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Introduction to Machine Learning with Python
by Andreas C. Mueller and Sarah Guido
Copyright © 2016 Sarah Guido, Andreas Mueller. All rights reserved.
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Andreas C. Mueller and Sarah Guido
Boston
Machine Learning with Python
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Table of Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Why machine learning? 9
Problems that machine learning can solve 10
Knowing your data 13
Why Python? 13
What this book will cover 13
What this book will not cover 14
Scikit-learn 14
Installing Scikit-learn 15
Essential Libraries and Tools 16
Python2 versus Python3 19
Versions Used in this Book 19
A First Application: Classifying iris species 20
Meet the data 22
Measuring Success: Training and testing data 24
First things first: Look at your data 25
Building your first model: k nearest neighbors 27
Making predictions 28
Evaluating the model 29
Summary 30
2. Supervised Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Classification and Regression 33
Generalization, Overfitting and Underfitting 35
Supervised Machine Learning Algorithms 37
k-Nearest Neighbor 42
k-Neighbors Classification 42
Analyzing KNeighborsClassifier 45
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k-Neighbors Regression 47
Analyzing k nearest neighbors regression 50
Strengths, weaknesses and parameters 51
Linear models 51
Linear models for regression 51
Linear Regression aka Ordinary Least Squares 53
Ridge regression 55
Lasso 57
Linear models for Classification 60
Linear Models for multiclass classification 66
Strengths, weaknesses and parameters 69
Naive Bayes Classifiers 70
Strengths, weaknesses and parameters 71
Decision trees 71
Building Decision Trees 73
Controlling complexity of Decision Trees 76
Analyzing Decision Trees 77
Feature Importance in trees 78
Strengths, weaknesses and parameters 81
Ensembles of Decision Trees 82
Random Forests 82
Gradient Boosted Regression Trees (Gradient Boosting Machines) 88
Kernelized Support Vector Machines 91
Linear Models and Non-linear Features 92
The Kernel Trick 96
Understanding SVMs 97
Tuning SVM parameters 98
Preprocessing Data for SVMs 101
Strengths, weaknesses and parameters 102
Neural Networks (Deep Learning) 102
The Neural Network Model 103
Tuning Neural Networks 106
Strengths, weaknesses and parameters 115
Uncertainty estimates from classifiers 116
The Decision Function 117
Predicting probabilities 119
Uncertainty in multi-class classification 121
Summary and Outlook 123
3. Unsupervised Learning and Preprocessing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Types of unsupervised learning 127
Challenges in unsupervised learning 128
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Preprocessing and Scaling 128
Different kinds of preprocessing 129
Applying data transformations 130
Scaling training and test data the same way 132
The effect of preprocessing on supervised learning 134
Dimensionality Reduction, Feature Extraction and Manifold Learning 135
Principal Component Analysis (PCA) 135
Non-Negative Matrix Factorization (NMF) 152
Manifold learning with t-SNE 157
Clustering 162
k-Means clustering 162
Agglomerative Clustering 173
DBSCAN 178
Summary of Clustering Methods 194
Summary and Outlook 195
4. Summary of scikit-learn methods and usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
The Estimator Interface 197
Fit resets a model 198
Method chaining 199
Shortcuts and efficient alternatives 200
Important Attributes 200
Summary and outlook 201
5. Representing Data and Engineering Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Categorical Variables 204
One-Hot-Encoding (Dummy variables) 205
Binning, Discretization, Linear Models and Trees 210
Interactions and Polynomials 215
Univariate Non-linear transformations 222
Automatic Feature Selection 225
Univariate statistics 225
Model-based Feature Selection 227
Iterative feature selection 229
Utilizing Expert Knowledge 230
Summary and outlook 237
6. Model evaluation and improvement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Cross-validation 240
Cross-validation in scikit-learn 241
Benefits of cross-validation 241
Stratified K-Fold cross-validation and other strategies 242
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More control over cross-validation 244
Leave-One-Out cross-validation 245
Shuffle-Split cross-validation 245
Cross-validation with groups 246
Grid Search 247
Simple Grid-Search 248
The danger of overfitting the parameters and the validation set 249
Grid-search with cross-validation 251
Analyzing the result of cross-validation 255
Using different cross-validation strategies with grid-search 259
Nested cross-validation 260
Parallelizing cross-validation and grid-search 261
Evaluation Metrics and scoring 262
Keep the end-goal in mind 262
Metrics for binary classification 263
Multi-class classification 285
Regression metrics 288
Using evaluation metrics in model selection 288
Summary and outlook 290
7. Algorithm Chains and Pipelines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Parameter Selection with Preprocessing 294
Building Pipelines 295
Using Pipelines in Grid-searches 296
The General Pipeline Interface 299
Convenient Pipeline creation with make_pipeline 300
Grid-searching preprocessing steps and model parameters 304
Summary and Outlook 306
8. Working with Text Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Types of data represented as strings 307
Example application: Sentiment analysis of movie reviews 309
Representing text data as Bag of Words 311
Bag-of-word for movie reviews 314
Stop-words 317
Rescaling the data with TFIDF 318
Investigating model coefficients 321
Bag of words with more than one word (n-grams) 322
Advanced tokenization, stemming and lemmatization 326
Topic Modeling and Document Clustering 329
Summary and Outlook 337
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CHAPTER 1
Introduction
Machine learning is about extracting knowledge from data. It is a research field at the
intersection of statistics, artificial intelligence and computer science, which is also
known as predictive analytics or statistical learning. The application of machine
learning methods has in recent years become ubiquitous in everyday life. From auto‐
matic recommendations of which movies to watch, to what food to order or which
products to buy, to personalized online radio and recognizing your friends in your
photos, many modern websites and devices have machine learning algorithms at their
core.
When you look at at complex websites like Facebook, Amazon or Netflix, it is very
likely that every part of the website you are looking at contains multiple machine
learning models.
Outside of commercial applications, machine learning has had a tremendous influ‐
ence on the way data driven research is done today. The tools introduced in this book
have been applied to diverse scientific questions such as understanding stars, finding
distant planets, analyzing DNA sequences, and providing personalized cancer treat‐
ments.
Your application doesn’t need to be as large-scale or world-changing as these exam‐
ples in order to benefit from machine learning. In this chapter, we will explain why
machine learning became so popular, and dicuss what kind of problem can be solved
using machine learning. Then, we will show you how to build your first machine
learning model, introducing important concepts on the way.
Why machine learning?
In the early days of “intelligent” applications, many systems used hand-coded rules of
if” and “else” decisions to process data or adjust to user input. Think of a spam filter
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whose job is to move an email to a spam folder. You could make up a black-list of
words that would result in an email marked as spam. This would be an example of
using an expert designed rule system to design an “intelligent” application. Designing
kind of manual design of decision rules is feasible for some applications, in particular
for those applications in which humans have a good understanding of how a decision
should be made. However, using hand-coded rules to make decisions has two major
disadvantages:
1. The logic required to make a decision is specific to a single domain and task.
Changing the task even slightly might require a rewrite of the whole system.
2. Designing rules requires a deep understanding of how a decision should be made
by a human expert.
One example of where this hand-coded approach will fail is in detecting faces in
images. Today every smart phone can detect a face in an image. However, face detec‐
tion was an unsolved problem until as recent as 2001. The main problem is that the
way in which pixels (which make up an image in a computer) are “perceived by” the
computer is very different from how humans perceive a face. This difference in repre‐
sentation makes it basically impossible for a human to come up with a good set of
rules to describe what constitutes a face in a digital image.
Using machine learning, however, simply presenting a program with a large collec‐
tion of images of faces is enough for an algorithm to determine what characteristics
are needed to identify a face.
Problems that machine learning can solve
The most successful kind of machine learning algorithms are those that automate a
decision making processes by generalizing from known examples. In this setting,
which is known as a supervised learning setting, the user provides the algorithm with
pairs of inputs and desired outputs, and the algorithm finds a way to produce the
desired output given an input.
In particular, the algorithm is able to create an output for an input it has never seen
before without any help from a human.
Going back to our example of spam classification, using machine learning, the user
provides the algorithm a large number of emails (which are the input), together with
the information about whether any of these emails are spam (which is the desired
output). Given a new email, the algorithm will then produce a prediction as to
whether or not the new email is spam.
Machine learning algorithms that learn from input-output pairs are called supervised
learning algorithms because a “teacher” provides supervision to the algorithm in the
form of the desired outputs for each example that they learn from.
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While creating a dataset of inputs and outputs is often a laborious manual process,
supervised learning algorithms are well-understood and their performance is easy to
measure. If your application can be formulated as a supervised learning problem, and
you are able to create a dataset that includes the desired outcome, machine learning
will likely be able to solve your problem.
Examples of supervised machine learning tasks include:
Identifying the ZIP code from handwritten digits on an envelope. Here the
input is a scan of the handwriting, and the desired output is the actual digits in
the zip code. To create a data set for building a machine learning model, you need
to collect many envelopes. Then you can read the zip codes yourself and store the
digits as your desired outcomes.
Determining whether or not a tumor is benign based on a medical image.
Here the input is the image, and the output is whether or not the tumor is
benign. To create a data set for building a model, you need a database of medical
images. You also need an expert opinion, so a doctor needs to look at all of the
images and decide which tumors are benign and which are not.
Detecting fraudulent activity in credit card transactions. Here the input is a
record of the credit card transaction, and the output is whether it is likely to be
fraudulent or not. Assuming that you are the entity distributing the credit cards,
collecting a dataset means storing all transactions, and recording if a user reports
any transaction as fraudulent.
An interesting thing to note about the three examples above is that although the
inputs and outputs look fairly straight-forward, the data collection process for these
three tasks is vastly different.
While reading envelopes is laborious, it is easy and cheap. Obtaining medical imaging
and expert opinions on the other hand not only requires expensive machinery but
also rare and expensive expert knowledge, not to mention ethical concerns and pri‐
vacy issues. In the example of detecting credit card fraud, data collection is much
simpler. Your customers will provide you with the desired output, as they will report
fraud. All you have to do to obtain the input output pairs of fraudulent and non-
fraudulent activity is wait.
The other type of algorithms that we will cover in this book is unsupervised algo‐
rithms. In unsupervised learning, only the input data is known and there is no known
output data given to the algorithm. While there are many successful applications of
these methods as well, they are usually harder to understand and evaluate.
Examples of unsupervised learning include:
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Identifying topics in a set of blog posts. If you have a large collection of text
data, you might want to summarize it and find prevalent themes in it. You might
not know beforehand what these topics are, or how many topics there might be.
Therefore, there are no known outputs.
Segmenting customers into groups with similar preferences. Given a set of cus‐
tomer records, you might want to identify which customers are similar, and
whether there are groups of customers with similar preferences. For a shopping
site these might be “parents, “bookworms” or “gamers. Since you dont know in
advanced what these groups might be, or even how many there are, you have no
known outputs.
Detecting abnormal access patterns to a website. To identify abuse or bugs, it is
often helpful to find access patterns that are different from the norm. Each
abnormal pattern might be very different, and you might not have any recorded
instances of abnormal behavior. Since in this example you only observe traffic,
and you dont know what constitutes normal and abnormal behavior, this is an
unsupervised problem.
For both supervised and unsupervised learning tasks, it is important to have a repre‐
sentation of your input data that a computer can understand. Often it is helpful to
think of your data as a table. Each data point that you want to reason about (each
email, each customer, each transaction) is a row, and each property that describes that
data point (say the age of a customer, the amount or location of a transaction) is a
column.
You might describe users by their age, their gender, when they created an account and
how often they bought from your online shop. You might describe the image of a
tumor by the gray-scale values of each pixel, or maybe by using the size, shape and
color of the tumor to describe it.
Each entity or row here is known as data point or sample in machine learning, while
the columns, the properties that describe these entities, are called features.
We will later go into more detail on the topic of building a good representation of
your data, which is called feature extraction or feature engineering. You should keep
in mind however that no machine learning algorithm will be able to make a predic‐
tion on data for which it has no information. For example, if the only feature that you
have for a patient is their last name, no algorithm will be able to predict their gender.
This information is simply not contained in your data. If you add another feature that
contains their first name, you will have much better luck, as it is often possible to tell
the gender by a persons first name.
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Knowing your data
Quite possibly the most important part in the machine learning process is under‐
standing the data you are working with. It will not be effective to randomly choose an
algorithm and throw your data at it. It is necessary to understand what is going on in
your dataset before you begin building a model. Each algorithm is different in terms
of what data it works best for, what kinds data it can handle, what kind of data it is
optimized for, and so on. Before you start building a model, it is important to know
the answers to most of, if not all of, the following questions:
How much data do I have? Do I need more?
How many features do I have? Do I have too many? Do I have too few?
Is there missing data? Should I discard the rows with missing data or handle
them differently?
What question(s) am I trying to answer? Do I think the data collected can answer
that question?
The last bullet point is the most important question, and certainly is not easy to
answer. Thinking about these questions will help drive your analysis.
Keeping these basics in mind as we move through the book will prove helpful,
because while scikit-learn is a fairly easy tool to use, it is geared more towards those
with domain knowledge in machine learning.
Why Python?
Python has become the lingua franca for many data science applications. It combines
the powers of general purpose programming languages with the ease of use of
domain specific scripting languages like matlab or R.
Python has libraries for data loading, visualization, statistics, natural language pro‐
cessing, image processing, and more. This vast toolbox provides data scientists with a
large array of general and special purpose functionality.
As a general purpose programming language, Python also allows for the creation of
complex graphic user interfaces (GUIs), web services and for integration into existing
systems.
What this book will cover
In this book, we will focus on applying machine learning algorithms for the purpose
of solving practical problems. We will focus on how to write applications using the
machine learning library scikit-learn for the Python programming language. Impor‐
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tant aspects that we will cover include formulating tasks as machine learning prob‐
lems, preprocessing data for use in machine learning algorithms, and choosing
appropriate algorithms and algorithmic parameters.
We will focus mostly on supervised learning techniques and algorithms, as these are
often the most useful ones in practice, and they are easy for beginners to use and
understand.
We will also discuss several common types of input, including text data.
What this book will not cover
This book will not cover the mathematical details of machine learning algorithms,
and we will keep the number of formulas that we include to a minimum. In particu‐
lar, we will not assume any familiarity with linear algebra or probability theory. As
mathematics, in particular probability theory, is the foundation upon which machine
learning is build, we will not be able to go into the analysis of the algorithms in great
detail. If you are interested in the mathematics of machine learning algorithms, we
recommend the text book “Elements of Statistical Learning” by Hastie, Tibshirani and
Friedman, which is available for free at the authors website[footnote: http://stat‐
web.stanford.edu/~tibs/ElemStatLearn/]. We will also not describe how to write
machine learning algorithms from scratch, and will instead focus on how to use the
large array of models already implemented in scikit-learn and other libraries.
We will not discuss reinforcement learning, which is about an agent learning from its
interaction with an environment, and we will only briefly touch upon deep learning.
Some of the algorithms that are implemented in scikit-learn but are outside the scope
of this book include Gaussian Processes, which are complex probabilistic models, and
semi-supervised models, which work with supervised information on only some of
the samples.
We will not also explicitly talk about how to work with time-series data, although
many of techniques we discuss are applicable to this kind of data as well. Finally, we
will not discuss how to do machine learning on natural images, as this is beyond the
scope of this book.
Scikit-learn
Scikit-learn is an open-source project, meaning that scikit-learn is free to use and dis‐
tribute, and anyone can easily obtain the source code to see what is going on behind
the scenes. The scikit-learn project is constantly being developed and improved, and
has a very active user community. It contains a number of state-of-the-art machine
learning algorithms, as well as comprehensive documentation about each algorithm
on the website [footnote http://scikit-learn.org/stable/documentation]. Scikit-learn is
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a very popular tool, and the most prominent Python library for machine learning. It
is widely used in industry and academia, and there is a wealth of tutorials and code
snippets about scikit-learn available online. Scikit-learn works well with a number of
other scientific Python tools, which we will discuss later in this chapter.
While studying the book, we recommend that you also browse the scikit-learn user
guide and API documentation for additional details, and many more options to each
algorithm. The online documentation is very thorough, and this book will provide
you with all the prerequisites in machine learning to understand it in detail.
Installing Scikit-learn
Scikit-learn depends on two other Python packages, NumPy and SciPy. For plotting
and interactive development, you should also install matplotlib, IPython and the
Jupyter notebook. We recommend using one of the following pre-packaged Python
distributions, which will provide the necessary packages:
Anaconda (https://store.continuum.io/cshop/anaconda/): a Python distribution
made for large-scale data processing, predictive analytics, and scientific comput‐
ing. Anaconda comes with NumPy, SciPy, matplotlib, IPython, Jupyter note‐
books, and scikit-learn. Anaconda is available on Mac OS X, Windows, and
Linux.
Enthought Canopy (https://www.enthought.com/products/canopy/): another
Python distribution for scientific computing. This comes with NumPy, SciPy,
matplotlib, and IPython, but the free version does not come with scikit-learn. If
you are part of an academic, degree-granting institution, you can request an aca‐
demic license and get free access to the paid subscription version of Enthought
Canopy. Enthought Canopy is available for Python 2.7.x, and works on Mac,
Windows, and Linux.
Python(x,y) (https://code.google.com/p/pythonxy/): a free Python distribution for
scientific computing, specifically for Windows. Python(x,y) comes with NumPy,
SciPy, matplotlib, IPython, and scikit-learn.
If you already have a python installation set up, you can use pip to install any of these
packages.
$ pip install numpy scipy matplotlib ipython scikit-learn
We do not recommended using pip to install NumPy and SciPy on Linux, as it
involves compiling the packages from source. See the scikit-learn website for more
detailed installation.
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Essential Libraries and Tools
Understanding what scikit-learn is and how to use it is important, but there are a few
other libraries that will enhance your experience. Scikit-learn is built on top of the
NumPy and SciPy scientific Python libraries. In addition to knowing about NumPy
and SciPy, we will be using Pandas and matplotlib. We will also introduce the Jupyter
Notebook, which is an browser-based interactive programming environment. Briefly,
here is what you should know about these tools in order to get the most out of scikit-
learn.
If you are unfamiliar with numpy or matplotlib, we recommend reading the first
chapter of the scipy lecture notes[footnote: http://www.scipy-lectures.org/].
Jupyter Notebook
The Jupyter Notebook is an interactive environment for running code in the browser.
It is a great tool for exploratory data analysis and is widely used by data scientists.
While Jupyter Notebook supports many programming languages, we only need the
Python support. The Jypyter Notebook makes it easy to incorporate code, text, and
images, and all of this book was in fact written as an IPython notebook.
All of the code examples we include can be downloaded from github [FIXME add git‐
hub footnote].
NumPy
NumPy is one of the fundamental packages for scientific computing in Python. It
contains functionality for multidimensional arrays, high-level mathematical func‐
tions such as linear algebra operations and the Fourier transform, and pseudo ran‐
dom number generators.
The NumPy array is the fundamental data structure in scikit-learn. Scikit-learn takes
in data in the form of NumPy arrays. Any data you’re using will have to be converted
to a NumPy array. The core functionality of NumPy is this “ndarray”, meaning it has
n dimensions, and all elements of the array must be the same type. A NumPy array
looks like this:
import numpy as np
x = np.array([[1, 2, 3], [4, 5, 6]])
x
array([[1, 2, 3],
[4, 5, 6]])
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SciPy
SciPy is both a collection of functions for scientific computing in python. It provides,
among other functionality, advanced linear algebra routines, mathematical function
optimization, signal processing, special mathematical functions and statistical distri‐
butions. Scikit-learn draws from SciPy’s collection of functions for implementing its
algorithms.
The most important part of scipy for us is scipy.sparse with provides sparse matri‐
ces, which is another representation that is used for data in scikit-learn. Sparse matri‐
ces are used whenever we want to store a 2d array that contains mostly zeros:
from scipy import sparse
# create a 2d numpy array with a diagonal of ones, and zeros everywhere else
eye = np.eye(4)
print("Numpy array:\n%s" % eye)
# convert the numpy array to a scipy sparse matrix in CSR format
# only the non-zero entries are stored
sparse_matrix = sparse.csr_matrix(eye)
print("\nScipy sparse CSR matrix:\n%s" % sparse_matrix)
Numpy array:
[[ 1. 0. 0. 0.]
[ 0. 1. 0. 0.]
[ 0. 0. 1. 0.]
[ 0. 0. 0. 1.]]
Scipy sparse CSR matrix:
(0, 0) 1.0
(1, 1) 1.0
(2, 2) 1.0
(3, 3) 1.0
More details on scipy sparse matrices can be found in the scipy lecture notes.
matplotlib
Matplotlib is the primary scientific plotting library in Python. It provides function for
making publication-quality visualizations such as line charts, histograms, scatter
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plots, and so on. Visualizing your data and any aspects of your analysis can give you
important insights, and we will be using matplotlib for all our visualizations.
%matplotlib inline
import matplotlib.pyplot as plt
# Generate a sequence of integers
x = np.arange(20)
# create a second array using sinus
y = np.sin(x)
# The plot function makes a line chart of one array against another
plt.plot(x, y, marker="x")
Pandas
Pandas is a Python library for data wrangling and analysis. It is built around a data
structure called DataFrame, that is modeled after the R DataFrame. Simply put, a
Pandas Pandas DataFrame is a table, similar to an Excel Spreadsheet. Pandas provides
a great range of methods to modify and operate on this table, in particular it allows
SQL-like queries and joins of tables. Another valuable tool provided by Pandas is its
ability to ingest from a great variety of file formats and databases, like SQL, Excel files
and comma separated value (CSV) files. Going into details about the functionality of
Pandas is out of the scope of this book. However, “Python for Data Analysis” by Wes
McKinney provides a great guide.
