pygame.display.update() function 255
pygame.draw.polygon(screen, (255, 0, 0), polygon_points[i]) function 266
pygame.event.get() function 255
pygame.font.SysFont() function 254
pygame.init() function 254
pygame.OPENGL|pygame.DOUBLEBUF) function
269
pygame.quit() function 255
pygame.surfarray.array2d() function 263
pygame.surfarray.blit_array() function 263
Pygame surfarray module 261
pylab mode, IPython 25
PyOpenGL
about 266
installing 266
Sierpinski gasket, drawing 267, 268
Python
about 9
installing, on Debian and Ubuntu 10
installing, on Mac 10
installing, on Windows 10
Q
quad function 242, 243
R
random numbers 147
rate function 161
[ 284 ]
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real attribute 47
Real random numbers 147
record data type
creating 34
reduceat method
applying, on add function 117
reduce method
applying, on add function 116
remainder function 121
reshape function 38
rint function 122
row_stack function 42
row stacking 42
rundocs function 195
S
sample comparison
stock log returns, comparing 230, 231
savemat function 225
savetxt function 52, 67
sawtooth and triangle waves
about 127
drawing 127, 128
formula 127
scatter function 211
scatter plots
about 211
price and volume returns, plotting 211
scikit-learn project 263
SciKits 230
scikits.statsmodels.stattools 230
SciPy
about 225
audio processing 247
Fourier analysis 235
image processing 245
installing, on Linux 13
installing, on Mac OS X 14
installing, on Windows 13
installing, with MacPorts or Fink 17
interpolation 243
mathematical optimization 238
MATLAB or Octave matrices, loading 226
numerical integration 242
SciPyscipy.stats 227
signal processing 232
statistics 227
stock log returns, comparing 230
SciPy channel 26
scipy.fftpack module 235, 237
scipy.interpolate function 243
scipy.interpolate module 244
scipy.io package 225
scipy.io.wavfile module 247
scipy.ndimage module 246
scipy.optimize module 238, 240
SciPy signal
about 233
trend, detecting in QQQ 233, 234
scipy.signal module 232
statistics module
about 227
random values, analyzing 227-229
scipy.stats
about 227
data generation, improving 229
random values, analyzing 227-229
scipy.stats.norm.rvs function 229
screen.blit() function 255
sctypeDict.keys() 33
SDL 251
searching, through arrays
argmax function 158
argmin function 158
argwhere function 159
extract function 159
nanargmax function 158
nanargmin function 158
searchsorted function 159
searchsorted function
about 159
using 159, 160
setastest decorator
applying, to methods 191, 192
applying, to test functions 191, 192
shape attribute 45
Sierpinski gasket
drawing 267, 268
signal processing
about 232
trend detecting, in QQQ 233
sign function 100
[ 285 ]
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Simple DirectMedia Layer. See SDL
simple game
creating 252, 253
simple moving average
about 72
computing 72, 73
simple plots
about 198
polynomial function, plotting 198, 199
simulation
about 102
loops, avoiding with vectorize 102, 103
sinc function 244
about 173
plotting 173, 174
sin function 124
singular value decomposition
about 139
matrix, decomposing 139, 140
size attribute 46
sklearn.cluster.AffinityPropagation().fit(S)
function 266
smoothing
hanning function, used 105-107
smoothing variations 109
sort_complex function 155
sort function 155
special mathematical functions
about 172
Bessel function 172
split function 44, 66
sqrt function 60
square waves
about 125
drawing 125, 126
formula 125
representing 125
Stack Overflow software development forum
URL 25
statistics
about 56
simple statistics, performing 57, 58
std function 59
stock log returns
comparing 230, 232
stock log returns, comparing
histograms plotting, Matplotlib used 231
Jarque Bera test 231
Kolmogorov Smirnov test 231
log returns, calculating 230
quotes, downloading 230
stock quotes
plotting 204-206
stock returns
analyzing 59, 60
stock volume
plotting 209, 210
strings
comparing 185
strip_zeroes function 108
subplot function 202
subplots
about 201
polynomial and its derivatives, plotting 201,
203
summarize function 66
surface pixel data
accessing, with NumPy 262, 263
sysFont.render() function 254
T
take function 63
T attribute 46
Test driven development (TDD) 177
three-by-three matrix
creating 29
three-dimensional plots
about 218
plotting 219, 220
Time-weighted average price. See TWAP
trend
detecting, in QQQ 233, 234
trend detecting, in QQQ
date, formatter 233
diagram 234
locators, creating 233
signal, detrending 233
X axis labels 234
trend line
about 82
drawing 82- 85
[ 286 ]
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true_divide function 119
TWAP
about 54
calculating 54
U
Unit of Least Precision (ULP)
comparing 185
unit tests
about 178, 187
writing 188, 189
universal function methods
accumulate 116
applying, on add function 116, 117
out 116
reduce 116
reduceat 116
universal functions
about 114
creating 115
methods 116
usecols parameter 53
V
ValueError 116
value range
about 55
highest value, finding 55
lowest value, finding 56
variance 58
vectorize function 102
vectors, NumPy
adding 18, 20
vertical splitting 44
vertical stacking 41
volume
about 99
balancing 100, 101
Volume-weighted average price. See VWAP
vsplit function 44
vstack function 41
VWAP
about 53
calculating 54
W
where function 60
window functions
about 166
bartlett 166
Bartlett window, plotting 167
blackman 166
hamming 166
hanning 166
kaiser 166
Windows
IPython, installing 13
Matplotlib, installing 13
NumPy, installing 10, 11, 12
Python, installing 10
SciPy, installing 13
write function 247
X
XOR operator 129
Y
Yahoo Finance
URL 204
[ 287 ]
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