Here is a small example of creating a DataFrame using a dictionary:
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import pandas as pd
# create a simple dataset of people
data = {'Name': ["John", "Anna", "Peter", "Linda"],
'Location' : ["New York", "Paris", "Berlin", "London"],
'Age' : [24, 13, 53, 33]
}
data_pandas = pd.DataFrame(data)
data_pandas
Age Location Name
024 New York John
113 Paris Anna
253 Berlin Peter
333 London Linda
Python2 versus Python3
There are two major versions of Python that are widely used at the moment: Python2
(more precisely 2.7) and Python3 (with the latest release being 3.5 at the time of writ‐
ing), which sometimes leads to some confusion. Python2 is no longer actively devel‐
oped, but because Python3 contains major changes, Python2 code does usually not
run without changes on Python3. If you are new to Python, or are starting a new
project from scratch, we highly recommend using the latests version of Python3.
If you have a large code-base that you rely on that is written for Python2, you are
excused from upgrading for now. However, you should try to migrate to Python3 as
soon as possible. Writing any new code, it is for the most part quite easy to write code
that runs under Python2 and Python3 [Footnote: The six package can be very handy
for that].
All the code in this book is written in a way that works for both versions. However,
the exact output might differ slightly under Python2.
Versions Used in this Book
We are using the following versions of the above libraries in this book:
import pandas as pd
print("pandas version: %s" % pd.__version__)
import matplotlib
print("matplotlib version: %s" % matplotlib.__version__)
import numpy as np
print("numpy version: %s" % np.__version__)
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import IPython
print("IPython version: %s" % IPython.__version__)
import sklearn
print("scikit-learn version: %s" % sklearn.__version__)
pandas version: 0.17.1
matplotlib version: 1.5.1
numpy version: 1.10.4
IPython version: 4.1.2
scikit-learn version: 0.18.dev0
While it is not important to match these versions exactly, you should have a version
of scikit-learn that is as least as recent as the one we used.
Now that we have everything set up, lets dive into our first appication of machine
learning.
A First Application: Classifying iris species
In this section, we will go through a simple machine learning application and create
our first model.
In the process, we will introduce some core concepts and nomenclature for machine
learning.
Lets assume that a hobby botanist is interested in distinguishing what the species is of
some iris flowers that she found. She has collected some measurements associated
with the iris: the length and width of the petals, and the length and width of the sepal,
all measured in centimeters.
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She also has the measurements of some irises that have been previously identified by
an expert botanist as belonging to the species Setosa, Versicolor or Virginica. For
these measurements, she can be certain of which species each iris belongs to. Let’s
assume that these are the only species our hobby botanist will encounter in the wild.
Our goal is to build a machine learning model that can learn from the measurements
of these irises whose species is known, so that we can predict the species for a new
iris.
Since we have measurements for which we know the correct species of iris, this is a
supervised learning problem. In this problem, we want to predict one of several
options (the species of iris). This is an example of a classication problem. The possi‐
ble outputs (different species of irises) are called classes.
Since every iris in the dataset belongs to one of three classes this problem is a three-
class classification problem.
The desired output for a single data point (an iris) is the species of this flower. For a
particular data point, the species it belongs to is called its label.
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Meet the data
The data we will use for this example is the iris dataset, a classical dataset in machine
learning an statistics.
It is included in scikit-learn in the dataset module. We can load it by calling the
load_iris function:
from sklearn.datasets import load_iris
iris = load_iris()
The iris object that is returned by load_iris is a Bunch object, which is very similar
to a dictionary. It contains keys and values:
iris.keys()
dict_keys(['DESCR', 'data', 'target_names', 'feature_names', 'target'])
The value to the key DESCR is a short description of the dataset. We show the begin‐
ning of the description here. Feel free to look up the rest yourself.
print(iris['DESCR'][:193] + "\n...")
Iris Plants Database
====================
Notes
-----
Data Set Characteristics:
:Number of Instances: 150 (50 in each of three classes)
:Number of Attributes: 4 numeric, predictive att
...
The value with key target_names is an array of strings, containing the species of
flower that we want to predict:
iris['target_names']
array(['setosa', 'versicolor', 'virginica'],
dtype='<U10')
The feature_names are a list of strings, giving the description of each feature:
iris['feature_names']
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['sepal length (cm)',
'sepal width (cm)',
'petal length (cm)',
'petal width (cm)']
The data itself is contained in the target and data fields. The data contains the
numeric measurements of sepal length, sepal width, petal length, and petal width in a
numpy array:
type(iris['data'])
numpy.ndarray
The rows in the data array correspond to flowers, while the columns represent the
four measurements that were taken for each flower:
iris['data'].shape
(150, 4)
We see that the data contains measurements for 150 different flowers.
Remember that the individual items are called samples in machine learning, and their
properties are called features.
The shape of the data array is the number of samples times the number of features.
This is a convention in scikit-learn, and your data will always be assumed to be in this
shape.
Here are the feature values for the first five samples:
iris['data'][:5]
array([[ 5.1, 3.5, 1.4, 0.2],
[ 4.9, 3. , 1.4, 0.2],
[ 4.7, 3.2, 1.3, 0.2],
[ 4.6, 3.1, 1.5, 0.2],
[ 5. , 3.6, 1.4, 0.2]])
The target array contains the species of each of the flowers that were measured, also
as a numpy array:
type(iris['target'])
numpy.ndarray
The target is a one-dimensional array, with one entry per flower:
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iris['target'].shape
(150,)
The species are encoded as integers from 0 to 2:
iris['target']
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])
The meaning of the numbers are given by the iris['target_names'] array: 0 means
Setosa, 1 means Versicolor and 2 means Virginica.
Measuring Success: Training and testing data
We want to build a machine learning model from this data that can predict the spe‐
cies of iris for a new set of measurements.
Before we can apply our model to new measurements, we need to know whether our
model actually works, that is whether we should trust its predictions.
Unfortunately, we can not use the data we use to build the model to evaluate it. This is
because our model can always simply remember the whole training set, and will
therefore always predict the correct label for any point in the training set. This
remembering” does not indicate to us whether our model will generalize well, in
other words whether it will also perform well on new data. So before we apply our
model to new measurements, we will want to know whether we can trust its predic‐
tions.
To assess the models’ performance, we show the model new data (that it hasn’t seen
before) for which we have labels. This is usually done by splitting the labeled data we
have collected (here our 150 flower measurements) into two parts.
The part of the data is used to build our machine learning model, and is called the
training data or training set. The rest of the data will be used to access how well the
model works and is called test data, test set or hold-out set.
Scikit-learn contains a function that shuffles the dataset and splits it for you, the
train_test_split function.
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This function extracts 75% of the rows in the data as the training set, together with
the corresponding labels for this data. The remaining 25% of the data, together with
the remaining labels are declared as the test set.
How much data you want to put into the training and the test set respectively is
somewhat arbitrary, but using a test-set containing 25% of the data is a good rule of
thumb.
In scikit-learn, data is usually denoted with a capital X, while labels are denoted by a
lower-case y.
Lets call train_test_split on our data and assign the outputs using this nomencla‐
ture:
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(iris['data'], iris['target'],
random_state=0)
The train_test_split function shuffles the dataset using a pseudo random number
generator before making the split. If we would take the last 25% of the data as a test
set, all the data point would have the label 2, as the data points are sorted by the label
(see the output for iris['target'] above). Using a tests set containing only one of
the three classes would not tell us much about how well we generalize, so we shuffle
our data, to make sure the test data contains data from all classes.
To make sure that we will get the same output if we run the same function several
times, we provide the pseudo random number generator with a fixed seed using the
random_state parameter. This will make the outcome deterministic, so this line will
always have the same outcome. We will always fix the random_state in this way when
using randomized procedures in this book.
The output of the train_test_split function are X_train, X_test, y_train and
y_test, which are all numpy arrays. X_train contains 75% of the rows of the dataset,
and X_test contains the remaining 25%:
X_train.shape
(112, 4)
X_test.shape
(38, 4)
First things rst: Look at your data
Before building a machine learning model, it is often a good idea to inspect the data,
to see if the task is easily solvable without machine learning, or if the desired infor‐
mation might not be contained in the data.
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Additionally, inspecting your data is a good way to find abnormalities and peculiari‐
ties. Maybe some of your irises were measured using inches and not centimeters, for
example. In the real world, inconsistencies in the data and unexpected measurements
are very common.
One of the best ways to inspect data is to visualize it. One way to do this is by using a
scatter plot.
A scatter plot of the data puts one feature along the x-axis, one feature along the y-
axis, and draws a dot for each data point.
Unfortunately, computer screens have only two dimensions, which allows us to only
plot two (or maybe three) features at a time. It is difficult to plot datasets with more
than three features this way.
One way around this problem is to do a pair plot, which looks at all pairs of two fea‐
tures. If you have a small number of features, such as the four we have here, this is
quite reasonable. You should keep in mind that a pair plot does not show the interac‐
tion of all of features at once, so some interesting aspects of the data may not be
revealed when visualizing it this way.
Here is a pair plot of the features in the training set. The data points are colored
according to the species the iris belongs to:
fig, ax = plt.subplots(3, 3, figsize=(15, 15))
plt.suptitle("iris_pairplot")
for i in range(3):
for j in range(3):
ax[i, j].scatter(X_train[:, j], X_train[:, i + 1], c=y_train, s=60)
ax[i, j].set_xticks(())
ax[i, j].set_yticks(())
if i == 2:
ax[i, j].set_xlabel(iris['feature_names'][j])
if j == 0:
ax[i, j].set_ylabel(iris['feature_names'][i + 1])
if j > i:
ax[i, j].set_visible(False)
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From the plots, we can see that the three classes seem to be relatively well separated
using the sepal and petal measurements. This means that a machine learning model
will likely be able to learn to separate them.
Building your rst model: k nearest neighbors
Now we can start building the actual machine learning model. There are many classi‐
fication algorithms in scikit-learn that we could use. Here we will use a k nearest
neighbors classifier, which is easy to understand.
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Building this model only consists of storing the training set. To make a prediction for
a new data point, the algorithm finds the point in the training set that is closest to the
new point. Then, it and assigns the label of this closest data training point to the new
data point.
The k in k nearest neighbors stands for the fact that instead of using only the closest
neighbor to the new data point, we can consider any fixed number k of neighbors in
the training (for example, the closest three or five neighbors). Then, we can make a
prediction using the majority class among these neighbors. We will go into more
details about this later.
Lets use only a single neighbor for now.
All machine learning models in scikit-learn are implemented in their own class,
which are called Estimator classes. The k nearest neighbors classification algorithm
is implemented in the KNeighborsClassifier class in the neighbors module.
Before we can use the model, we need to instantiate the class into an object. This is
when we will set any parameters of the model. The single parameter of the KNeighbor
sClassifier is the number of neighbors, which we will set to one:
from sklearn.neighbors import KNeighborsClassifier
knn = KNeighborsClassifier(n_neighbors=1)
The knn object encapsulates the algorithm to build the model from the training data,
as well the algorithm to make predictions on new data points.
It will also hold the information the algorithm has extracted from the training data.
In the case of KNeighborsClassifier, it will just store the training set.
To build the model on the training set, we call the fit method of the knn object,
which takes as arguments the numpy array X_train containing the training data and
the numpy array y_train of the corresponding training labels:
knn.fit(X_train, y_train)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
metric_params=None, n_jobs=1, n_neighbors=1, p=2,
weights='uniform')
Making predictions
We can now make predictions using this model on new data, for which we might not
know the correct labels.
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Imagine we found an iris in the wild with a sepal length of 5cm, a sepal width of
2.9cm, a petal length of 1cm and a petal width of 0.2cm. What species of iris would
this be?
We can put this data into a numpy array, again with the shape number of samples
(one) times number of features (four):
X_new = np.array([[5, 2.9, 1, 0.2]])
X_new.shape
(1, 4)
To make prediction we call the predict method of the knn object:
prediction = knn.predict(X_new)
prediction
array([0])
iris['target_names'][prediction]
array(['setosa'],
dtype='<U10')
Our model predicts that this new iris belongs to the class 0, meaning its species is
Setosa.
But how do we know whether we can trust our model? We dont know the correct
species of this sample, which is the the whole point of building the model!
Evaluating the model
This is where the test set that we created earlier comes in. This data was not used to
build the model, but we do know what the correct species are for each iris in the test
set.
We can make a prediction for an iris in the test data, and compare it against its label
(the known species). We can measure how well the model works by computing the
accuracy, which is the fraction of flowers for which the right species was predicted:
y_pred = knn.predict(X_test)
np.mean(y_pred == y_test)
0.97368421052631582
We can also use the score method of the knn object, which will compute the test set
accuracy for us:
knn.score(X_test, y_test)
0.97368421052631582
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For this model, the test set accuracy is about 0.97, which means we made the right
prediction for 97% of the irises in the test set. Under some mathematical assump‐
tions, this means that we can expect our model to be correct 97% of the time for new
irises.
For our hobby botanist application, this high level of accuracy means that our models
may be trustworthy enough to use. In later chapters we will discuss how we can
improve performance, and what caveats there are in tuning a model.
Summary
Lets summarize what we learned in this chapter. We started off formulating a task of
predicting which species of iris a particular flower belongs to by using physical meas‐
urements of the flower. We used a dataset of measurements that was annotated by an
expert with the correct species to build our model, making this a supervised learning
task. There were three possible species, Setosa, Versicolor or Virginica, which made
the task a three-class classication problem. The possible species are called classes in
the classification problem, and the species of a single iris is called its label.
The dataset consists of two numpy arrays, one containing the data, which is referred
to as X in scikit-learn, and one containing the correct or desired outputs, which is
called y. The array X is a two-dimensional array of features, with one row per data
point, and one column per feature. The array y is a one-dimensional array, which
here contained one class label from 0 to 2 for each of the samples.
We split our dataset into a training set, to build our model, and a test set, to evaluate
how well our model will generalize to new, unseen data.
We chose the k nearest neighbors classification algorithm, which makes predictions
for a new data point by considering its closest neighbor(s) in the training set.
The algorithm is implemented in the KNeighborsClassifier class, which contains
the algorithm to build the model, as well as the algorithm to make a prediction using
the model. We instantiated the class, setting parameters. Then, we built the model by
calling the fit method, passing the training data X_train and training outputs
y_train as parameters.
We evaluated the model using the score method, that computes the accuracy of the
model. We applied the score method to the test set data and the test set labels, and
found that our model is about 97% accurate, meaning it is correct 97% of the time on
the test set.
This gave us the confidence to apply the model to new data (in our example, new
flower measurements), and trust that the model will be correct about 97% of the time.
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Here is a summary of the code needed for the whole training and evaluation proce‐
dure:
X_train, X_test, y_train, y_test = train_test_split(iris['data'], iris['target'],
random_state=0)
knn = KNeighborsClassifier(n_neighbors=1)
knn.fit(X_train, y_train)
knn.score(X_test, y_test)
0.97368421052631582
This snippet contains the core code for applying any machine learning algorithms
using scikit-learn. The fit, predict and score methods are the common interface to
supervised models in scikit-learn, and with the concepts introduced in this chapter,
you can apply these models to many machine learning tasks.
In the next chapter, we will go into more depth about the different kinds of super‐
vised models in scikit-learn, and how to apply them successfully.
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CHAPTER 2
Supervised Learning
As we mentioned in the introduction, supervised machine learning is one of the most
commonly used and successful types of machine learning. In this chapter, we will
describe supervised learning in more detail, and explain several popular supervised
learning algorithms.
We already saw an application of supervised machine learning in the last chapter:
classifying iris flowers into several species using physical measurements of the flow‐
ers.
Remember that supervised learning is used whenever we want to predict a certain
outcome from a given input, and we have examples of input-output pairs. We build a
machine learning model from these input-output pairs, which comprise our training
set. Our goal is to make accurate predictions to new, never-before seen data.
Supervised learning often requires human effort to build the training set, but after‐
wards automates and often speeds up an otherwise laborious or infeasible task.
Classication and Regression
There are two major types of supervised machine learning algorithms, called classi‐
cation and regression.
In classification, the goal is to predict a class label, which is a choice from a predefined
list of possibilities. In Chapter 1 (Introduction) we used the example of classifying iri‐
ses into one of three possible species. Classification is sometimes separated into
binary classication, which is the special case of distinguishing between exactly two
classes, and multi-class classication which is classificaFItion between more than two
classes. You can think of binary classification as trying to answer a “yes” or “no” ques‐
tion.
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Classifying emails into either spam or not spam is an example of a binary classifica‐
tion problem. In this binary classification task, the yes or no question being asked
would be “Is this email spam?”.
[info box] In binary classification we often speak of one class being the positive class
and the other class being the negative class. Here, positive dont represent benefit or
value, but rather what the object of study is. So when looking for spam, “positive
could mean the spam class. Which of the two classes is called positive is often a sub‐
jective manner, and specific to the domain.FI
[/info box]
The iris example on the other hand is an example of a multi-class classification prob‐
lem.
Another example of a multi-class classification problem is predicting what language a
website is in from the text on the website. The classes here would be a pre-defined list
of possible languages.
For regression tasks, the goal is to predict a continuous number, or a oating point
number in programming terms (a real number in mathematical terms). Predicting a
persons annual income from their education, their age and where they live, is a[n
example of a] regression task. When predicting income, the predicted value is an
amount, and can be any number in a given range. Another example of a regression
task is predicting the yield of a corn farm, given attributes such as previous yields,
weather and number of employees working on the farm. The yield again can be an
arbitrary number.
An easy way to distinguish between classification and regression tasks is to ask
whether there is some kind of ordering or continuity in the output. If there is an
ordering, or a continuity between possible outcomes, then the problem is a regression
problem.
Think about predicting annual income. There is a clear ordering of “making more
money” or “making less money”. There is a natural understanding that 40.000$ per
year is between 50.000$ per year and 30.000$ per year. There is also a continuity in the
output. Whether a person makes 40,000$ or 40,001$ a year does not make a tangible
difference, even though they are different amounts of money. So if our algorithm pre‐
dicts 39,999$ or 40,001$ when it should have predicted 40,000$, we don’t mind that
much.
Contrastively, for the task of recognizing the language of a website (which is a classifi‐
cation problem), there is no matter of degree. A website is in one language, or it is in
another. There is no continuity between languages, and there is no language that is
between English and French [footnote: We ask linguists to excuse the simplified pre‐
sentation of languages as distinct and fixed entities].
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Generalization, Overtting and Undertting
In supervised learning, we want to built a model on the training data, and then be
able to make accurate predictions on new, unseen data, that has the same characteris‐
tics as the training set that we used. If a model is able to make accurate predictions on
unseen data, we say it is able to generalize from the training set to the test set.
We want to build a model that is able to generalize as well as possible.
Usually we build a model in such a way that it can make accurate predictions on the
training set. If the training and test set have enough in common, we expect the model
to also be accurate on the test set.
However, there are some cases where this can go wrong. For example, if we allow our‐
selves to build very complex models, we can always be as accurate as we like on the
training set.
Lets take a look at a made-up example. Say a novice data scientist wants to predict a
persons salary, and for each person, the only characteristic he has is the date of birth.
The dataset might look like this:
|Date of Birth|Annual salary ($)|
|-|-|
|30/4/1950|50500|
|05/8/1964|41000|
|09/2/2001|35200|
|17/5/1989|36000|
Because our novice data scientist knows he needs to present a machine learning algo‐
rithm with numbers, he replaces the date of birth with each persons age at the time of
analysis, in 2016. That seems very little to go by, so our novice data scientist also adds
the last four digits of their social security number, their house number, their zip code,
and the number of their children.
Now the data looks like this:
|Age|SSN|House|ZIP|Children|Annual salary ($)|
|-|-|-|-|-|-|
|66|1882|19|10030|2|50500|
|52|1337|2|10028|0|41000|
|22|3467|8|10041|1|35200|
|25|8391|27|10009|4|36000|
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Now he builds a machine learning model using the first three rows as a training set.
Lets save how the algorithm works for later. The algorithm produces the following
formula for the annual salary:
salary = 333 * x[0] + 1 * x[1] + 237 * x[2] - 20 * x[3] + 26 * x[4] +
225866
Here x[0] to x[4] contain the age, last digits of the SSN, the house number, ZIP code
and number of children.
The formula works very well on the training set, the first three rows of the dataset.
The predictions for the training set are 53681, 44433 and 37761 which are very close
to the true values.
However, the prediction the formula makes for the fourth point in the dataset, which
was not part of the training set, is 48905, which is quite far from the 36000 which was
the desired output.
So what happened here? The data scientist allowed his machine learning algorithm to
build a relatively complex interaction between the five features and the output (the
annual salary) without a lot of support for this model in the data. The result is a
model that doesn’t reflect a real world relationship. For example, this model predicts
that you would make $237 more if you move to the house next door (237 is the coef‐
ficient for x[2]!
Building a complex model that does well on the training set but does not generalize to
new data is known as overtting, because we are focusing too much on the particular‐
ities of the training data. Avoiding overfitting is a crucial aspect of building a success‐
ful machine learning model. A good way to avoid overfitting is to restrict ourselves to
building very simple models.
A much simpler model for the salary prediction task is to always predict the average
salary of the three people in the training set, which is
Predicting that everybody’s salary is 42233 is clearly too simple, and does not capture
the variation in our training set very well. Using too simple a model is called undert‐
ting, because we don’t explain the target output for the training data well enough.
A middle ground for the salary prediction would be to use age as a single feature,
which restricts us to very simple models, but still allows us to capture some trends in
our data.
A model only including the age feature is:
salary = 323 * age + 27146
This model makes predictions of 48464, 43942, and 34252 for the training set, which
is not as good as our previous model.
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However, it generalizes much better to the test set when compared to the complex
model we used before. It predicts 35221 for the fourth row in the table.
The trade-off between overfitting and underfitting is illustrated in Figure
model_complexity.
If we choose use a model that is too simple, we will do badly on the training set, and
similarly badly on the test set, as we would using only the mean prediction.
The more complex we allow our model to be, the better we will be able to predict on
the training data. However, if our model becomes too complex, we start focusing too
much on the particularities of our training set, and the model will not generalize well
to new data.
There is a sweet spot in between, which will yield the best generalization perfor‐
mance. This is the model we want to find.
Understanding the implications of model complexity is hard, and has different impli‐
cations for each kind of machine learning model.
Supervised Machine Learning Algorithms
We will now go through the most popular machine learning algorithms and explain
how they learn from data and how they make predictions. We will also discuss how
the concept of model complexity plays out for each of these models.
While an in-depth discussion of each algorithm is beyond the scope of this book, we
will try to give some intuition about how each algorithm builds a model.
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We will also discuss strength and weaknesses of each algorithm, and what kind of
data they can be best applied to. We will also explain the meaning of the most impor‐
tant parameters and options. Discussing all of them is beyond the scope of the book,
and we refer you to the scikit-learn documentation for more details.
Many algorithms have a classification and a regression variant, and we will describe
both.
It is not necessary to read through the description of each algorithm in detail, but
understanding the models will give you a better feeling for the different ways
machine learning algorithms can work. This chapter can also be used as a reference
guide, and you can come back to it when you are unsure about the workings of any of
the algorithms.
We will use several datasets to illustrate the different algorithms. Some of the datasets
will be small synthetic (meaning made-up) datasets, designed to highlight particular
aspects of the algorithms. Other datasets will be larger, real world examples datasets.
An example of a synthetic two-class classification dataset is the forge dataset, which
has two features. Below is a scatter plot visualizing all of the data points in this data‐
set. The plot has the first feature on the x-axis and the second feature on the y-axis.
As is always the case in in scatter plots, each data point is represented as one dot. The
color of the dot indicates its class, with red meaning class 0 and blue meaning class 1.
X, y = mglearn.datasets.make_forge()
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
print("X.shape: %s" % (X.shape,))
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X.shape: (26, 2)
As you can see from X.shape, this dataset consists of 26 data points, with 2 features.
To illustrate regression algorithms, we will use the synthetic wave dataset shown
below. The wave dataset only has a single input feature, and a continuous target
variable (or response) that we want to model.
The plot below is showing the single feature on the x-axis, with the data points as
green dots. For each data point, the target output is plotted in blue on the y-axis.
X, y = mglearn.datasets.make_wave(n_samples=40)
plt.plot(X, y, 'o')
plt.plot(X, -3 * np.ones(len(X)), 'o')
plt.ylim(-3.1, 3.1)
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We are using these very simple, low-dimensional datasets as we can easily visualize
them -- a computer monitor has two dimensions, so data with more than two fea‐
tures is hard to show. Any intuition derived from datasets with few features (also
called low-dimensional datasets) might not hold in datasets with many features (high
dimensional datasets). As long as you keep that in mind, inspecting algorithms on
low-dimensional datasets can be very instructive.
We will complement these small synthetic dataset with two real-world datasets that
are included in scikit-learn. One is the Wisconsin breast cancer dataset (or cancer for
short), which records clinical measurements of breast cancer tumors. Each tumor is
labeled as “benign” (for harmless tumors) or “malignant” (for cancerous tumors), and
the task is to learn to predict whether a tumor is malignant based on the measure‐
ments of the tissue.
The data can be loaded using the load_breast_cancer from scikit-learn. Datasets
that are included in scikit-learn are usually stored as Bunch objects, which contain
some information about the dataset as well as the actual data.
All you need to know about Bunch objects is that they behave like dictionaries, with
the added benefit that you can access values using a dot (as in bunch.key instead of
bunch['key']).
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from sklearn.datasets import load_breast_cancer
cancer = load_breast_cancer()
cancer.keys()
dict_keys(['DESCR', 'target_names', 'data', 'target', 'feature_names'])
The dataset consists of 569 data points, with 30 features each:
print(cancer.data.shape)
(569, 30)
Of these 569 data points, 212 are labeled as malignant, and 357 as benign:
print(cancer.target_names)
np.bincount(cancer.target)
array([212, 357])
['malignant' 'benign']
To get a description of the semantic meaning of each feature, we can have a look at
the feature_names attribute:
cancer.feature_names
array(['mean radius', 'mean texture', 'mean perimeter', 'mean area',
'mean smoothness', 'mean compactness', 'mean concavity',
'mean concave points', 'mean symmetry', 'mean fractal dimension',
'radius error', 'texture error', 'perimeter error', 'area error',
'smoothness error', 'compactness error', 'concavity error',
'concave points error', 'symmetry error', 'fractal dimension error',
'worst radius', 'worst texture', 'worst perimeter', 'worst area',
'worst smoothness', 'worst compactness', 'worst concavity',
'worst concave points', 'worst symmetry', 'worst fractal dimension'],
dtype='<U23')
You can find out more about the data by reading cancer.DESCR if you are interested.
We will also be using a real-world regression dataset, the Boston Housing dataset.
The task associated with this dataset is to predict the median value of homes in sev‐
eral Boston neighborhoods in the 1970s, using information about the neighborhoods
such as crime rate, proximity to the Charles River, highway accessibility and so on.
The datasets contains 506 data points, described by 13 features:
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from sklearn.datasets import load_boston
boston = load_boston()
print(boston.data.shape)
(506, 13)
Again, you can get more information about the dataset by reading the DESCR attribute
of boston.
For our purposes here, we will actually expand this dataset, by not only considering
these 13 measurements as input features, but also looking at all products (also called
interactions) between features.
In other words, we will not only consider crime rate and highway accessibility as a
feature, but also the product of crime rate and highway accessibility. Including
derived feature like these is called feature engineering, which we will discuss in more
detail in Chapter 5 (Representing Data).
This derived dataset can be loaded using the load_extended_boston function:
X, y = mglearn.datasets.load_extended_boston()
print(X.shape)
(506, 105)
The resulting 105 features are the 13 original features, the 13 choose 2 = 91 (Footnote:
the number of ways to pick 2 elements out of 13 elements) features that are product
of two features, and one constant feature.
We will use these datasets to explain and illustrate the properties of the different
machine learning algorithms. But for now, let’s get to the algorithms themselves.
First, we will revisit the k-Nearest Neighbor algorithm, that we already saw in the last
chapter.
k-Nearest Neighbor
The k-Nearest Neighbors (kNN) algorithm is arguably the simplest machine learning
algorithm. Building the model only consists of storing the training dataset. To make a
prediction for a new data point, the algorithm finds the closest data points in the
training dataset, it “nearest neighbors.
k-Neighbors Classication
In its simplest version, the algorithm only considers exactly one nearest neighbor,
which is the closest training data point to the point we want to make a prediction for.
The prediction is then simply the known output for this training point.
Figure forge_one_neighbor illustrates this for the case of classification on the forge
dataset.
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mglearn.plots.plot_knn_classification(n_neighbors=1)
plt.title("forge_one_neighbor");
Here, we added three new data points, shown as crosses. For each of them, we
marked the closest point in the training set. The prediction of the one-nearest-
neighbor algorithm is the label of that point (shown by the color of the cross).
Instead of considering only the closest neighbor, we can also consider an arbitrary
number $k$ of neighbors. This is where the name of the $k$ neighbors algorithm
comes from. When considering more than one neighbor, we use voting to assign a
label. This means, for each test point, we count how many neighbors are red, and
how many neighbors are blue. We then assign the class that is more frequent: in other
words, the majority class among the k neighbors.
Below is an illustration using the three closest neighbors. Again, the prediction is
shown as the color of the cross. You can see that the prediction changed for the point
in the top left from using only one neighbor.
mglearn.plots.plot_knn_classification(n_neighbors=3)
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While this illustration is for a binary classification problem, you can imagine this
working with any number of classes. For more classes, we count how many neighbors
belong to each class, and again predict the most common class.
Now lets look at how we can apply the $k$ nearest neighbors algorithm using scikit-
learn.
First, we split our data into a training and a test set, so we can evaluate generalization
performance, as discussed in Chapter 1 (Introduction).
from sklearn.model_selection import train_test_split
X, y = mglearn.datasets.make_forge()
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
Next we import and instantiate the class. This is when we can set parameters, like the
number of neighbors to use. Here, we set it to three.
from sklearn.neighbors import KNeighborsClassifier
clf = KNeighborsClassifier(n_neighbors=3)
Now, we fit the classifier using the training set. For KNeighborsClassifier this
means storing the dataset, so we can compute neighbors during prediction.
clf.fit(X_train, y_train)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
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metric_params=None, n_jobs=1, n_neighbors=3, p=2,
weights='uniform')
To make predictions on the test data, we call the predict method. This computes the
nearest neighbors in the training set and finds the most common class among these:
clf.predict(X_test)
array([1, 0, 1, 0, 1, 0, 0])
To evaluate how well our model generalizes, we can call the score method with the
test data together with the test labels:
clf.score(X_test, y_test)
0.8571428571428571
We see that our model is about 86% accurate, meaning the model predicted the class
correctly for 85% of the samples in the test dataset.
Analyzing KNeighborsClassier
For two-dimensional datasets, we can also illustrate the prediction for all possible test
point in the xy-plane. We color the plane red in regions where points would be
assigned the red class, and blue otherwise. This lets us view the decision boundary,
which is the divide between where the algorithm assigns class red versus where it
assigns class blue.
Here is a visualization of the decision boundary for one, three and five neighbors:
fig, axes = plt.subplots(1, 3, figsize=(10, 3))
for n_neighbors, ax in zip([1, 3, 9], axes):
clf = KNeighborsClassifier(n_neighbors=n_neighbors).fit(X, y)
mglearn.plots.plot_2d_separator(clf, X, fill=True, eps=0.5, ax=ax, alpha=.4)
ax.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
ax.set_title("%d neighbor(s)" % n_neighbors)
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As you can see in the left figure, using a single neighbor results in a decision bound‐
ary that follows the training data closely. Considering more and more neighbors leads
to a smoother decision boundary. A smoother boundary corresponds to a simple
model. In other words, using few neighbors corresponds to high model complexity
(as shown on the right side of Figure model_complexity), and using many neighbors
corresponds to low model complexity (as shown on the left side of Figure
model_complexity).
Lets investigate whether we can confirm the connection between model complexity
and generalization that we discussed above.
We will do this on the real world breast cancer dataset.
We begin by splitting the dataset into a training and a test set. Then we will evaluate
training and test set performance with different numbers of neighbors.
from sklearn.datasets import load_breast_cancer
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, stratify=cancer.target, random_state=66)
training_accuracy = []
test_accuracy = []
# try n_neighbors from 1 to 10.
neighbors_settings = range(1, 11)
for n_neighbors in neighbors_settings:
# build the model
clf = KNeighborsClassifier(n_neighbors=n_neighbors)
clf.fit(X_train, y_train)
# record training set accuracy
training_accuracy.append(clf.score(X_train, y_train))
# record generalization accuracy
test_accuracy.append(clf.score(X_test, y_test))
plt.plot(neighbors_settings, training_accuracy, label="training accuracy")
plt.plot(neighbors_settings, test_accuracy, label="test accuracy")
plt.legend()
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The plot shows the training and test set accuracy on the y axis against the setting of
n_neighbors on the x axis. While the real world plots are rarely very smooth, we can
still recognize some of the characteristics of overfitting and underfitting. As consider‐
ing fewer neighbors corresponds to a more complex model, the plot is horizontally
flipped relative to the illustration in Figure model_complexity.
Considering a single nearest neighbor, the prediction on the training set is perfect.
Considering more neighbors, the model becomes more simple, and the training accu‐
racy drops.
The test set accuracy for using a single neighbor is lower then when using more
neighbors, indicating that using a single nearest neighbor leads to a model that is too
complex. On the other hand, when considering 10 neighbors, the model is too sim‐
ple, and performance is even worse. The best performance is somewhere in the mid‐
dle, around using six neighbors.
Still, it is good to keep the scale of the plot in mind. The worst performance is around
88% accuracy, which might still be acceptable.
k-Neighbors Regression
There is also a regression variant of the k-nearest neighbors algorithm. Again, lets
start by using a single nearest neighbor, this time using the wave dataset. We added
three test data points as green crosses on the x axis.
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The prediction using a single neighbor is just the target value of the nearest neighbor,
shown as the blue cross:
mglearn.plots.plot_knn_regression(n_neighbors=1)
Again, we can also use more than one nearest neighbor for regression. When using
multiple nearest neighbors for regression, the prediction is the average (or mean) of
the relevant neighbors:
mglearn.plots.plot_knn_regression(n_neighbors=3)
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The k nearest neighbors algorithm for regression is implemented in the KNeighbors
Regressor class in scikit-learn.
Using it looks much like the KNeighborsClassifier above:
from sklearn.neighbors import KNeighborsRegressor
X, y = mglearn.datasets.make_wave(n_samples=40)
# split the wave dataset into a training and a test set
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
# Instantiate the model, set the number of neighbors to consider to 3:
reg = KNeighborsRegressor(n_neighbors=3)
# Fit the model using the training data and training targets:
reg.fit(X_train, y_train)
KNeighborsRegressor(algorithm='auto', leaf_size=30, metric='minkowski',
metric_params=None, n_jobs=1, n_neighbors=3, p=2,
weights='uniform')
Now, we can make predictions on the test set:
reg.predict(X_test)
array([-0.05396539, 0.35686046, 1.13671923, -1.89415682, -1.13881398,
-1.63113382, 0.35686046, 0.91241374, -0.44680446, -1.13881398])
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We can also evaluate the model using the score method, which for regressors returns
the $R^2$ score.
The $R^2$ score, also known as coefficient of determination, is a measure of good‐
ness of a prediction for a regression model, and yields a score up to 1. A value of 1
corresponds to a perfect prediction, and a value of 0 corresponds to a constant model
that just predicts the mean of the training set responses y_train.
reg.score(X_test, y_test)
0.83441724462496036
Here, the score is 0.83 which indicates a relatively good model fit.
Analyzing k nearest neighbors regression
For our one-dimensional dataset, we can see what the predictions look like for all
possible feature values. To do this, we create a test-dataset consisting of many points
on the line.
fig, axes = plt.subplots(1, 3, figsize=(15, 4))
# create 1000 data points, evenly spaced between -3 and 3
line = np.linspace(-3, 3, 1000).reshape(-1, 1)
plt.suptitle("nearest_neighbor_regression")
for n_neighbors, ax in zip([1, 3, 9], axes):
# make predictions using 1, 3 or 9 neighbors
reg = KNeighborsRegressor(n_neighbors=n_neighbors).fit(X, y)
ax.plot(X, y, 'o')
ax.plot(X, -3 * np.ones(len(X)), 'o')
ax.plot(line, reg.predict(line))
ax.set_title("%d neighbor(s)" % n_neighbors)
In the plots above, the blue points are again the responses for the training data, while
the red line is the prediction made by the model for all points on the line.
Using only a single neighbor, each point in the training set has an obvious influence
on the predictions, and the predicted values go through all of the data points. This
leads to a very unsteady prediction. Considering more neighbors leads to smoother
predictions, but these do not fit the training data as well.
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Strengths, weaknesses and parameters
In principal, there are two important parameters to the KNeighbors classifier: the
number of neighbors and how you measure distance between data points. In practice,
using a small number of neighbors like 3 or 5 often works well, but you should cer‐
tainly adjust this parameter. Choosing the right distance measure is somewhat
beyond the scope of this book. By default, Euclidean distance is used, which works
well in many settings.
One of the strengths of nearest neighbors is that the model is very easy to understand,
and often gives reasonable performance without a lot of adjustments. Using nearest
neighbors is a good baseline method to try before considering more advanced techni‐
ques. Building the nearest neighbors model is usually very fast, but when your train‐
ing set is very large (either in number of features or in number of samples) prediction
can be slow.
When using nearest neighbors, its important to preprocess your data (see Chapter 3
Unsupervised Learning). Nearest neighbors often does not perform well on dataset
with very many features, in particular sparse datasets, a common type of data in
which there are many features, but only few of the features are non-zero for any given
data point.
So while the nearest neighbors algorithm is easy to understand, it is not often used in
practice, due to prediction being slow, and its inability to handle many features. The
method we discuss next has neither of these drawbacks.
Linear models
Linear models are a class of models that are widely used in practice, and have been
studied extensively in the last few decades, with roots going back over a hundred
years.
Linear models are models that make a prediction that using a linear function of the
input features, which we will explain below.
Linear models for regression
For regression, the general prediction formula for a linear model looks as follows:
\begin{align*}
&\hat{y} = w[0] x[0] + w[1] x[1] + \dotsc + w[p] x[p] + b \text{ (1) linear regression}
\end{align*}
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Here, $x[0]$ to $x[p]$ denotes the features (here the number of features is $p$) of a
single data point, $w$ and $b$ are parameters of the model that are learned, and $
\hat{y}$ is the prediction the model makes. For a dataset with a single feature, this is
which you might remember as the equation for a line from high school mathematics.
Here, $w[0]$ is the slope, and $b$ is the y-axis offset. For more features, w contains
the slopes along each feature axis. Alternatively, you can think of the predicted
response as being a weighted sum of the input features, with weights (which can be
negative) given by the entries of w.
Trying to learn the parameters $w[0]$ and $b$ on our one-dimensional wave dataset
might lead to the following line:
mglearn.plots.plot_linear_regression_wave()
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w[0]: 0.393906 b: -0.031804
We added a coordinate cross into the plot to make it easier to understand the line.
Looking at w[0] we see that the slope should be roughly around .4, which we can con‐
firm visually in the plot above. The intercept is where the prediction line should cross
the y-axis, which is slightly below 0, which you can also confirm in the image.
Linear models for regression can be characterized as regression models for which the
prediction is a line for a single feature, a plane when using two features, or a hyper‐
plane in higher dimensions (that is when having more features).
If you compare the predictions made by the red line with those made by the KNeigh‐
borsRegressor in Figure nearest_neighbor_regression, using a straight line to make
predictions seems very restrictive. It looks like all the fine details of the data are lost.
In a sense this is true. It is a strong (and somewhat unrealistic) assumption that our
target $y$ is a linear combination of the features. But looking at one-dimensional
data gives a somewhat skewed perspective. For datasets with many features, linear
models can be very powerful. In particular, if you have more features than training
data points, any target $y$ can be perfectly modeled (on the training set) as a linear
function (FOOTNOTE This is easy to see if you know some linear algebra).
There are many different linear models for regression. The difference between these
models is how the model parameters $w$ and $b$ are learned from the training data,
and how model complexity can be controlled. We will now go through the most pop‐
ular linear models for regression.
Linear Regression aka Ordinary Least Squares
Linear regression or Ordinary Least Squares (OLS) is the simplest and most classic
linear method for regression.
Linear regression finds the parameters $w$ and $b$ that minimize the mean squared
error between predictions and the true regression targets $y$ on the training set. The
mean squared error is the sum of the squared differences between the predictions and
the true values. Linear regression has no parameters, which is a benefit, but it also has
no way to control model complexity.
Here is the code that produces the model you can see in figure XX.
from sklearn.linear_model import LinearRegression
X, y = mglearn.datasets.make_wave(n_samples=60)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
lr = LinearRegression().fit(X_train, y_train)
The “slope” parameters w, also called weights or coecients are stored in the coef_
attribute, while the offset or intercept b is stored in the intercept_ attribute. [Foot‐
note: you might notice the strange-looking trailing underscore. Scikit-learn always
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stores anything that is derived from the training data in attributes that end with a
trailing underscore. That is to separate them from parameters that are set by the
user.]
print("lr.coef_: %s" % lr.coef_)
print("lr.intercept_: %s" % lr.intercept_)
lr.coef_: [ 0.39390555]
lr.intercept_: -0.0318043430268
The intercept_ attribute is always a single float number, while the coef_ attribute is
a numpy array with one entry per input feature. As we only have a single input fea‐
ture in the wave dataset, lr.coef_ only has a single entry.
Lets look at the training set and test set performance:
print("training set score: %f" % lr.score(X_train, y_train))
print("test set score: %f" % lr.score(X_test, y_test))
training set score: 0.670089
test set score: 0.659337
An $R^2$ of around .66 is not very good, but we can see that the score on training
and test set are very close together. This means we are likely underfitting, not overfit‐
ting. For this one-dimensional dataset, there is little danger of overfitting, as the
model is very simple (or restricted).
However, with higher dimensional datasets (meaning a large number of features), lin‐
ear models become more powerful, and there is a higher chance of overfitting.
Lets take a look at how LinearRegression performs on a more complex dataset, like
the Boston Housing dataset. Remember that this dataset has 506 samples and 105
derived features.
We load the dataset and split it into a training and a test set. Then we build the linear
regression model as before:
X, y = mglearn.datasets.load_extended_boston()
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
lr = LinearRegression().fit(X_train, y_train)
When comparing training set and test set score, we find that we predict very accu‐
rately on the training set, but the $R^2$ on the test set is much worse:
print("training set score: %f" % lr.score(X_train, y_train))
print("test set score: %f" % lr.score(X_test, y_test))
training set score: 0.952353
test set score: 0.605775
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This is a clear sign of overfitting, and therefore we should try to find a model that
allows us to control complexity.
One of the most commonly used alternatives to standard linear regression is Ridge
regression, which we will look into next.
Ridge regression
Ridge regression is also a linear model for regression, so the formula it uses to make
predictions is still Formula (1), as for ordinary least squares. In Ridge regression,the
coefficients w are chosen not only so that they predict well on the training data, but
there is an additional constraint. We also want the magnitude of coefficients to be as
small as possible; in other words, all entries of w should be close to 0.
Intuitively, this means each feature should have as little effect on the outcome as pos‐
sible (which translates to having a small slope), while still predicting well.
This constraint is an example of what is called regularization. Regularization means
explicitly restricting a model to avoid overfitting. The particular kind used by Ridge
regression is known as l2 regularization. [footnote: Mathematically, Ridge penalizes
the l2 norm of the coefficients, or the Euclidean length of w.]
Ridge regression is implemented in linear_model.Ridge. Lets see how well it does
on the extended Boston dataset:
from sklearn.linear_model import Ridge
ridge = Ridge().fit(X_train, y_train)
print("training set score: %f" % ridge.score(X_train, y_train))
print("test set score: %f" % ridge.score(X_test, y_test))
training set score: 0.886058
test set score: 0.752714
As you can see, the training set score of Ridge is lower than for LinearRegression,
while the test set score is higher. This is consistent with our expectation. With linear
regression, we were overfitting to our data. Ridge is a more restricted model, so we
are less likely to overfit. A less complex model means worse performance on the
training set, but better generalization.
As we are only interested in generalization performance, we should choose the Ridge
model over the LinearRegression model.
The Ridge model makes a trade-off between the simplicity of the model (near zero
coefficients) and its performance on the training set. How much importance the
model places on simplicity versus training set performance can be specified by the
user, using the alpha parameter. Above, we used the default parameter alpha=1.0.
There is no reason why this would give us the best trade-off, though. Increasing alpha
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forces coefficients to move more towards zero, which decreases training set perfor‐
mance, but might help generalization.
ridge10 = Ridge(alpha=10).fit(X_train, y_train)
print("training set score: %f" % ridge10.score(X_train, y_train))
print("test set score: %f" % ridge10.score(X_test, y_test))
training set score: 0.788346
test set score: 0.635897
Decreasing alpha allows the coefficients to be less restricted, meaning we move right
on the figure XXX.
For very small values of alpha, coefficients are barely restricted at all, and we end up
with a model that resembles LinearRegression.
ridge01 = Ridge(alpha=0.1).fit(X_train, y_train)
print("training set score: %f" % ridge01.score(X_train, y_train))
print("test set score: %f" % ridge01.score(X_test, y_test))
training set score: 0.928578
test set score: 0.771793
Here, alpha=0.1 seems to be working well. We could try decreasing alpha even more
to improve generalization. For now, notice how the parameter alpha corresponds to
the model complexity as shown in Figure model_complexity. We will discuss meth‐
ods to properly select parameters in Chapter 6 (Model Selection).
We can also get a more qualitative insight into how the alpha parameter changes the
model by inspecting the coef_ attribute of models with different values of alpha. A
higher alpha means a more restricted model, so we expect that the entries of coef_
have smaller magnitude for a high value of alpha than for a low value of alpha.
This is confirmed in the plot below:
plt.title("ridge_coefficients")
plt.plot(ridge.coef_, 'o', label="Ridge alpha=1")
plt.plot(ridge10.coef_, 'o', label="Ridge alpha=10")
plt.plot(ridge01.coef_, 'o', label="Ridge alpha=0.1")
plt.plot(lr.coef_, 'o', label="LinearRegression")
plt.ylim(-25, 25)
plt.legend()
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Here, the x-axis enumerates the entries of coef_: x=0 shows the coefficient associated
with the first feature, x=1 the coefficient associated with the second feature, and so on
up to x=100. The y-axis shows the numeric value of the corresponding value of the
coefficient. The main take-away here is that for alpha=10 (as shown by the green
dots), the coefficients are mostly between around -3 and 3. The coefficients for the
ridge model with alpha=1 (as shown by the blue dots), are somewhat larger. The red
dots have larger magnitude still, and many of the teal dots, corresponding to linear
regression without any regularization (which would be alpha=0) are so large they are
even outside of the chart.
Lasso
An alternative to Ridge for regularizing linear regression is the Lasso. The lasso also
restricts coefficients to be close to zero, similarly to Ridge regression, but in a slightly
different way, called “l1” regularization.[footnote: The Lasso penalizes the l1 norm of
the coefficient vector, or in other words the sum of the absolute values of the coeffi‐
cients].
The consequence of l1 regularization is that when using the Lasso, some coefficients
are exactly zero. This means some features are entirely ignored by the model. This can
be seen as a form of automatic feature selection. Having some coefficients be exactly
zero often makes a model easier to interpret, and can reveal the most important fea‐
tures of your model.
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Lets apply the lasso to the extended Boston housing dataset:
from sklearn.linear_model import Lasso
lasso = Lasso().fit(X_train, y_train)
print("training set score: %f" % lasso.score(X_train, y_train))
print("test set score: %f" % lasso.score(X_test, y_test))
print("number of features used: %d" % np.sum(lasso.coef_ != 0))
training set score: 0.293238
test set score: 0.209375
number of features used: 4
As you can see, the Lasso does quite badly, both on the training and the test set. This
indicates that we are
underfitting. We find that it only used four of the 105 features. Similarly to Ridge, the
Lasso also has a regularization parameter alpha that controls how strongly coeffi‐
cients are pushed towards zero . Above, we used the default of alpha=1.0. To dimin‐
ish underfitting, lets try decreasing alpha:
lasso001 = Lasso(alpha=0.01).fit(X_train, y_train)
print("training set score: %f" % lasso001.score(X_train, y_train))
print("test set score: %f" % lasso001.score(X_test, y_test))
print("number of features used: %d" % np.sum(lasso001.coef_ != 0))
/home/andy/checkout/scikit-learn/sklearn/linear_model/coordinate_descent.py:474: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations
ConvergenceWarning)
A lower alpha allowed us to fit a more complex model, which worked better on the
training and the test data. The performance is slightly better than using Ridge, and we
are using only 32 of the 105 features. This makes this model potentially easier to
understand.
If we set alpha too low, we again remove the effect of regularization and end up with a
result similar to LinearRegression.
lasso00001 = Lasso(alpha=0.0001).fit(X_train, y_train)
print("training set score: %f" % lasso00001.score(X_train, y_train))
print("test set score: %f" % lasso00001.score(X_test, y_test))
print("number of features used: %d" % np.sum(lasso00001.coef_ != 0))
/home/andy/checkout/scikit-learn/sklearn/linear_model/coordinate_descent.py:474: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations
ConvergenceWarning)
Again, we can plot the coefficients of the different models, similarly to Figure
ridge_coefficients.
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For alpha=1, with coefficients shown as blue dots, we not only see that most of the
coefficients are zero (which we already knew), but that the remaining coefficients are
also small in magnitude. Decreasing alpha to 0.01 we obtain the solution shown as
the green dots, which causes most features be exactly zero. Using alpha=0.00001, we
get a model that is quite unregularized, with most coefficients nonzero and of large
magnitude.
For comparison, the best Ridge solution is shown in teal. The ridge model with
alpha=0.1 has similar predictive performance as the lasso model with alpha=0.01, but
using Ridge, all coefficients are non-zero.
plt.plot(lasso.coef_, 'o', label="Lasso alpha=1")
plt.plot(lasso001.coef_, 'o', label="Lasso alpha=0.01")
plt.plot(lasso00001.coef_, 'o', label="Lasso alpha=0.0001")
plt.plot(ridge01.coef_, 'o', label="Ridge alpha=0.1")
plt.ylim(-25, 25)
plt.legend()
In practice, Ridge regression is usually the first choice between these two models.
However, if you have a large amount of features and expect only a few of them to be
important, Lasso might be a better choice. Similarly, if you would like to have a
model that is easy to interpret, Lasso will provide a model that is easier to under‐
stand, as it will select only a subset of the input features.
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Linear models for Classication
Linear models are also extensively used for classification. Lets look at binary classifi‐
cation first. In this case, a prediction is made using the following formula:
\begin{align*}
&\hat{y} = w[0] x[0] + w[1] x[1] + \dotsc + w[p] * x[p] + b > 0 &\text{ (2) linear
binary classification}
\end{align*}
The formula looks very similar to the one for linear regression, but instead of just
returning the weighted sum of the features, we threshold the predicted value at zero.
If the function was smaller than zero, we predict the class -1, if it was larger than zero,
we predict the class +1.
This prediction rule is common to all linear models for classification. Again, there are
many different ways to find the coefficients w and the intercept b.
For linear models for regression, the output y was a linear function of the features: a
line, plane, or hyperplane (in higher dimensions). For linear models for classification,
the decision boundary is a linear function of the input. In other words, a (binary) lin‐
ear classifier is a classifier that separates two classes using a line, a plane or a hyper‐
plane. We will see examples of that below.
There are many algorithms for learning linear models. These algorithms all differ in
the following two ways:
1. How they measure how well a particular combination of coefficients and inter‐
cept fits the training data.
1. If and what kind of regularization they use.
Different algorithms choose different ways to measure what “fitting the training set
well” means in 1. For technical mathematical reasons, it is not possible to adjust w and
b to minimize the number of misclassifications the algorithms produce, as one might
hope. For our purposes, and many applications, the different choices for 1. (called loss
function) is of little significance.
The two most common linear classification algorithms are logistic regression, imple‐
mented in linear_model.LogisticRegression and linear support vector machines
(linear SVMs), implemented in svm.LinearSVC (SVC stands for Support Vector Clas‐
sifier). Despite its name, LogisticRegression is a classification algorithm and not a
regression algorithm, and should not be confused with LinearRegression.
We can apply the LogisticRegression and LinearSVC models to the forge dataset,
and visualize the decision boundary as found by the linear models:
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from sklearn.linear_model import LogisticRegression
from sklearn.svm import LinearSVC
X, y = mglearn.datasets.make_forge()
fig, axes = plt.subplots(1, 2, figsize=(10, 3))
plt.suptitle("linear_classifiers")
for model, ax in zip([LinearSVC(), LogisticRegression()], axes):
clf = model.fit(X, y)
mglearn.plots.plot_2d_separator(clf, X, fill=False, eps=0.5, ax=ax, alpha=.7)
ax.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
ax.set_title("%s" % clf.__class__.__name__)
In this figure, we have the first feature of the forge dataset on the x axis and the sec‐
ond feature on the y axis as before. We display the decision boundaries found by Lin‐
earSVC and LogisticRegression respectively as straight lines, separating the area
classified as blue on the top from the area classified as red on the bottom.
In other words, any new data point that lies above the black line will be classified as
blue by the respective classifier, while any point that lies below the black line will be
classified as red.
The two models come up with similar decision boundaries. Note that both misclas‐
sify two of the points. By default, both models apply an l2 regularization, in the same
way that Ridge does for regression.
For LogisticRegression and LinearSVC the trade-off parameter that determines the
strength of the regularization is called C, and higher values of C correspond to less
regularization. In other words, when using a high value of the parameter C, Logisti
cRegression and LinearSVC try to fit the training set as best as possible, while with
low values of the parameter C, the model put more emphasis on finding a coefficient
vector w that is close to zero.
There is another interesting intuition of how the parameter C acts. Using low values
of C will cause the algorithms try to adjust to the “majority” of data points, while
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using a higher value of C stresses the importance that each individual data point be
classified correctly. Here is an illustration using LinearSVC.
mglearn.plots.plot_linear_svc_regularization()
On the left hand side, we have a very small C corresponding to a lot of regularization.
Most of the blue points are at the top, and most of the red points are at the bottom.
The strongly regularized model chooses a relatively horizontal line, misclassifying
two points.
In the center plot, C is slightly higher, and the model focuses more on the two mis‐
classified samples, tilting the decision boundary. Finally, on the right hand side, a very
high value of C in the model tilts the decision boundary a lot, now correctly classify‐
ing all red points. One of the blue points is still misclassified, as it is not possible to
correctly classify all points in this dataset using a straight line. The model illustrated
on the right hand side tries hard to correctly classify all points, but might not capture
the overall layout of the classes well. In other words, this model is likely overfitting.
Similarly to the case of regression, linear models for classification might seem very
restrictive in low dimensional spaces, only allowing for decision boundaries which
are straight lines or planes. Again, in high dimensions, linear models for classification
become very powerful, and guarding against overfitting becomes increasingly impor‐
tant when considering more features.
Lets analyze LinearLogistic in more detail on the breast_cancer dataset:
from sklearn.datasets import load_breast_cancer
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, stratify=cancer.target, random_state=42)
logisticregression = LogisticRegression().fit(X_train, y_train)
print("training set score: %f" % logisticregression.score(X_train, y_train))
print("test set score: %f" % logisticregression.score(X_test, y_test))
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training set score: 0.953052
test set score: 0.958042
The default value of C=1 provides quite good performance, with 95% accuracy on
both the training and the test set. As training and test set performance are very close,
it is likely that we are underfitting. Lets try to increase C to fit a more flexible model.
logisticregression100 = LogisticRegression(C=100).fit(X_train, y_train)
print("training set score: %f" % logisticregression100.score(X_train, y_train))
print("test set score: %f" % logisticregression100.score(X_test, y_test))
training set score: 0.971831
test set score: 0.965035
Using C=100 results in higher training set accuracy, and also a slightly increased test
set accuracy, confirming our intuition that a more complex model should perform
better.
We can also investigate what happens if we use an even more regularized model than
the default of C=1, by setting C=0.01.
logisticregression001 = LogisticRegression(C=0.01).fit(X_train, y_train)
print("training set score: %f" % logisticregression001.score(X_train, y_train))
print("test set score: %f" % logisticregression001.score(X_test, y_test))
training set score: 0.934272
test set score: 0.930070
As expected, when moving more to the left in Figure model_complexity from an
already underfit model, both training and test set accuracy decrease relative to the
default parameters.
Finally, lets look at the coefficients learned by the models with the three different set‐
tings of the regularization parameter C.
plt.plot(logisticregression.coef_.T, 'o', label="C=1")
plt.plot(logisticregression100.coef_.T, 'o', label="C=100")
plt.plot(logisticregression001.coef_.T, 'o', label="C=0.001")
plt.xticks(range(cancer.data.shape[1]), cancer.feature_names, rotation=90)
plt.ylim(-5, 5)
plt.legend()
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As LogisticRegression applies an L2 regularization by default, the result looks simi‐
lar to Ridge in Figure ridge_coefficients. Stronger regularization pushes coefficients
more and more towards zero, though coefficients never become exactly zero.
Inspecting the plot more closely, we can also see an interesting effect in the third coef‐
ficient, for “mean perimeter”. For C=100 and C=1, the coefficient is negative, while
for C=0.001, the coefficient is positive, with a magnitude that is even larger as for
C=1. Interpreting a model like this, one might think the coefficient tells us which
class a feature is associated with. For example, one might think that a high “texture
error” feature is related to a sample being “malignant. However, the change of sign in
the coefficient for “mean perimeter” means that depending on which model we look
at, high “mean perimeter” could be either taken as being indicative of “benign” or
indicative of “malignant. This illustrates that interpretations of coefficients of linear
models should always be taken with a grain of salt.
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If we desire a more interpretable model, using L1 regularization might help, as it lim‐
its the model to only using a few features. Here is the coefficient plot and classifica‐
tion accuracies for L1 regularization:
for C in [0.001, 1, 100]:
lr_l1 = LogisticRegression(C=C, penalty="l1").fit(X_train, y_train)
print("training accuracy of L1 logreg with C=%f: %f"
% (C, lr_l1.score(X_train, y_train)))
print("test accuracy of L1 logreg with C=%f: %f"
% (C, lr_l1.score(X_test, y_test)))
plt.plot(lr_l1.coef_.T, 'o', label="C=%f" % C)
plt.xticks(range(cancer.data.shape[1]), cancer.feature_names, rotation=90)
plt.ylim(-5, 5)
plt.legend(loc=2)
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training accuracy of L1 logreg with C=0.001000: 0.913146
test accuracy of L1 logreg with C=0.001000: 0.923077
training accuracy of L1 logreg with C=1.000000: 0.960094
test accuracy of L1 logreg with C=1.000000: 0.958042
training accuracy of L1 logreg with C=100.000000: 0.985915
test accuracy of L1 logreg with C=100.000000: 0.979021
Linear Models for multiclass classication
Many linear classification models are binary models, and dont extend naturally to the
multi-class case (with the exception of Logistic regression). A common technique to
extend a binary classification algorithm to a multi-class classification algorithm is the
one-vs-rest approach. In the one-vs-rest approach, a binary model is learned for each
class, which tries to separate this class from all of the other classes, resulting in as
many binary models as there are classes.
To make a prediction, all binary classifiers are run on a test point. The classifier that
has the highest score on its single class “wins” and this class label is returned as pre‐
diction.
Having one binary classifier per class results in having one vector of coefficients $w$
and one intercept $b$ for each class. The class for which the result of formula
\begin{align*}
& w[0] x[0] + w[1] x[1] + \dotsc + w[p] * x[p] + b & \text{ (3) classification confi‐
dence}
\end{align*}
is highest is the assigned class label.
The mathematics behind logistic regression are somewhat different from the one-vs-
rest approach, but they also result in one coefficient vector and intercept per class,
and the same method of making a prediction is applied.
Lets apply the one-vs-rest method to a simple three-class classification dataset.
We use a two-dimensional dataset, where each class is given by data sampled from a
Gaussian distribution.
from sklearn.datasets import make_blobs
X, y = make_blobs(random_state=42)
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm3)
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Now, we train a LinearSVC classifier on the dataset.
linear_svm = LinearSVC().fit(X, y)
print(linear_svm.coef_.shape)
print(linear_svm.intercept_.shape)
(3, 2)
(3,)
We see that the shape of the coef_ is (3, 2), meaning that each row of coef_ con‐
tains the coefficient vector for one of the three classes. Each row has two entries, cor‐
responding to the two features in the dataset.
The intercept_ is now a one-dimensional array, storing the intercepts for each class.
Lets visualize the lines given by the three binary classifiers:
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm3)
line = np.linspace(-15, 15)
for coef, intercept in zip(linear_svm.coef_, linear_svm.intercept_):
plt.plot(line, -(line * coef[0] + intercept) / coef[1])
plt.ylim(-10, 15)
plt.xlim(-10, 8)
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The red line shows the decision boundary for the binary classifier for the red class,
and so on.
You can see that all the red points in the training data are below the red line, which
means they are on the “red” side of this binary classifier. The red points are left of the
green line, which means they are classified as “rest” by the binary classifier for the
green class. The red points are below the blue line, which means the binary classifier
for the blue class also classifies them as “rest. Therefore, any point in this area will be
classified as red by the final classifier (Formula (3) of the red classifier is greater than
zero, while it is smaller than zero for the other two classes).
But what about the triangle in the middle of the plot? All three binary classifiers clas‐
sify points there as “rest. Which class would a point there be assigned to? The answer
is the one with the highest value in Formula (3): the class of the closest line.
The following figure shows the prediction shown for all regions of the 2d space:
mglearn.plots.plot_2d_classification(linear_svm, X, fill=True, alpha=.7)
plt.scatter(X[:, 0], X[:, 1], c=y, s=60)
line = np.linspace(-15, 15)
for coef, intercept in zip(linear_svm.coef_, linear_svm.intercept_):
plt.plot(line, -(line * coef[0] + intercept) / coef[1])
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Strengths, weaknesses and parameters
The main parameter of linear models is the regularization parameter, called alpha in
the regression models and C in LinearSVC and LogisticRegression. Large alpha or
small C mean simple models. In particular for the regression models, tuning this
parameter is quite important. Usually C and alpha are searched for on a logarithmic
scale.
The other decision you have to make is whether you want to use L1 regularization or
L2 regularization. If you assume that only few of your features are actually important,
you should use L1. Otherwise, you should default to L2.
L1 can also be useful if interpretability of the model is important. As L1 will use only
a few features, it is easier to explain which features are important to the model, and
what the effect of these features is.
Linear models are very fast to train, and also fast to predict. They scale to very large
datasets and work well with sparse data. If your data consists of hundreds of thou‐
sands or millions of samples, you might want to investigate SGDClassifier and
SGDRegressor, which implement even more scalable versions of the linear models
described above.
Another strength of linear models is that they make i] relatively easy to understand
how a prediction is made, using Formula (1) for regression and Formula (2) for clas‐
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sification. Unfortunately, it is often not entirely clear why coefficients are the way they
are. This is particularly true if your dataset has highly correlated features; in these
cases, the coefficients might be hard to interpret.
Linear models often perform well when the number of features is large compared to
the number of samples. They are also often used on very large datasets, simply
because other models are not feasible to train. However, on smaller dataset, other
models might yield better generalization performance.
Naive Bayes Classiers
Naive Bayes classifiers are a family of classifiers that are quite similar to the linear
models discussed above. However, they tend to be even faster in training. The price
paid for this efficiency is that naive Bayes models often provide generalization perfor‐
mance that is slightly worse than linear classifiers like LogisticRegression and Line
arSVC.
The reason that naive Bayes models are so efficient is that they learn parameters by
looking at each feature individually, and collect simple per-class statistics from each
feature.
There are three kinds of naive Bayes classifiers implemented in scikit-learn, Gaus
sianNB, BernoulliNB and MultinomialNB.
GaussianNB can be applied to any continuous data, while BernoulliNB assumes
binary data and MultinomialNB assumes count data (that is each feature represents an
integer count of something, like how often a word appears in a sentence). Bernoul
liNB and MultinomialNB are mostly used in text data classification, and we will revisit
them in Chapter 7 (Text Data).
The BernoulliNB classifier counts how often every feature of each class is not zero.
This is most easily understood with an example:
X = np.array([[0, 1, 0, 1],
[1, 0, 1, 1],
[0, 0, 0, 1],
[1, 0, 1, 0]])
y = np.array([0, 1, 0, 1])
Here, we have four data points, with four binary features each. There are two classes,
0 and 1.
For class 0 (the first and third data point), the first feature is zero 2 times and non-
zero 0 times, the second features is zero 1 time and non-zero 1 time, and so on.
These same counts are then calculated for the data points in the second class.
Counting the non-zero entries per class in essence looks like this:
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counts = {}
for label in np.unique(y):
# iterate over each class
# count (sum) entries of 1 per feature
counts[label] = X[y == label].sum(axis=0)
print(counts)
{0: array([0, 1, 0, 2]), 1: array([2, 0, 2, 1])}
The other two naive Bayes models, MultinomialNB and GaussianNB are slightly dif‐
ferent in what kind of statistics they compute. MultinomialNB takes into account the
average value of each feature for each class, while GaussianNB stores the average value
as well as the standard deviation of each feature for each class.
To make a prediction, a data point is compared to the statistics for each of the classes,
and the best matching class is predicted. Interestingly, for MultinomialNB and Ber
noulliNB, this leads to a prediction formula that is of the same form as in the linear
models (Formula (2)). Unfortunately, coef_ for the naive Bayes models has a slightly
different meaning then in the linear models, in that coef_ is not the same as w.
Strengths, weaknesses and parameters
The MultinomialNB and BernoulliNB have a single parameter alpha, which controls
model complexity. The way alpha works is that the algorithm adds alpha many vir‐
tual data points to the data, that have positive values for all the features. This results
in a “smoothing” of the statistics. A large alpha means more smoothing, resulting in
less complex models. The algorithms performance is relatively robust to the setting of
alpha, meaning that setting alpha is not critical for good performance However, tun‐
ing it usually improves accuracy somewhat.
The GaussianNB model seems to be rarely used by practitioners, while the other two
variants of naive Bayes are widely used for sparse count data such as text. Multino‐
mialNB usually performs better than BinaryNB, in particular on datasets with a rela‐
tively large number of non-zero features (i.e. large documents).
The naive Bayes models share many of the strengths and weaknesses of the linear
models. They are very fast to train and to predict, and the training procedure is easy
to understand. The models work very well with high-dimensional sparse data, and
are relatively robust to the parameters. Naive Bayes models are great baseline models,
and are often used on very large datasets, where training even a linear model might
take too long.
Decision trees
Decision trees are a widely used models for classification and regression tasks.
Essentially, they learn a hierarchy of “if-else” questions, leading to a decision.
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These questions are similar to the questions you might ask in a game of twenty ques‐
tions.
Imagine you want to distinguish between the following four animals: bears, hawks,
penguins and dolphins.
Your goal is to get to the right answer b] asking as few if-else questions as possible.
You might start off by asking whether the animal has feathers, a question that nar‐
rows down your possible animals to just two animals.
If the answer is yes, you can ask another question that could help you distinguish
between hawks and penguins. For example, you could ask whether or not the animal
can fly. If the animal doesnt have feathers, your possible animal choices are dolphins
and bears, and you will need to ask a question to distinguish between these two ani‐
mals, for example, asking whether the animal has fins.
This series of questions can be expressed as a decision tree, as shown in Figure ani‐
mal_tree.
In this illustration, each node in the tree either represents a question, or a terminal
node (also called a leaf) which contains the answer. The edges connect the answers to
a question with the next question you would ask.
mglearn.plots.plot_animal_tree()
plt.suptitle("animal_tree");
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In machine learning parlance, we built a model to distinguish between four classes of
animals (hawks, penguins, dolphins and bears) using the three features “has feathers,
can fly” and “has fins.
Instead of building these models by hand, we can learn them from data using super‐
vised learning.
Building Decision Trees
Lets go through the process of building a decision tree for the 2d classification dataset
shown at the top of Figure tree_building. The dataset consists of two half-moon
shapes of blue and red points, consisting of 75 data points each. We will refer to this
dataset as two_moons.
mglearn.plots.plot_tree_progressive()
plt.suptitle("tree_building");
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Learning a decision tree means learning a sequence of if/else questions that gets us to
the true answer most quickly.
In the machine learning setting, these questions are called tests (not to be confused
with the test set, which is the data we use to test to see how generalizable our model
is).
Usually data does not come in the form of binary yes/no features as in the animal
example, but is instead represented as continuous features such as in the 2d dataset
shown in the figure. The tests that are used on continuous data are of the from “is
feature i larger than value a.
To build a tree, the algorithm searches over all possible tests, and finds the one that is
most informative about the target variable.
The second row in Figure tree_building shows the first test that is picked. Splitting
the dataset vertically at x[1]=0.2372 yields the most information; it best separates the
blue points from the red points. The top node, also called the root, represents the
whole dataset, consisting of 75 red and 75 blue points. The split is done by testing
whether x[1] <= 0.2372, indicated by a black line. If the test is true, a point is
assigned to the left node, which contains 8 blue points and 58 red points. Otherwise
the point is assigned to the right node, which contains 67 red points and 17 blue
points. These two nodes correspond to the top and bottom region shown in Figure
tree_building.
Even though the first split did a good job of separating the blue and red points, the
bottom region still contains blue points, and the top region still contains red points.
We can build a more accurate model by repeating the process of looking for the best
test in both regions.
Figure tree_building shows that the most informative next split for the left and the
right region are based on x[0].
This recursive process yields a binary tree of decisions, with each node containing a
test.
Alternatively, you can think of each test as splitting the part of the data that is cur‐
rently considered along one axis. This yields a view of the algorithm as building a
hierarchical partition. As each test concerns only a single feature, the regions in the
resulting partition always have axis-parallel boundaries.
Figure tree_building illustrates the partitioning of the data in the left hand column,
and the resulting tree in the right hand column.
The recursive partitioning of the data is usually repeated until each region in the par‐
tition (each leaf in the decision tree) only contains a single target value (a single class
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or a single regression value). A leaf of the tree containing only one target value is
called pure.
A prediction on a new data point is made by checking which region of the partition
of the feature space the point lies in, and then predicting the majority target (or the
single target in the case of pure leaves) in that region. The region can be found by
traversing the tree from the root and going left or right, depending on whether the
test is fulfilled or not.
Controlling complexity of Decision Trees
Typically, building a tree as described above, and continuing until all leaves are pure
leads to models that are very complex and highly overfit to the training data. The
presence of pure leaves mean that a tree is 100% accurate on the training set; each
data point in the training set is in a leaf that has the correct majority class. The over‐
fitting can be seen on the left of Figure tree_building in the bottom column. You can
see the regions determined to be red in the middle of all the blue points. On the other
hand, there is a small strip of blue around the single blue point to the very right. This
is not how one would imagine the decision boundary to look, and the decision
boundary focuses a lot on single outlier points that are far away from the other points
in that class.
There are two common strategies to prevent overfitting: stopping the creation of the
tree early, also called pre-pruning, or building the tree but then removing or collaps‐
ing nodes that contain little information, also called post-pruning or just pruning. Pos‐
sible criteria for pre-pruning include limiting the maximum depth of the tree,
limiting the maximum number of leaves, or requiring a minimum number of points
in a node to keep splitting it.
Decision trees in scikit-learn are implemented in the DecisionTreeRegressor and
DecisionTreeClassifier classes. Scikit-learn only implements pre-pruning, not post-
pruning.
Lets look at the effect of pre-pruning in more detail on the breast cancer dataset.
As always, we import the dataset and split it into a training and test part.
Then we build a model using the default setting of fully developing the tree (growing
the tree until all leaves are pure). We fix the random_state in the tree, which is used
for tie-breaking internally.
from sklearn.tree import DecisionTreeClassifier
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, stratify=cancer.target, random_state=42)
tree = DecisionTreeClassifier(random_state=0)
tree.fit(X_train, y_train)
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print("accuracy on training set: %f" % tree.score(X_train, y_train))
print("accuracy on test set: %f" % tree.score(X_test, y_test))
accuracy on training set: 1.000000
accuracy on test set: 0.937063
As expected, the accuracy on the training set is 100% as the leaves are pure.
The test-set accuracy is slightly worse than the linear models above, which had
around 95% accuracy.
Now lets apply pre-pruning to the tree, which will stop developing the tree before we
perfectly fit to the training data.
One possible way is to stop building the tree after a certain depth has been reached.
Here we set max_depth=4, meaning only four consecutive questions can be asked (cf.
Figure tree_building).
tree = DecisionTreeClassifier(max_depth=4, random_state=0)
tree.fit(X_train, y_train)
print("accuracy on training set: %f" % tree.score(X_train, y_train))
print("accuracy on test set: %f" % tree.score(X_test, y_test))
accuracy on training set: 0.988263
accuracy on test set: 0.951049
Limiting the depth of the tree decreases overfitting. This leads to a lower accuracy on
the training set, but an improvement on the test set.
Analyzing Decision Trees
We can visualize the tree using the export_graphviz function from the tree module.
This writes a file in the dot file format, which is a text file format for storing graphs.
We set an option to color the nodes to reflect the majority class in each node and pass
the class and features names so the tree can be properly labeled.
from sklearn.tree import export_graphviz
export_graphviz(tree, out_file="mytree.dot", class_names=["malignant", "benign"],
feature_names=cancer.feature_names, impurity=False, filled=True)
We can read this file and visualize it using the graphviz module (or you can use any
program that can read dot files):
import graphviz
with open("mytree.dot") as f:
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dot_graph = f.read()
graphviz.Source(dot_graph)
The visualization of the tree provides a great in-depth view of how the algorithm
makes predictions, and is a good example of a machine learning algorithm that is
easily explained to non-experts. However, even with a tree of depth four, as seen here,
the tree can become a bit overwhelming. Deeper trees (depth ten is not uncommon)
are even harder to grasp.
One method of inspecting the tree that may be helpful is to find out which path most
of the data actually takes.
The n_samples shown in each node in the figure gives the number of samples in each
node, while value provides the number of samples per class.
Following the branches to the right, we see that texture_error <= 0.4732 creates a
node that only contains 8 benign but 134 malignant samples. The rest of this side of
the tree then uses some finer distinctions to split off these 8 remaining benign sam‐
ples. Of the 142 samples that went to the right in the initial split, nearly all of them
(132) end up in the leaf to the very right.
Taking a left at the root, for texture_error > 0.4732, we end up with 25 malignant
and 259 benign samples.
Nearly all of the benign samples end up in the second leave from the right, with most
of the other leaves only containing very few samples.
Feature Importance in trees
Instead of looking at the whole tree, which can be taxing, there are some useful statis‐
tics that we can derive properties that we can derive to summarize the workings of
the tree. The most commonly used summary is feature importance, which rates how
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important each feature is for the decision a tree makes. It is a number between 0 and
1 for each feature, where 0 means “not used at all” and 1 means “perfectly predicts the
target.
The feature importances always sum to one.
tree.feature_importances_
array([ 0. , 0. , 0. , 0. , 0. ,
0. , 0. , 0. , 0. , 0. ,
0.01019737, 0.04839825, 0. , 0. , 0.0024156 ,
0. , 0. , 0. , 0. , 0. ,
0.72682851, 0.0458159 , 0. , 0. , 0.0141577 ,
0. , 0.018188 , 0.1221132 , 0.01188548, 0. ])
We can visualize the feature importances in a way that is similar to the way we visual‐
ize the coefficients in the linear model.
plt.plot(tree.feature_importances_, 'o')
plt.xticks(range(cancer.data.shape[1]), cancer.feature_names, rotation=90)
plt.ylim(0, 1)
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Here, we see that the feature used at the top split (“worst radius”) is by far the most
important feature. This confirms our observation in analyzing the tree, that the first
level already separates the two classes fairly well.
However, if a feature has a low feature_importance, it doesn’t mean that this feature
is uninformative. It only means that this feature was not picked by the tree, likely
because another feature encodes the same information.
In contrast to the coefficients in linear models, feature importances are always posi‐
tive, and dont encode which class a feature is indicative of. The feature importances
tell us that worst radius is important, but it does not tell us whether a high radius is
indicative of a sample being “benign” or “malignant. In fact, there might not be such
a simple relationship between features and class, as you can see in the example below:
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tree = mglearn.plots.plot_tree_not_monotone()
plt.suptitle("tree_not_monotone")
tree
Feature importances: [ 0. 1.]
The plot shows a dataset with two features and two classes. Here, all the information
is contained in X[1], and X[0] is not used at all. But the relation between X[1] and
the output class is not monotonous, meaning we cannot say “a high value of X[0]
means class red, and a low value means class blue” or the other way around.
While we focuses our discussion here on decision trees for classification, all that was
said is similarly true for decision trees for regression, as implemented in Decision
TreeRegressor. Both the usage and the analysis of regression trees are very similar to
classification trees, so we won’t go into any more detail here.
Strengths, weaknesses and parameters
As discussed above, the parameters that control model complexity in decision trees
are the pre-pruning parameters that stop the building of the tree before it is fully
developed. Usually picking one of the pre-pruning strategies, either setting
min_depth, max_leaf_nodes or min_samples_leaf is to prevent overfitting.
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Decision trees have two advantages over many of the algorithms we discussed so far:
The resulting model can easily be visualized and understood by non-experts (at least
for smaller trees), and the algorithms is completely invariant to scaling of the data: As
each feature is processed separately, and the possible splits of the data don’t depend
on scaling, no preprocessing like normalization or standardization of features is
needed for decision tree algorithms.
In particular, decision trees work well when you have features that are on completely
different scales, or a mix of binary and continuous features.
The main down-side of decision trees is that even with the use of pre-pruning, deci‐
sion trees tend to overfit, and provide poor generalization performance. Therefore, in
most applications, the ensemble methods we discuss below are usually used in place
of a single decision tree.
Ensembles of Decision Trees
Ensembles are methods that combine multiple machine learning models to create
more powerful models.
There are many models in the machine learning literature that belong to this cate‐
gory, but there are two ensemble models that have proven to be effective on a wide
range of datasets for classification and regression, both of which use decision trees as
their building block: Random Forests and Gradient Boosted Decision Trees.
Random Forests
As observed above, a main drawback of decision trees is that they tend to overfit the
training data. Random forests
are one way to address this problem. Random forests are essentially a collection of
decision trees, where each tree is slightly different from the others.
The idea of random forests is that each tree might do a relatively good job of predict‐
ing, but will likely overfit on part of the data.
If we build many trees, all of which work well and overfit in different ways, we can
reduce the amount of overfitting by averaging their results. This reduction in overfit‐
ting, while retaining the predictive power of the trees, can be shown using rigorous
mathematics.
To implement this strategy, we need to build many decision tree. Each tree should do
an acceptable job of predicting the target, and should also be different from the other
trees. Random forests get their name from injecting randomness into the tree build‐
ing to ensure each tree is different. There are two ways in which the trees in a random
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forest are randomized: by selecting the data points used to build a tree and by select‐
ing the features in each split test. Lets go into this process in more detail.
Building Random Forests
To build a random forest model, you need to decide on the number of trees to build
(the n_estimator parameter of RandomForestRegressor or RandomForestClassi
fier). Lets say we want to build ten trees. These trees will be built completely inde‐
pendent from each other, and [will?] make random choices to make sure they are
distinct [the trees make random choices?].
To build a tree, we first take what is called a bootstrap sample of our data. A bootstrap
sample means from our n_samples data points, we repeatedly draw an example ran‐
domly with replacement (i.e. the same sample can be picked multiple times), n_sam
ples times. This will create a dataset that is as big as the original dataset, but some
data points will be missing from it, and some will be repeated.
To illustrate, lets say we want to create a bootstrap sample of the list ['a', 'b', 'c',
'd']. A possible bootstrap sample would be ['b', 'd', 'd', 'c']. Another possi‐
ble sample would be ['d', 'a', 'd', 'a'].
Next, a decision tree is built based on this newly created dataset. However, the algo‐
rithm we described for the decision tree is slightly modified. Instead of looking for
the best test for each node, in each node the algorithm randomly selects a subset of
the features, and looks for the best possible test involving one of these features. The
amount of features that is selected is controlled by the max_features parameter.
This selection of a subset of features is repeated separately in each node, so that each
node in a tree can make a decision using a different subset of the features.
The bootstrap sampling leads to each decision tree in the random forest being built
on a slightly different dataset. Because of the selection of features in each node, each
split in each tree operates on a different subset of features. Together these two mecha‐
nisms ensure that all the trees in the random forests are different.
A critical parameter in this process is max_features. If we set max_features to n_fea
tures, that means that each split can look at all features in the dataset, and no ran‐
domness will be injected. If we set max_features to one, that means that the splits
have no choice at all on which feature to test, and can only search over different
thresholds for the feature that was selected randomly.
Therefore, a high max_features means that the trees in the random forest will be
quite similar, and they will be able to fit the data easily, using the most distinctive fea‐
tures. A low max_features means that the trees in the random forest will be quite
different, and that each tree might need to be very deep in order to fit the data well.
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To make a prediction using the random forest, the algorithm first makes a prediction
for every tree in the forest. For regression, we can average these results to get our final
prediction. For classification, a “soft voting” strategy is used. This means each algo‐
rithm makes a “soft” prediction, providing a probability for each possible output
label. The probabilities predicted by all the trees are averaged, and the class with the
highest label is predicted.
Analyzing Random Forests
Lets apply a random forest consisting of five trees to the two_moon data we studied
above.
from sklearn.ensemble import RandomForestClassifier
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=100, noise=0.25, random_state=3)
X_train, X_test, y_train, y_test = train_test_split(X, y, stratify=y, random_state=42)
forest = RandomForestClassifier(n_estimators=5, random_state=2)
forest.fit(X_train, y_train)
RandomForestClassifier(bootstrap=True, class_weight=None, criterion='gini',
max_depth=None, max_features='auto', max_leaf_nodes=None,
min_samples_leaf=1, min_samples_split=2,
min_weight_fraction_leaf=0.0, n_estimators=5, n_jobs=1,
oob_score=False, random_state=2, verbose=0, warm_start=False)
The trees that are built as part of the random forest are stored in the estimator_
attribute. Lets visualize the decision boundaries learned by each tree, together with
their aggregate prediction, as made by the forest.
fig, axes = plt.subplots(2, 3, figsize=(20, 10))
for i, (ax, tree) in enumerate(zip(axes.ravel(), forest.estimators_)):
ax.set_title("tree %d" % i)
mglearn.plots.plot_tree_partition(X_train, y_train, tree, ax=ax)
mglearn.plots.plot_2d_separator(forest, X_train, fill=True, ax=axes[-1, -1], alpha=.4)
axes[-1, -1].set_title("random forest")
plt.scatter(X_train[:, 0], X_train[:, 1], c=np.array(['r', 'b'])[y_train], s=60)
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You can clearly see that the decisions learned by the five trees are quite different. Each
of them makes some mistakes, as some of the training points that are plotted here
were not actually included in the training set of the tree, due to the bootstrap sam‐
pling.
The random forest overfit less than any of the trees individually, and provides a much
more intuitive decision boundary. In any real application, we would use many more
trees (often hundreds or thousands), leading to even smoother boundaries.
Lets apply a random forest consisting of 100 trees on the breast cancer dataset:
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, random_state=0)
forest = RandomForestClassifier(n_estimators=100, random_state=0)
forest.fit(X_train, y_train)
print("accuracy on training set: %f" % forest.score(X_train, y_train))
print("accuracy on test set: %f" % forest.score(X_test, y_test))
accuracy on training set: 1.000000
accuracy on test set: 0.972028
The random forest gives us an accuracy of 97%, better than the linear models or a
single decision tree, without tuning any parameters. We could adjust the max_fea
tures setting, or apply pre-pruning as we did for the single decision tree.
However, often the default parameters of the random forest already work quite well.
Similarly to the decision tree, the random forest provides feature importances, which
are computed by aggregating the feature importances over the trees in the forest. Typ‐
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ically the feature importances provided by the random forest are more reliable than
the ones provided by a single tree.
plt.plot(forest.feature_importances_, 'o')
plt.xticks(range(cancer.data.shape[1]), cancer.feature_names, rotation=90);
As you can see, the random forest gives non-zero importance to many more features
than the single tree.Similarly to the single decision tree, the random forest also gives a
lot of importance to the “worst radius, but it actually chooses “worst perimeter” to be
the most informative feature overall. The randomness in building the random forest
forces the algorithm to consider many possible explanations, the result of which
being that the random forest captures a much broader picture of the data than a sin‐
gle tree.
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Strengths, weaknesses and parameters
Random forests for regression and classification are currently among the most widely
used machine learning methods.
They are very powerful, often work well without heavy tuning of the parameters, and
dont require scaling of the data.
Essentially, random forests share all of the benefits of decision trees, while making up
for some of their deficiencies.
One reason to still use decision trees is if you need a compact representation of the
decision making process. It is basically impossible to interpret tens or hundreds of
trees in detail, and trees in random forests tend to be deeper than decision trees
(because of the use of feature subsets). Therefore, if you need to summarize the pre‐
diction making in a visual way to non-experts, a single decision tree might be a better
choice.
While building random forests on large dataset might be somewhat time-consuming,
it an be parallelized across multiple CPU cores within a computer easily. If you are
using a multi-core processor (as nearly all modern computers do), you can use the
n_jobs parameter to adjust the number of cores to use. Using more CPU cores will
result in linear speed-ups (using two cores, the training of the random forest will be
twice as fast), but specifying n_jobs larger than the number of cores will not help.
You can set n_jobs=-1 to use all the cores in your computer.
You should keep in mind that random forests, by their nature, are random, and set‐
ting different random states (or not setting the random_state at all) can drastically
change the model that is built. The more trees there are in the forest, the more robust
it will be against the choice of random state. If you want to have reproducible results,
it is important to fix the random_state.
Random forests dont tend to perform well on very high dimensional, sparse data,
such as text data. For this kind of data, linear models might be more appropriate.
Random forests usually work well even on very large datasets, and training can easily
be parallelized over many CPU cores within a powerful computer. However, random
forests require more memory and are slower to train and to predict than linear mod‐
els. If time and memory are important in an application, it might make sense to use a
linear model instead.
The important parameters to adjust are n_estimators, max_features and possibly
pre-pruning options like max_depth. For n_estimators, larger is always better. Aver‐
aging more trees will yield a more robust ensemble. However, there are diminishing
returns, and more trees need more memory and more time to train. A common rule
of thumb is to build “as many as you have time / memory for”.
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As described above max_features determines how random each tree is, and a smaller
max_features reduces overfitting. The default values, and a good rule of thumb, are
max_features=sqrt(n_features) for classification and max_features=log2(n_fea
tures) for regression.
Adding max_features or max_leaf_nodes might sometimes improve performance. It
can also drastically reduce space and time requirements for training and prediction.
Gradient Boosted Regression Trees (Gradient Boosting Machines)
Gradient boosted regression trees is another ensemble method that combines multi‐
ple decision trees to a more powerful model. Despite the “regression” in the name,
these models can be used for regression and classification.
In contrast to random forests, gradient boosting works by building trees in a serial
manner, where each tree tries to correct the mistakes of the previous one. There is no
randomization in gradient boosted regression trees; instead, strong pre-pruning is
used. Gradient boosted trees often use very shallow trees, of depth one to five, often
making the model smaller in terms of memory, and making predictions faster.
The main idea behind gradient boosting is to combine many simple models (in this
context known as weak learners), like shallow trees. Each tree can only provide good
predictions on part of the data, and so more and more trees are added to iteratively
improve performance.
Gradient boosted trees are frequently the winning entries in machine learning com‐
petitions, and are widely used in industry. They are generally a bit more sensitive to
parameter settings than random forests, but can provide better accuracy if the param‐
eter are set correctly.
Apart from the pre-pruning and the number of trees in the ensemble, another impor‐
tant parameter of gradient boosting is the learning_rate which controls how
strongly each tree tries to correct the mistakes of the previous trees. A higher learning
rate means each tree can make stronger corrections, allowing for more complex mod‐
els. Similarly, adding more trees to the ensemble, which can be done by increasing
n_estimators, also increases the model complexity, as the model has more chances
to correct mistakes on the training set.
Here is an example of using GradientBoostingClassifier on the breast cancer data‐
set.
By default, 100 trees of maximum depth three are used, with a learning rate of 0.1.
from sklearn.ensemble import GradientBoostingClassifier
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, random_state=0)
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gbrt = GradientBoostingClassifier(random_state=0)
gbrt.fit(X_train, y_train)
print("accuracy on training set: %f" % gbrt.score(X_train, y_train))
print("accuracy on test set: %f" % gbrt.score(X_test, y_test))
accuracy on training set: 1.000000
accuracy on test set: 0.958042
As the training set accuracy is 100%, we are likely to be overfitting. To reduce overfit‐
ting, we could either apply stronger pre-pruning by limiting the maximum depth or
lower the learning rate:
gbrt = GradientBoostingClassifier(random_state=0, max_depth=1)
gbrt.fit(X_train, y_train)
print("accuracy on training set: %f" % gbrt.score(X_train, y_train))
print("accuracy on test set: %f" % gbrt.score(X_test, y_test))
accuracy on training set: 0.990610
accuracy on test set: 0.972028
gbrt = GradientBoostingClassifier(random_state=0, learning_rate=0.01)
gbrt.fit(X_train, y_train)
print("accuracy on training set: %f" % gbrt.score(X_train, y_train))
print("accuracy on test set: %f" % gbrt.score(X_test, y_test))
accuracy on training set: 0.988263
accuracy on test set: 0.965035
Both methods of decreasing the model complexity decreased the training set accuracy
as expected. In this case, lowering the maximum depth of the trees provided a signifi‐
cant improvement of the model, while lowering the learning rate only
increased the generalization performance slightly.
As for the other decision tree based models, we can again visualize the feature impor‐
tances to get more insight into our model. As we used 100 trees, it is impractical to
inspect them all, even if they are all of depth 1.
gbrt = GradientBoostingClassifier(random_state=0, max_depth=1)
gbrt.fit(X_train, y_train)
plt.plot(gbrt.feature_importances_, 'o')
plt.xticks(range(cancer.data.shape[1]), cancer.feature_names, rotation=90);
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We can see that the feature importances of the gradient boosted trees are somewhat
similar to the feature importances of the random forests, though the gradient boost‐
ing completely ignored some of the features.
As gradient boosting and random forest perform well on similar kinds of data, a
common approach is to first try random forests, which work quite robustly. If ran‐
dom forests work well, but prediction time is at a premium, or it is important to
squeeze out the last percentage of accuracy from the machine learning model, mov‐
ing to gradient boosting often helps.
If you want to apply gradient boosting to a large scale problem, it might be worth
looking into the xgboost package and its python interface, which at the time of writ‐
ing is faster (and sometimes easier to tune) than the scikit-learn implementation of
gradient boosting on many datasets.
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Strengths, weaknesses and parameters
Gradient boosted decision trees are among the most powerful and widely used mod‐
els for supervised learning.
Their main drawback is that they require careful tuning of the parameters, and may
take a long time to train.
Similarly to other tree-based models, the algorithm works well without scaling and
on a mixture of binary and continuous features. As other tree-based models, it also
often does not work well on high-dimensional sparse data.
The main parameters of the gradient boosted tree models are the number of trees
n_estimators, and the learning_rate, which controls how much each tree is
allowed to correct the mistakes of the previous trees.
These two parameters are highly interconnected, as a lower learning_rate means
that more trees are needed to build a model of similar complexity. In contrast to ran‐
dom forests, where higher n_estimators is always better, increasing n_estimators in
gradient boosting leads to a more complex model, which may lead to overfitting.
A common practice is to fit n_estimators depending on the time and memory
budget, and then search over different learning_rates.
Another important parameter is max_depth, which is usually very low for gradient
boosted models, often not deeper than five splits.
Kernelized Support Vector Machines
The next type of supervised model we will discuss is kernelized support vector
machines (SVMs).
We already saw linear support vector machines for classification in the linear model
section. Kernelized support vector machines (often just referred to as SVMs) are an
extension that allows for more complex models which are not defined simply by
hyperplanes in the input space. While there are support vector machines for classifi‐
cation and regression, we will restrict ourself to the classification case, as imple‐
mented in SVC. Similar concepts apply to support vector regression, as implemented
in SVR.
The math behind kernelized support vector machines is a bit involved, and is mostly
beyond the scope of this book.
However, we will try to give you some intuitions about the idea behind the method.
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Linear Models and Non-linear Features
As you saw in Figure linear_classifiers, linear models can be quite limiting in low-
dimensional spaces, as lines or hyperplanes have limited flexibility. One way to make
a linear model more flexible is by adding more features, for example by adding inter‐
actions or polynomials of the input features.
Lets look at the synthetic dataset we used in Figure tree_not_monotone:
X, y = make_blobs(centers=4, random_state=8)
y = y % 2
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
plt.xlabel("feature1")
plt.ylabel("feature2")
A linear model for classification can only separate points using a line, and will not be
able to do a very good job on this dataset:
from sklearn.svm import LinearSVC
linear_svm = LinearSVC().fit(X, y)
mglearn.plots.plot_2d_separator(linear_svm, X)
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
plt.xlabel("feature1")
plt.ylabel("feature2")
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Now, let’s expand the set of input features, say by also adding feature2 ** 2, the
square of the second feature, as a new feature. Instead of representing each data point
as a two-dimensional point (feature1, feature2), we now represent it as a three-
dimensional point (feature1, feature2, feature2 ** 2) (Footnote: We picked
this particular feature to add for illustration purposes. The choice is not particular
important.). This new representation is illustrated below in a three-dimensional scat‐
ter plot:
# add the squared first feature
X_new = np.hstack([X, X[:, 1:] ** 2])
from mpl_toolkits.mplot3d import Axes3D, axes3d
figure = plt.figure()
# visualize in 3D
ax = Axes3D(figure, elev=-152, azim=-26)
ax.scatter(X_new[:, 0], X_new[:, 1], X_new[:, 2], c=y, cmap=mglearn.cm2, s=60)
ax.set_xlabel("feature1")
ax.set_ylabel("feature2")
ax.set_zlabel("feature1 ** 2")
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In the new, three-dimensional representation of the data, it is now indeed possible to
separate the red and the blue points
using a linear model, a plane in three dimensions. We can confirm this by fitting a
linear model to the augmented data:
linear_svm_3d = LinearSVC().fit(X_new, y)
coef, intercept = linear_svm_3d.coef_.ravel(), linear_svm_3d.intercept_
# show linear decision boundary
figure = plt.figure()
ax = Axes3D(figure, elev=-152, azim=-26)
xx = np.linspace(X_new[:, 0].min(), X_new[:, 0].max(), 50)
yy = np.linspace(X_new[:, 1].min(), X_new[:, 1].max(), 50)
XX, YY = np.meshgrid(xx, yy)
ZZ = (coef[0] * XX + coef[1] * YY + intercept) / -coef[2]
ax.scatter(X_new[:, 0], X_new[:, 1], X_new[:, 2], c=y, cmap=mglearn.cm2, s=60)
ax.plot_surface(XX, YY, ZZ, rstride=8, cstride=8, alpha=0.3)
ax.set_xlabel("feature1")
ax.set_ylabel("feature2")
ax.set_zlabel("feature1 ** 2")
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As a function of the original features, the linear SVM model is not actually linear any‐
more. It is not a line, but more of an ellipse.
ZZ = YY ** 2
dec = linear_svm_3d.decision_function(np.c_[XX.ravel(), YY.ravel(), ZZ.ravel()])
plt.contourf(XX, YY, dec.reshape(XX.shape), levels=[dec.min(), 0, dec.max()],
cmap=mglearn.cm2, alpha=0.5)
plt.scatter(X[:, 0], X[:, 1], c=y, s=60, cmap=mglearn.cm2)
plt.xlabel("feature1")
plt.ylabel("feature2")
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The Kernel Trick
The lesson here is that adding non-linear features to the representation of our data
can make linear models much more powerful. However, often we dont know which
features to add, and adding many features (like all possible interactions in a 100
dimensional feature space) might make computation very expensive.
Luckily, there is a clever mathematical trick that allows us to learn a classifier in a
higher dimensional space without actually computing the new, possibly very large
representation. This trick is known as the kernel trick.
The kernel trick works by directly computing the distance (more precisely, the scalar
products) of the data points for the expanded feature representation, without ever
actually computing the expansion.
There are two ways to map your data into a higher dimensional space that are com‐
monly used with support vector machines: the polynomial kernel, which computes all
possible polynomials up to a certain degree of the original features (like feature1 **
2 * feature2 ** 5), and the radial basis function (rbf) kernel, also known as Gaus‐
sian kernel.
The Gaussian kernel is a bit harder to explain, as it corresponds to an infinite dimen‐
sional feature space. One way to explain the Gaussian kernel is that it considers all
possible polynomials of all degrees, but the importance of the features decreases for
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higher degrees. [Footnote: this follows from the Taylor expansion of the exponential
map].
If all of this is too much math talk for you, don’t worry. You can still use SVMs
without trying to imagine infinite dimensional feature spaces. In practice, how a
SVM with an rbf kernel makes a decision can be summarized quite easily.
Understanding SVMs
During training, the SVM learns how important each of the training data points is to
represent the decision boundary between the two classes. Typically only a subset of
the training points matter for defining the decision boundary: the ones that lie on the
border between the classes. These are called support vectors and give the support vec‐
tor machine its name.
To make a prediction for a new point, the distance to the support vectors is measured.
A classification decision is made based on the distance to the support vectors, and the
importance of the support vectors that was learned during training (stored in the
dual_coef_ attribute of SVC).
The way distance between data points is measured by the Gaussian kernel:
\begin{align*}
&k_\text{rbf}(x_1, x_2) = \exp(\gamma||x_1 - x_2||^2) &\text{ (4) Gaussian kernel}
\end{align*}
Here, $x_1$ and $x_2$ are data points, $||x_1 - x_2 ||$ denotes Euclidean distance
and $\gamma$ is a parameter that controls the width of the Gaussian kernel.
Below is the result of training an support vector machine on a two-dimensional two-
class dataset.
The decision boundary is shown in black, and the support vectors are the points with
wide black circles.
from sklearn.svm import SVC
X, y = mglearn.tools.make_handcrafted_dataset()
svm = SVC(kernel='rbf', C=10, gamma=0.1).fit(X, y)
mglearn.plots.plot_2d_separator(svm, X, eps=.5)
# plot data
plt.scatter(X[:, 0], X[:, 1], s=60, c=y, cmap=mglearn.cm2)
# plot support vectors
sv = svm.support_vectors_
plt.scatter(sv[:, 0], sv[:, 1], s=200, facecolors='none', zorder=10, linewidth=3)
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In this case, the SVM yields a very smooth and non-linear (not a straight line)
boundary.
There are two parameters we adjusted here: The C parameter and the gamma parame‐
ter, which we will now discuss in detail.
Tuning SVM parameters
The gamma parameter is the one shown in Formula (4), which controls the width of
the Gaussian kernel. It determines the scale of what it means for points to be close
together.
The C parameter is a regularization parameter similar to the linear models. It limits
the importance of each point (or more precisely, their dual_coef_).
Lets have a look at what happens when we vary these parameters:
fig, axes = plt.subplots(3, 3, figsize=(15, 10))
for ax, C in zip(axes, [-1, 0, 3]):
for a, gamma in zip(ax, range(-1, 2)):
mglearn.plots.plot_svm(log_C=C, log_gamma=gamma, ax=a)
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Going from left to right, we increase the parameter gamma from 0.1 to 10. A small
gamma means a large radius for the Gaussian kernel, which means that many points
are considered close-by. This is reflected in very smooth decision boundaries on the
left, and boundaries that focus more on single points further to the right. A low value
of gamma means that the decision boundary will vary slowly, which yields a model of
low complexity, while a high value of gamma yields a more complex model.
Going from top to bottom, we increase the C parameter from 0.1 to 1000. As with the
linear models, a small C means a very restricted model, where each data point can
only have very limited influence. You can see that in the top left, the decision bound‐
ary looks nearly linear, with the red and blue points that are misclassified barely
changing the line.
Increasing C, as shown on the bottom right, allows these points to have a stronger
influence on the model, and makes the decision boundary bend to correctly classify
them.
Lets apply the rbf kernel SVM to the breast cancer dataset. By default, C=1 and
gamma=1./n_features.
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, random_state=0)
svc = SVC()
svc.fit(X_train, y_train)
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print("accuracy on training set: %f" % svc.score(X_train, y_train))
print("accuracy on test set: %f" % svc.score(X_test, y_test))
accuracy on training set: 1.000000
accuracy on test set: 0.629371
The model overfit quite substantially, with a perfect score on the training set and only
62% accuracy on the test set.
While SVMs often perform quite well, they are very sensitive to the settings of the
parameters, and to the scaling of the data. In particular, they require all the features to
vary on a similar scale. Lets look at the minimum and maximum values for each fea‐
ture, plotted in log-space:
plt.plot(X_train.min(axis=0), 'o', label="min")
plt.plot(X_train.max(axis=0), 'o', label="max")
plt.legend(loc="best")
plt.yscale("log")
From this plot we can determine that features in the breast cancer dataset are of com‐
pletely different orders of magnitude.
This can be somewhat of a problem for other models (like linear models), but it has
devastating effects for the kernel SVM.
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Preprocessing Data for SVMs
One way to resolve this problem is by rescaling each feature, so that they are approxi‐
mately on the same scale.
A common rescaling methods for kernel SVMs is to scale the data such that all fea‐
tures are between zero and one. We will see how to do this using the MinMaxScaler
preprocessing method in Chapter 3 (Unsupervised Learning), where we’ll give more
details.
For now, lets do this “by hand”:
# Compute the minimum value per feature on the training set
min_on_training = X_train.min(axis=0)
# Compute the range of each feature (max - min) on the training set
range_on_training = (X_train - min_on_training).max(axis=0)
# subtract the min, divide by range
# afterwards min=0 and max=1 for each feature
X_train_scaled = (X_train - min_on_training) / range_on_training
print("Minimum for each feature\n%s" % X_train_scaled.min(axis=0))
print("Maximum for each feature\n %s" % X_train_scaled.max(axis=0))
Minimum for each feature
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
Maximum for each feature
[ 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. 1. 1.]
# use THE SAME transformation on the test set,
# using min and range of the training set. See Chapter 3 (unsupervised learning) for details.
X_test_scaled = (X_test - min_on_training) / range_on_training
svc = SVC()
svc.fit(X_train_scaled, y_train)
print("accuracy on training set: %f" % svc.score(X_train_scaled, y_train))
print("accuracy on test set: %f" % svc.score(X_test_scaled, y_test))
accuracy on training set: 0.948357
accuracy on test set: 0.951049
Scaling the data made a huge difference! Now we are actually in an underfitting
regime, where training and test set performance are quite similar. From here, we can
try increasing either C or gamma to fit a more complex model:
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svc = SVC(C=1000)
svc.fit(X_train_scaled, y_train)
print("accuracy on training set: %f" % svc.score(X_train_scaled, y_train))
print("accuracy on test set: %f" % svc.score(X_test_scaled, y_test))
accuracy on training set: 0.988263
accuracy on test set: 0.972028
Here, increasing C allows us to improve the model significantly, resulting in 97.2%
accuracy.
Strengths, weaknesses and parameters
Kernelized support vector machines are very powerful models and perform very well
on a variety of datasets.
SVMs allow for very complex decision boundaries, even if the data has only a few fea‐
tures. SVMs work well on low-dimensional and high-dimensional data (i.e. few and
many features), but dont scale very well with the number of samples. Running on
data with up to 10000 samples might work well, but working with datasets of size
100000 or more can become challenging in terms of runtime and memory usage.
Another downside of SVMs is that they require careful preprocessing of the data and
tuning of the parameters.
For this reason, SVMs have been replaced by tree-based models such as random for‐
ests (that require little or no preprocessing) in many applications. Furthermore, SVM
models are hard to inspect; it can be difficult to understand why a particular predic‐
tion was made, and it might be tricky to explain the model to a non-expert.
Still it might be worth trying SVMs, particularly if all of your features represent meas‐
urements in similar units (i.e. all are pixel intensities) and are on similar scales.
The important parameters in kernel SVMs are the regularization parameter C, the
choice of the kernel, and the kernel-specific parameters. We only talked about the rbf
kernel in any depth above, but other choices are available in scikit-learn. The rbf ker‐
nel has only one parameter, gamma, which is the inverse of the width of the Gaussian
kernel. gamma and C both control the complexity of the model, with large values in
either resulting in a more complex model. Therefore, good settings for the two
parameters are usually strongly correlated, and C and gamma should be adjusted
together.
Neural Networks (Deep Learning)
A family of algorithms known as neural networks has recently seen a revival under
the name “deep learning”.
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While deep learning shows great promise in many machine learning applications,
many deep learning algorithms are tailored very carefully to a specific use-case. Here,
we will only discuss some relatively simple methods, namely multilayer perceptrons
for classification and regression, that can serve as a starting point for more involved
deep learning methods. Multilayer perceptrons (MLPs) are also known as (vanilla)
feed-forward neural networks, or sometimes just neural networks.
The Neural Network Model
MLPs can be viewed as generalizations of linear models which perform multiple
stages of processing to come to a decision.
Remember that the prediction by a linear regressor is given as:
In words, y is a weighted sum of the input features x[0] to x[p], weighted by the
learned coefficients w[0] to w[p]. We could visualize this graphically as:
mglearn.plots.plot_logistic_regression_graph()
where each node on the left represents an input feature, the connecting lines repre‐
sent the learned coefficients, and the node on the right represents the output, which is
a weighted sum of the inputs.
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In an MLP, this process of computing weighted sums is repeated multiple times, first
computing hidden units that represent an intermediate processing step, which are
again combined using weighted sums, to yield the final result:
print("Figure single_hidden_layer")
mglearn.plots.plot_single_hidden_layer_graph()
Figure single_hidden_layer
This model has a lot more coefficients (also called weights) to learn: there is one
between every input and every hidden unit (which make up the hidden layer), and
one between every unit in the hidden layer and the output.
Computing a series of weighted sums is mathematically the same as computing just
one weighted sum, so to make this model truly more powerful than a linear model,
there is one extra trick we need. After computing a weighted sum for each hidden
unit, a non-linear function is applied to the result, usually the rectifying nonlinearity
(also known as rectified linear unit or relu) or the tangens hyperbolicus (tanh). The
result of this function is then used in the weighted sum that computes the output y.
The two functions are visualized in Figure activation_functions. The relu cuts off val‐
ues below zero, while tanh saturates to -1 for low input values and +1 for high input
values. Either non-linear function allows the neural network to learn much more
complicated function than a linear model could.
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line = np.linspace(-3, 3, 100)
plt.plot(line, np.tanh(line), label="tanh")
plt.plot(line, np.maximum(line, 0), label="relu")
plt.legend(loc="best")
plt.title("activation_functions")
For the small neural network pictures in Figure single_hidden_layer above, the full
formula for computing y in the case of regression would be (when using a tanh non-
linearity):
Here, w are the weights between the input x and the hidden layer h, and v are the
weights between the hidden layer h and the output y. The weights v and w are learned
from data, x are the input features, y is the computed output, and h are intermediate
computations.
An important parameter that needs to be set by the user is the number of nodes in the
hidden layer, and can be as small as 10 for very small or simple datasets, and can be as
big as 10000 for very complex data.
It is also possible add additional hidden layers, as in Figure two_hidden_layers below.
Having large neural networks made up of many of these layers of computation is
what inspired the term “deep learning”.
print("Figure two_hidden_layers")
mglearn.plots.plot_two_hidden_layer_graph()
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Figure two_hidden_layers
Tuning Neural Networks
Lets look into the workings of the MLP by applying the MLPClassifier to the
two_moons dataset we saw above.
from sklearn.neural_network import MLPClassifier
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=100, noise=0.25, random_state=3)
X_train, X_test, y_train, y_test = train_test_split(X, y, stratify=y, random_state=42)
mlp = MLPClassifier(algorithm='l-bfgs', random_state=0).fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3)
plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
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As you can see, the neural network learned a very nonlinear but relatively smooth
decision boundary.
We used algorithm='l-bfgs' which we will discuss later.
By default, the MLP uses 100 hidden nodes, which is quite a lot for this small dataset.
We can reduce the number (which reduces the complexity of the model) and still get
a good result:
mlp = MLPClassifier(algorithm='l-bfgs', random_state=0, hidden_layer_sizes=[10])
mlp.fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3)
plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
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With only 10 hidden units, the decision boundary looks somewhat more ragged. The
default nonlinearity is ‘relu', shown in Figure activation_function. With a single hid‐
den layer, this means the decision function will be made up of 10 straight line seg‐
ments. If we want a smoother decision boundary, we could either add more hidden
units (as in the figure above), add second hidden layer, or use the “tanh” nonlinearity:
# using two hidden layers, with 10 units each
mlp = MLPClassifier(algorithm='l-bfgs', random_state=0, hidden_layer_sizes=[10, 10])
mlp.fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3)
plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
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# using two hidden layers, with 10 units each, now with tanh nonlinearity.
mlp = MLPClassifier(algorithm='l-bfgs', activation='tanh',
random_state=0, hidden_layer_sizes=[10, 10])
mlp.fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3)
plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
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Finally, we can also control the complexity of a neural network by using an “l2” pen‐
alty to shrink the weights towards zero, as we did in ridge regression and the linear
classifiers. The parameter for this in the MLPClassifier is alpha (as in the linear
regression models), and is set to a very low value (little regularization) by default.
Here is the effect of different values of alpha on the two_moons dataset, using two
hidden layers of 10 or 100 units each:
fig, axes = plt.subplots(2, 4, figsize=(20, 8))
for ax, n_hidden_nodes in zip(axes, [10, 100]):
for axx, alpha in zip(ax, [0.0001, 0.01, 0.1, 1]):
mlp = MLPClassifier(algorithm='l-bfgs', random_state=0,
hidden_layer_sizes=[n_hidden_nodes, n_hidden_nodes],
alpha=alpha)
mlp.fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3, ax=axx)
axx.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
axx.set_title("n_hidden=[%d, %d]\nalpha=%.4f"
% (n_hidden_nodes, n_hidden_nodes, alpha))
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As you probably have realized by now, there are many ways to control the complexity
of a neural network: the number of hidden layers, the number of units in each hidden
layer, and the regularization (alpha). There are actually even more, which we wont go
into here.
An important property of neural networks is that their weights are set randomly
before learning is started, and this random initialization affects the model that is
learned. That means that even when using exactly the same parameters, we can
obtain very different models when using different random seeds.
If the networks are large, and their complexity is chosen properly, this should not
affect accuracy too much, but it is worth keeping in mind (particularly for smaller
networks).
Here are plots of several models, all learned with the same settings of the parameters:
fig, axes = plt.subplots(2, 4, figsize=(20, 8))
for i, ax in enumerate(axes.ravel()):
mlp = MLPClassifier(algorithm='l-bfgs', random_state=i,
hidden_layer_sizes=[100, 100])
mlp.fit(X_train, y_train)
mglearn.plots.plot_2d_separator(mlp, X_train, fill=True, alpha=.3, ax=ax)
ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, s=60, cmap=mglearn.cm2)
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To get a better understanding of neural networks on real-world data, lets apply the
MLPClassifier to the breast cancer dataset. We start with the default parameters:
X_train, X_test, y_train, y_test = train_test_split(
cancer.data, cancer.target, random_state=0)
mlp = MLPClassifier()
mlp.fit(X_train, y_train)
print("accuracy on training set: %f" % mlp.score(X_train, y_train))
print("accuracy on test set: %f" % mlp.score(X_test, y_test))
accuracy on training set: 0.373239
accuracy on test set: 0.370629
As you can see, the result on both the training and the test set are devastatingly bad
(even worse than random guessing!). As in the SVC example above, this is likely due
to scaling of the data. Neural networks also expect all input features to vary in a simi‐
lar way, and ideally should have a mean of zero, and a variance of one.
We [must] rescale our data so that it fulfills these requirements. Again, we will do this
“by hand” here, but introduce the StandardScaler to do this automatically in Chap‐
ter 3 (Unsupervised Learning).
# compute the mean value per feature on the training set
mean_on_train = X_train.mean(axis=0)
# compute the standard deviation of each feature on the training set
std_on_train = X_train.std(axis=0)
# subtract the mean, scale by inverse standard deviation
# afterwards, mean=0 and std=1
X_train_scaled = (X_train - mean_on_train) / std_on_train
# use THE SAME transformation (using training mean and std) on the test set
X_test_scaled = (X_test - mean_on_train) / std_on_train
mlp = MLPClassifier(random_state=0)
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mlp.fit(X_train_scaled, y_train)
print("accuracy on training set: %f" % mlp.score(X_train_scaled, y_train))
print("accuracy on test set: %f" % mlp.score(X_test_scaled, y_test))
/home/andy/checkout/scikit-learn/sklearn/neural_network/multilayer_perceptron.py:560: ConvergenceWarning: Stochastic Optimizer: Maximum iterations reached and the optimization hasn't converged yet.
% (), ConvergenceWarning)
The results are much better after scaling, and already quite competative. We got a
warning from the model, though, that tells us that the maximum number of iterations
has been reached. This is part of the adam algorithm for learning the model, and tells
us that we should increase the number of iterations:
mlp = MLPClassifier(max_iter=1000, random_state=0)
mlp.fit(X_train_scaled, y_train)
print("accuracy on training set: %f" % mlp.score(X_train_scaled, y_train))
print("accuracy on test set: %f" % mlp.score(X_test_scaled, y_test))
accuracy on training set: 0.995305
accuracy on test set: 0.965035
Increasing the number of iterations only increased the training set performance, but
not the generalization performance. Still, the model is performing quite well. As there
is some gap between the training
and the test performance, we might try to decrease the model complexity to get better
generalization performance. Here, we choose to increase the alpha parameter (quite
aggressively, from 0.0001 to 1), to add stronger regularization of the weights.
mlp = MLPClassifier(max_iter=1000, alpha=1, random_state=0)
mlp.fit(X_train_scaled, y_train)
print("accuracy on training set: %f" % mlp.score(X_train_scaled, y_train))
print("accuracy on test set: %f" % mlp.score(X_test_scaled, y_test))
accuracy on training set: 0.988263
accuracy on test set: 0.972028
This leads to a performance on par with the best models so far. [Footnote: You might
have noticed at this point that many of the well-performing models achieved exactly
the same accuracy of 0.972. This means that all of the models make exactly the same
number of mistakes, which is four. If you comparing the actual predictions, you can
even see that they make exactly the same mistakes! This might be either a conse‐
quence of data being very small, or may be because these points are really different
from the rest.]
While it is possible to analyze what a neural network learned, this is usually much
trickier than analyzing a linear model or a tree-based model. One way to introspect
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what was learned is to look at the weights in the model. You can see an example of
this in the scikit-learn example gallery on the website. For the breast cancer dataset,
this might be a bit hard to understand.
The plot below shows the weights that were learned connecting the input to the first
hidden layer.
The rows in this plot correspond to the 30 input features, while the columns corre‐
spond to the 100 hidden units.
Light green represents large positive values, while dark blue represents negative val‐
ues.
plt.figure(figsize=(20, 5))
plt.imshow(mlp.coefs_[0], interpolation='none', cmap='viridis')
plt.yticks(range(30), cancer.feature_names)
plt.colorbar()
One possible inference we can make is that features that have very small weights for
all of the hidden units are “less important” to the model. We can see that “mean
smoothness” and “mean compactness” in addition to the features found between
smoothness error” and “fractal dimension error” have relatively low weights com‐
pared to other features. This could mean that these are less important features, or,
possibly, that we didnt represent them in a way that the neural network could use.
We could also visualize the weights connecting the hidden layer to the output layer,
but those are even harder to interpret.
While the MLPClassifier and MLPRegressor provide easy-to-use interfaces for the
most common neural network architectures, they only capture a small subset of what
is possible with neural networks. If you are interested in working with more flexible
or larger models, we encourage you to look beyond scikit-learn into the fantastic deep
learning libraries that are our there. For python users, the most well-established are
keras, lasagna and tensor-flow. Keras and lasagna both build on the theano library.
These libraries provide a much more flexible interface to build neural networks, and
track the rapid process in deep learning research. All of the popular deep learning
libraries also allow the use of high-performance graphic processing units (GPUs),
which scikit-learn does not support.
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Using GPUs allows to accelerate computations by factors of 10x to 100x, and are
essential for applying deep learning methods to large-scale datasets.
Strengths, weaknesses and parameters
Neural networks have re-emerged as state of the art models in many applications of
machine learning. One of their main advantages is that they are able to capture infor‐
mation contained in large amounts of data and build incredibly complex models.
Given enough computation time, data, and careful tuning of the parameters, neural
networks often beat other machine learning algorithms (for classification and regres‐
sion tasks).
This brings us to the downsides; neural networks, in particular the large and powerful
ones, often take a long time to train. They also require careful preprocessing of the
data, as we saw above. Similarly to SVMs, they work best with “homogeneous” data,
where all the features have similar meanings. For data that has very different kinds of
features, tree-based models might work better.
Tuning neural network parameters is also an art onto itself. In our experiments
above, we barely scratched the surface of possible ways to adjust neural network
models, and how to train them.
Estimating complexity in neural networks
The most important parameters are the number of layers and the number of hidden
units per layer. You should start with one or two hidden layers, and possibly expand
from there. The number of nodes per hidden layer is often around the number of the
input features, but rarely higher than in the low to mid thousands.
A helpful measure when thinking about model complexity of a neural network is the
number of weights or coefficients that are learned. If you have a binary classification
dataset with 100 features, and you have 100 hidden units, then there are 100 * 100 =
10,000 weights between the input and the first hidden layer. There are also 100 * 1
= 100 weights between the hidden layer and the output layer, for a total of around
10,100 weights. If you add a second hidden layer with 100 hidden units, there will be
another 100 * 100 = 10,000 weights from the first hidden layer to the second hid‐
den layer, resulting in a total of 20,100 weights.
If instead, you use one layer with 1000 hidden units, you are learning 100 * 1000 =
100,000 weights from the input to the hidden layer, and 1000 x 1 weights from the
hidden to the output layer, for a total of 101,000.
If you add a second hidden layer, you add 1000 * 1000 = 1,000,000 weights, for a
whopping 1,101,000, which is 50 times larger than the model with two hidden layers
of size 100.
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A common way to adjust parameters in a neural network is to first create a network
that is large enough to overfit, making sure that the task can actually be learned by
the network. Once you know the training data can be learned, either shrink the net‐
work or increase alpha to add regularization, which will improve generalization per‐
formance.
During our experiments above, we focused mostly on the definition of the model: the
number of layers and nodes per layer, the regularization, and the nonlinearity. These
define the model we want to learn. There is also the question of how to learn the
model, or the algorithm that is used for learning of the parameters, which is set using
the algorithm parameter.
There are two easy-to-use choices for the algorithm. The default is 'adam', which
works well in most situations but is quite sensitive to the scaling of the data (so it is
important to always scale your data to zero mean and unit variance). The other one is
'l-bfgs', which is quite robust, but might take a long time on larger models or larger
datasets.
There is also the more advanced 'sgd' option, which is what many deep learning
researchers use. The 'sgd' option comes with many additional parameters that need
to be tuned for best results. You can find all of these parameters and their definitions
in the user-guide. When starting to work with MLPs, we recommend sticking to adam
and l-bfgs.
Uncertainty estimates from classiers
Another useful part of the scikit-learn interface that we havent talked about yet is the
ability of classifiers to provide uncertainty estimates of predictions.
Often, you are not only interested in which class a classifier predicts for a certain test
point, but also how certain it is that this is the right class. In practice, different kinds
of mistakes lead to very different outcomes in real world applications. Imagine a
medical application testing for cancer. Making a false positive prediction might lead
to a patient undergoing additional tests, while a false negative prediction might lead
to a serious disease not being treated.
We will go into this topic in more detail in Chapter 6 (Model Selection).
There are two different functions in scikit-learn that can be used to obtain uncer‐
tainty estimates from classifiers, decision_function and predict_proba. Most (but
not all) classifiers have at least one of them, and many classifiers have both. Lets look
at what these two functions do on as synthetic two-dimensional dataset, when build‐
ing a GradientBoostingClassifier classifier. GradientBoostingClassifier has
both a decision_function method and a predict_proba.
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# create and split a synthetic dataset
from sklearn.ensemble import GradientBoostingClassifier
from sklearn.datasets import make_blobs, make_circles
# X, y = make_blobs(centers=2, random_state=59)
X, y = make_circles(noise=0.25, factor=0.5, random_state=1)
# we rename the classes "blue" and "red" for illustration purposes:
y_named = np.array(["blue", "red"])[y]
# we can call train test split with arbitrary many arrays
# all will be split in a consistent manner
X_train, X_test, y_train_named, y_test_named, y_train, y_test = \
train_test_split(X, y_named, y, random_state=0)
# build the gradient boosting model model
gbrt = GradientBoostingClassifier(random_state=0)
gbrt.fit(X_train, y_train_named)
GradientBoostingClassifier(init=None, learning_rate=0.1, loss='deviance',
max_depth=3, max_features=None, max_leaf_nodes=None,
min_samples_leaf=1, min_samples_split=2,
min_weight_fraction_leaf=0.0, n_estimators=100,
presort='auto', random_state=0, subsample=1.0, verbose=0,
warm_start=False)
The Decision Function
In the binary classification case, the return value of decision_function is of shape
(n_samples,), it returns one floating point number for each sample:
print(X_test.shape)
print(gbrt.decision_function(X_test).shape)
(25, 2)
(25,)
This value encodes how strongly the model believes a data point to belong to the
positive” class, in this case class 1.
Positive values indicate a preference for the positive class, negative values indicate
preference for the “negative, that is the other class:
# show the first few entries of decision_function
gbrt.decision_function(X_test)[:6]
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array([ 4.13592629, -1.68343075, -3.95106099, -3.6261613 , 4.28986668,
3.66166106])
We can recover the prediction by looking only at the sign of the decision function:
print(gbrt.decision_function(X_test) > 0)
print(gbrt.predict(X_test))
[ True False False False True True False True True True False True
True False True False False False True True True True True False
False]
['red' 'blue' 'blue' 'blue' 'red' 'red' 'blue' 'red' 'red' 'red' 'blue'
'red' 'red' 'blue' 'red' 'blue' 'blue' 'blue' 'red' 'red' 'red' 'red'
'red' 'blue' 'blue']
For binary classification, the “negative” class is always the first entry of the classes_
attribute, and the “positive” class is the second entry of classes_. So if you want to
fully recover the output of predict, you need to make use of the classes_ attribute:
# make the boolean True/False into 0 and 1
greater_zero = (gbrt.decision_function(X_test) > 0).astype(int)
# use 0 and 1 as indices into classes_
pred = gbrt.classes_[greater_zero]
# pred is the same as the output of gbrt.predict
np.all(pred == gbrt.predict(X_test))
True
The range of decision_function can be arbitrary, and depends on the data and the
model parameters:
decision_function = gbrt.decision_function(X_test)
np.min(decision_function), np.max(decision_function)
(-7.6909717730121798, 4.289866676868515)
This arbitrary scaling makes the output of decision_function often hard to inter‐
pret.
Below we plot the decision_function for all points in the 2d plane using a color
coding, next to a visualization of the decision boundary, as we saw it in Chapter 2. We
show training points as circles and test data as triangles.
Encoding not only the predicted outcome, but also how certain the classifier is pro‐
vides additional information. However, in this visualization, it is hard to make out the
boundary between the two classes.
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fig, axes = plt.subplots(1, 2, figsize=(13, 5))
mglearn.tools.plot_2d_separator(gbrt, X, ax=axes[0], alpha=.4, fill=True, cm=mglearn.cm2)
scores_image = mglearn.tools.plot_2d_scores(gbrt, X, ax=axes[1], alpha=.4, cm='bwr')
for ax in axes:
# plot training and test points
ax.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap=mglearn.cm2, s=60, marker='^')
ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=mglearn.cm2, s=60)
plt.colorbar(scores_image, ax=axes.tolist())
Predicting probabilities
The output of predict_proba however is a probability for each class, and is often
more easily understood. It is always of shape (n_samples, 2) for binary classifica‐
tion:
gbrt.predict_proba(X_test).shape
(25, 2)
The first entry in each row is the estimated probability of the first class, the second
entry is the estimated probability of the second class. Because it is a probability, the
output of predict_proba is always between zero and 1, and the sum of the entries for
both classes is always 1:
np.set_printoptions(suppress=True, precision=3)
# show the first few entries of predict_proba
gbrt.predict_proba(X_test[:6])
array([[ 0.016, 0.984],
[ 0.843, 0.157],
[ 0.981, 0.019],
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[ 0.974, 0.026],
[ 0.014, 0.986],
[ 0.025, 0.975]])
Because the probabilities for the two classes sum to one, exactly one of the classes is
above 50% certainty. That class is the one that is predicted.
You can see in the output above, that the classifier is relatively certain for most points.
How well the uncertainty actually reflects uncertainty in the data depends on the
model and parameters. A model that is more overfit tends to make more certain pre‐
dictions, even if they might be wrong. A model with less complexity usually has more
uncertainty in predictions. A model is called calibrated if the reported uncertainty
actually matches how correct it is - in a calibrated model, a prediction made with 70%
certainty would be correct 70% of the time.
Below we show again the decision boundary on the dataset, next to the class proba‐
bilities for the blue class:
fig, axes = plt.subplots(1, 2, figsize=(13, 5))
mglearn.tools.plot_2d_separator(gbrt, X, ax=axes[0], alpha=.4,
fill=True, cm=mglearn.cm2)
scores_image = mglearn.tools.plot_2d_scores(gbrt, X, ax=axes[1], alpha=.4,
cm='bwr', function='predict_proba')
for ax in axes:
# plot training and test points
ax.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap=mglearn.cm2, s=60, marker='^')
ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=mglearn.cm2, s=60)
plt.colorbar(scores_image, ax=axes.tolist())
The boundaries in this this plot are much more well-defined, and the small areas of
uncertainty are clearly visible.
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The scikit-learn website [Footnote: http://scikit-learn.org/stable/auto_examples/clas‐
sification/plot_classifier_comparison.html] has a great comparison of many models,
and how their uncertainty estimates look like.
We reproduced the figure below, and encourage you to go though the example there.
Uncertainty in multi-class classication
Above we only talked about uncertainty estimates in binary classification. But the
decision_function and predict_proba methods also work in the multi-class setting.
Lets apply them on the iris dataset, which is a three-class classification dataset:
from sklearn.datasets import load_iris
iris = load_iris()
X_train, X_test, y_train, y_test = train_test_split(
iris.data, iris.target, random_state=42)
gbrt = GradientBoostingClassifier(learning_rate=0.01, random_state=0)
gbrt.fit(X_train, y_train)
GradientBoostingClassifier(init=None, learning_rate=0.01, loss='deviance',
max_depth=3, max_features=None, max_leaf_nodes=None,
min_samples_leaf=1, min_samples_split=2,
min_weight_fraction_leaf=0.0, n_estimators=100,
presort='auto', random_state=0, subsample=1.0, verbose=0,
warm_start=False)
print(gbrt.decision_function(X_test).shape)
# plot the first few entries of the decision function
print(gbrt.decision_function(X_test)[:6, :])
(38, 3)
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[[-0.529 1.466 -0.504]
[ 1.512 -0.496 -0.503]
[-0.524 -0.468 1.52 ]
[-0.529 1.466 -0.504]
[-0.531 1.282 0.215]
[ 1.512 -0.496 -0.503]]
In the multi-class case, the decision_function has the shape (n_samples,
n_classes), and each column provides a “certainty score” for each class, where a
large score means that a class is more likely, and a small score means the class is less
likely. You can recover the prediction from these scores by finding the maximum
entry for each data point:
print(np.argmax(gbrt.decision_function(X_test), axis=1))
print(gbrt.predict(X_test))
[1 0 2 1 1 0 1 2 1 1 2 0 0 0 0 1 2 1 1 2 0 2 0 2 2 2 2 2 0 0 0 0 1 0 0 2 1
0]
[1 0 2 1 1 0 1 2 1 1 2 0 0 0 0 1 2 1 1 2 0 2 0 2 2 2 2 2 0 0 0 0 1 0 0 2 1
0]
The output of predict_proba has the same shape, (n_samples, n_classes). Again,
the probabilities for the possible classes for each data point sum to one:
# show the first few entries of predict_proba
print(gbrt.predict_proba(X_test)[:6])
# show that sums across rows are one
print("sums: %s" % gbrt.predict_proba(X_test)[:6].sum(axis=1))
[[ 0.107 0.784 0.109]
[ 0.789 0.106 0.105]
[ 0.102 0.108 0.789]
[ 0.107 0.784 0.109]
[ 0.108 0.663 0.228]
[ 0.789 0.106 0.105]]
sums: [ 1. 1. 1. 1. 1. 1.]
We can again recover the predictions by computing the argmax of predict_proba:
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print(np.argmax(gbrt.decision_function(X_test), axis=1))
print(gbrt.predict(X_test))
[1 0 2 1 1 0 1 2 1 1 2 0 0 0 0 1 2 1 1 2 0 2 0 2 2 2 2 2 0 0 0 0 1 0 0 2 1
0]
[1 0 2 1 1 0 1 2 1 1 2 0 0 0 0 1 2 1 1 2 0 2 0 2 2 2 2 2 0 0 0 0 1 0 0 2 1
0]
To summarize, predict_proba and decision_function always have shape (n_sam
ples, n_classes) -- apart from the special case of decision_function in the binary
case. In the binary case, decision_function only has one column, corresponding to
the “positive” class classes_[1]. This is mostly for historical reasons.
You can recover the prediction when there are n_classes many columns by simply
computing the argmax across columns.
Be careful, though, if your classes are strings, or you use integers, but they are not
consecutive and starting from 0. If you want to compare results obtained with pre
dict to results obtained via decision_function or predict_proba make sure to use
the classes_ attribute of the classifier to get the actual class names.
Summary and Outlook
We started this chapter with a discussion of model complexity, and discussed general‐
ization, or learning a model that is able to perform well on new, unseen data. This led
us to the concepts of underfitting, which describe a model that can not capture the
variations present in the training data, and overfitting, which describe a model that
focuses too much on the training data, and is not able to generalize to new data very
well.
We then discussed a wide array of machine learning models for classification and
regression, what their advantages and disadvantages are, and how to control model
complexity for each of them.
We saw that for many of the algorithms, setting the right parameters is important for
good performance. Some of the algorithms are also sensitive to how we represent the
input data, in particular to how the features are scaled.
Therefore, blindly applying an algorithm to a dataset without understanding the
assumptions the models makes and the meaning of the parameter settings will rarely
lead to an accurate model.
This chapter contains a lot of information about the algorithms, and it is not neces‐
sary for you to remember all of these details for the following chapters. However,
knowing the models described above, and knowing which to use in a specific situa‐
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tion, is important for successfully applying machine learning in practice. Here is a
quick summary of when to use which model:
Nearest neighbors: for small datasets, good as a baseline, easy to explain.
Linear models: Go-to as a first algorithm to try, good for very large datasets, good
for very high-dimensional data.
Naive Bayes: Only for classification. Even faster than linear models, good for very
large, high-dimensional data. Often less accurate than linear models.
Decision trees: Very fast, dont need scaling of the data, can be visualized and
easily explained.
Random forests: Nearly always perform better than a single decision tree, very
robust and powerful. Dont need scaling of data. Not good for very high-
dimensional sparse data.
Gradient Boosted Decision Trees: Often slightly more accurate than random for‐
est. Slower to train but faster to predict than random forest, and smaller in mem‐
ory. Need more parameter tuning than random forest.
Support Vector Machines: Powerful for medium-sized datasets of features with
similar meaning. Needs scaling of data, sensitive to parameters.
Neural Networks: Can build very complex models, in particular for large data‐
sets. Sensitive to scaling of the data, and to the choice of parameters. Large mod‐
els need a long time to train.
When working with a new dataset, it is in general a good idea to start with a simple
model, such as a linear model, naive Bayes or nearest neighbors and see how far you
can get. After understanding more about the data, you can consider moving to an
algorithm that can build more complex models, such as random forests, gradient
boosting, SVMs or neural networks.
You should now be in a position where you have some idea how to apply, tune, and
analyze the models we discussed above.
In this chapter, we focused on the binary classification case, as this is usually easiest to
understand.
Most of the algorithms presented above have classification and regression variants,
however, and all of the classification algorithms support both binary and multi-class
classification.
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Try to apply any of these algorithms to the build-in datasets in scikit-learn, like the
boston_housing or diabetes datasets for regression, or the digits dataset for multi-
class classification.
Playing around with the algorithms on different datasets will give you a better feel on
how long they need to train, how easy it is to analyze the model, and how sensitive
they are to the representation of the data.
While we analyzed the consequences of different parameter settings for the algo‐
rithms we investigated, building a model that actually generalizes well to new data in
production is a bit trickier than that. We will see how to properly adjust parameters,
and how to find good parameters automatically in Chapter 6 Model Selection.
Before we do this, we will dive in more detail into preprocessing and unsupervised
learning in the next chapter.
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CHAPTER 3
Unsupervised Learning and Preprocessing
The second family of machine learning algorithms that we will discuss is unsuper‐
vised learning.
Unsupervised learning subsumes all kinds of machine learning where there is no
known output, no teacher to instruct the learning algorithm. In unsupervised learn‐
ing, the learning algorithm is just shown the input data, and asked to extract knowl‐
edge from this data.
Types of unsupervised learning
We will look into two kinds of unsupervised learning in this chapter: transformations
of the dataset, and clustering.
Unsupervised transformations of a dataset are algorithms that create a new representa‐
tion of the data which might be easier for humans or other machine learning algo‐
rithms to understand.
A common application of unsupervised transformations is dimensionality reduction,
which takes a high-dimensional representation of the data, consisting of many fea‐
tures, and finding a new way to represent this data that summarizes the essential
characteristics about the data with fewer features. A common application for dimen‐
sionality reduction is reduction to two dimensions for visualization purposes.
Another application for unsupervised transformations is finding the parts or compo‐
nents that “make up” the data. An example of this is topic extraction on collections of
text documents. Here, the task is to find the unknown topics that are talked about in
each document, and to learn what topics appear in each document.
This can be useful for tracking the discussion of themes like elections, gun control or
talk about pop-stars on social media.
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Clustering algorithms on the other hand partition data into distinct groups of similar
items.
Consider the example of uploading photos to a social media site. To allow you to
organize your pictures, the site might want to group together pictures that show the
same person. However, the site doesn’t know which pictures show whom, and it
doesnt know how many different people appear in your photo collection. A sensible
approach would be to extract all faces, and divide them into groups of faces that look
similar. Hopefully, these correspond to the same person, and can be grouped together
for you.
Challenges in unsupervised learning
A major challenge in unsupervised learning is evaluating whether the algorithm
learned something useful. Unsupervised learning algorithms are usually applied to
data that does not contain any label information, so we don’t know what the right
output should be. Therefore it is very hard to say whether a model “did well”. For
example, the clustering algorithm could have grouped all face pictures that are shown
in profile together, and all the face pictures that are face-forward together.
This would certainly be a possible way to divide a collection of face pictures, but not
the one we were looking for. However, there is no way for us to “tell” the algorithm
what we are looking for, and often the only way to evaluate the result of an unsuper‐
vised algorithm is to inspect it manually.
As a consequence, unsupervised algorithms are used often in an exploratory setting,
when a data scientist wants to understand the data better, rather than as part of a
larger automatic system. Another common application for unsupervised algorithms
is as a preprocessing step for supervised algorithms. Learning a new representation of
the data can sometimes improve the accuracy of supervised algorithms, or can lead to
reduced memory and time consumption.
Before we start with “real” unsupervised algorithms, we will briefly discuss some sim‐
ple preprocessing methods that often come in handy. Even though preprocessing and
scaling are often used in tandem with supervised learning algorithms, scaling meth‐
ods dont make use of the supervised information, making them unsupervised.
Preprocessing and Scaling
In the last chapter we saw that some algorithms, like neural networks and SVMs, are
very sensitive to the scaling of the data. Therefore a common practice is to adjust the
features so that the data representation is more suitable for these algorithms. Often,
this is a simple per-feature rescaling and shift of the data. A simple example is shown
in Figure scaling_data.
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mglearn.plots.plot_scaling()
plt.suptitle("scaling_data");
Dierent kinds of preprocessing
The first plot shows a synthetic two-class classification dataset with two features. The
first feature (the x-axis value) is between 10 and 15. The second feature (the y-axis
value) is between around 1 and 9.
The following four plots show four different ways to transform the data that yield
more standard ranges.
The StandardScaler in scikit-learn ensures that for each feature, the mean is zero,
and the variance is one, bringing all features to the same magnitude. However, this
scaling does not ensure any particular minimum and maximum values for the fea‐
tures.
The RobustScaler works similarly to the StandardScaler in that it ensures statistical
properties for each feature that guarantee that they are on the same scale. However,
the RobustScaler uses the median and quartiles [Footnote: the median of a set of
numbers is the number x such that half of the numbers are smaller than x and half of
the numbers are larger than x. The lower quartile is the number x such that 1/4th of
the numbers are smaller than x, the upper quartile is so that 1/4th of the numbers is
larger than x], instead of mean and variance. This makes the RobustScaler ignore
data points that are very different from the rest (like measurement errors). These odd
data points are also called outliers, and might often lead to trouble for other scaling
techniques.
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The MinMaxScaler on the other hand shifts the data such that all features are exactly
between 0 and 1. For the two-dimensional dataset this means all of the data is con‐
tained within the rectangle created by the x axis between 0 and 1 and the y axis
between zero and one.
Finally, the Normalizer does a very different kind of rescaling. It scales each data
point such that the feature vector has a euclidean length of one. In other words, it
projects a data point on the circle (or sphere in the case of higher dimensions) with a
radius of 1. This means every data point is scaled by a different number (by the
inverse of its length).
This normalization is often used when only the direction (or angle) of the data mat‐
ters, not the length of the feature vector.
Applying data transformations
After seeing what the different kind of transformations do, lets apply them using
scikit-learn.
We will use the cancer dataset that we saw in chapter 2. Preprocessing methods like
the scalers are usually applied before applying a supervised machine learning algo‐
rithm. As an example, say we want to apply the kernel SVM (SVC) to the cancer data‐
set, and use MinMaxScaler for preprocessing the data. We start by loading and
splitting our dataset into a training set and a test set. We need a separate training and
test set to evaluate the supervised model we will build after the preprocessing:
from sklearn.datasets import load_breast_cancer
from sklearn.model_selection import train_test_split
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data, cancer.target,
random_state=1)
print(X_train.shape)
print(X_test.shape)
(426, 30)
(143, 30)
As a reminder, the data contains 150 data points, each represented by four measure‐
ments. We split the dataset into 112 samples for the training set and 38 samples for
the test set.
As with the supervised models we built earlier, we first import the class implementing
the preprocessing, and then instantiate it:
from sklearn.preprocessing import MinMaxScaler
scaler = MinMaxScaler()
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We then fit the scaler using the fit method, applied to the training data. For the Min
MaxScaler, the fit method computes the minimum and maximum value of each fea‐
ture on the training set. In contrast to the classifiers and regressors of chapter 2, the
scaler is only provided with the data X_train when fit is called, and y_train is not
used:
scaler.fit(X_train)
MinMaxScaler(copy=True, feature_range=(0, 1))
To apply the transformation that we just learned, that is, to actually scale the training
data, we use the transform method of the scaler. The transform method is used in
scikit-learn whenever a model returns a new representation of the data:
# don't print using scientific notation
np.set_printoptions(suppress=True, precision=2)
# transform data
X_train_scaled = scaler.transform(X_train)
# print data set properties before and after scaling
print("transformed shape: %s" % (X_train_scaled.shape,))
print("per-feature minimum before scaling:\n %s" % X_train.min(axis=0))
print("per-feature maximum before scaling:\n %s" % X_train.max(axis=0))
print("per-feature minimum after scaling:\n %s" % X_train_scaled.min(axis=0))
print("per-feature maximum after scaling:\n %s" % X_train_scaled.max(axis=0))
transformed shape: (426, 30)
per-feature minimum before scaling:
[ 6.98 9.71 43.79 143.5 0.05 0.02 0. 0. 0.11
0.05 0.12 0.36 0.76 6.8 0. 0. 0. 0.
0.01 0. 7.93 12.02 50.41 185.2 0.07 0.03 0.
0. 0.16 0.06]
per-feature maximum before scaling:
[ 28.11 39.28 188.5 2501. 0.16 0.29 0.43 0.2
0.3 0.1 2.87 4.88 21.98 542.2 0.03 0.14
0.4 0.05 0.06 0.03 36.04 49.54 251.2 4254.
0.22 0.94 1.17 0.29 0.58 0.15]
per-feature minimum after scaling:
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
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per-feature maximum after scaling:
[ 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. 1. 1.]
The transformed data has the same shape as the original data - the features are simply
shifted and scaled.
You can see that all of the feature are now between zero and one, as desired.
To apply the SVM to the scaled data, we also need to transform the test set. This is
done by again calling the transform method, this time on X_test:
# transform test data
X_test_scaled = scaler.transform(X_test)
# print test data properties after scaling
print("per-feature minimum after scaling: %s" % X_test_scaled.min(axis=0))
print("per-feature maximum after scaling: %s" % X_test_scaled.max(axis=0))
per-feature minimum after scaling: [ 0.03 0.02 0.03 0.01 0.14 0.04 0. 0. 0.15 -0.01 -0. 0.01
0. 0. 0.04 0.01 0. 0. -0.03 0.01 0.03 0.06 0.02 0.01
0.11 0.03 0. 0. -0. -0. ]
per-feature maximum after scaling: [ 0.96 0.82 0.96 0.89 0.81 1.22 0.88 0.93 0.93 1.04 0.43 0.5
0.44 0.28 0.49 0.74 0.77 0.63 1.34 0.39 0.9 0.79 0.85 0.74
0.92 1.13 1.07 0.92 1.21 1.63]
Maybe somewhat surprisingly, you can see that for the test set, after scaling, the mini‐
mum and maximum are not zero and one. Some of the features are even outside the
0-1 range!
The explanation is that the MinMaxScaler (and all the other scalers) always applies
exactly the same transformation to the training and the test set. So the transform
method always subtracts the training set minimum, and divides by the training set
range, which might be different than the minimum and range for the test set.
Scaling training and test data the same way
It is important that exactly the same transformation is applied to the training set and
the test set for the supervised model to make sense on the test set. The following fig‐
ure illustrates what would happen if we would use the minimum and range of the test
set instead:
from sklearn.datasets import make_blobs
# make synthetic data
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X, _ = make_blobs(n_samples=50, centers=5, random_state=4, cluster_std=2)
# split it into training and test set
X_train, X_test = train_test_split(X, random_state=5, test_size=.1)
# plot the training and test set
fig, axes = plt.subplots(1, 3, figsize=(13, 4))
axes[0].scatter(X_train[:, 0], X_train[:, 1],
c='b', label="training set", s=60)
axes[0].scatter(X_test[:, 0], X_test[:, 1], marker='^',
c='r', label="test set", s=60)
axes[0].legend(loc='upper left')
axes[0].set_title("original data")
# scale the data using MinMaxScaler
scaler = MinMaxScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# visualize the properly scaled data
axes[1].scatter(X_train_scaled[:, 0], X_train_scaled[:, 1],
c='b', label="training set", s=60)
axes[1].scatter(X_test_scaled[:, 0], X_test_scaled[:, 1], marker='^',
c='r', label="test set", s=60)
axes[1].set_title("scaled data")
# rescale the test set separately, so that test set min is 0 and test set max is 1
# DO NOT DO THIS! For illustration purposes only
test_scaler = MinMaxScaler()
test_scaler.fit(X_test)
X_test_scaled_badly = test_scaler.transform(X_test)
# visualize wrongly scaled data
axes[2].scatter(X_train_scaled[:, 0], X_train_scaled[:, 1],
c='b', label="training set", s=60)
axes[2].scatter(X_test_scaled_badly[:, 0], X_test_scaled_badly[:, 1], marker='^',
c='r', label="test set", s=60)
axes[2].set_title("improperly scaled data")
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The first panel is an unscaled two-dimensional dataset, with the training set shown in
blue and the test set shown in red. The second figure is the same data, but scaled
using the MinMaxScaler. Here, we called fit on the training set, and then transform
on the training and the test set. You can see that the dataset in the second panel looks
identical to the first, only the ticks on the axes changed. Now all the features are
between 0 and 1.
You can also see that the minimum and maximum feature values for the test data (the
red points) are not 0 and 1.
The third panel shows what would happen if we scaled training and test set sepa‐
rately. In this case, the minimum and maximum feature values for both the training
and the test set are 0 and 1. But now the dataset looks different. The test points
moved incongruously to the training set, as they were scaled differently. We changed
the arrangement of the data in an arbitrary way. Clearly this is not what we want to
do.
Another way to reason about this is the following: Imagine your test set was a single
point. There is no way to scale a single point correctly, to fulfill the minimum and
maximum requirements of the MinMaxScaler. But the size of your test set should not
change your processing.
The eect of preprocessing on supervised learning
Now lets go back to the cancer dataset and see what the effect of using the Min
MaxScaler is on learning the SVC (this is a different way of doing the same scaling we
did in chapter 2).
First, lets fit the SVC on the original data again for comparison:
from sklearn.svm import SVC
X_train, X_test, y_train, y_test = train_test_split(cancer.data, cancer.target,
random_state=0)
svm = SVC(C=100)
svm.fit(X_train, y_train)
print(svm.score(X_test, y_test))
0.629370629371
Now, let’s scale the data using MinMaxScaler before fitting the SVC:
# preprocessing using 0-1 scaling
scaler = MinMaxScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# learning an SVM on the scaled training data
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svm.fit(X_train_scaled, y_train)
# scoring on the scaled test set
svm.score(X_test_scaled, y_test)
0.965034965034965
As we saw before, the effect of scaling the data is quite significant. Even though scal‐
ing the data doesn’t involve any complicated math, it is good practice to use the scal‐
ing mechanisms provided by scikit-learn, instead of reimplementing them yourself,
as making mistakes even in these simple computations is easy.
You can also easily replace one preprocessing algorithm by another by changing the
class you use, as all of the preprocessing classes have the same interface, consisting of
the fit and transform methods:
# preprocessing using zero mean and unit variance scaling
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
# learning an SVM on the scaled training data
svm.fit(X_train_scaled, y_train)
# scoring on the scaled test set
svm.score(X_test_scaled, y_test)
0.95804195804195802
Now that we’ve seen how simple data transformations for preprocessing work, lets
move on to more interesting transformations using unsupervised learning.
Dimensionality Reduction, Feature Extraction and
Manifold Learning
As we discussed above, transforming data using unsupervised learning can have
many motivations. The most common motivations are visualization, compressing the
data, and finding a representation that is more informative for further processing.
One of the simplest and most widely used algorithms for all of these is Principal
Component Analysis.
Principal Component Analysis (PCA)
Principal component analysis (PCA) is a method that rotates the dataset in a way
such that the rotated features are statistically uncorrelated. This rotation is often fol‐
lowed by selecting only a subset of the new features, according to how important they
are for explaining the data.
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mglearn.plots.plot_pca_illustration()
plt.suptitle("pca_illustration");
The plot above shows a simple example on a synthetic two-dimensional dataset. The
first plot shows the original data points, colored to distinguish the points. The algo‐
rithm proceeds by first finding the direction of maximum variance, labeled as “Com‐
ponent 1”. This is the direction in the data that contains most of the information, or
in other words, the direction along which the features are most correlated with each
other.
Then, the algorithm finds the direction that contains the most information while
being orthogonal (is at a right angle) to the first direction. In two dimensions, there is
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only one possible orientation that is at a right angle, but in higher dimensional spaces
there would be (infinitely) many orthogonal directions.
Although the two components are drawn as arrows, it doesnt really matter where the
head and the tail is; we could have drawn the first component from the center up to
the top left instead of to the bottom right.The directions found using this process are
called principal components, as they are the main directions of variance in the data. In
general, there are as many principal components as original features.
The second plot shows the same data, but now rotated so that the first principal com‐
ponent aligns with the x axis, and the second principal component aligns with the y
axis. Before the rotation, the mean was subtracted from the data, so that the trans‐
formed data is centered around zero. In the rotated representation found by PCA, the
two axes are uncorrelated, meaning that the correlation matrix of the data in this rep‐
resentation is zero except for the diagonal.
We can use PCA for dimensionality reduction by retaining only some of the principal
components. In this example, we might keep only the first principal component, as
shown in the third panel in Figure X.
This reduced the data from a two-dimensional dataset to a one-dimensional dataset.
But instead of keeping only one of the original features, we found the most interest‐
ing direction (top left to bottom right in the first panel) and kept this direction, the
first principal component.
Finally, we can undo the rotation, and add the mean back to the data. This will result
in the data shown in the last panel. These points are in the original feature space, but
we kept only the information contained in the first principal component. This trans‐
formation is sometimes used to remove noise effects from the data, or visualize what
part of the information is kept in the PCA.
Applying PCA to the cancer dataset for visualization
One of the most common applications of PCA is visualizing high-dimensional data‐
sets. As we already saw in Chapter 1, it is hard to create scatter plots of data that has
more than two features. For the iris dataset, we could create a pair plot (Figure
iris_pairplot in Chapter 1), which gave us us a partial picture of the data by showing
us all combinations of two features. If we want to look at the breast cancer dataset,
even using a pair-plot is tricky. The breast cancer dataset has 30 features, which
would result in 30 * 14 = 420 scatter plots! Youd never be able to look at all these plots
in detail, let alone try to understand them.
We could go for an even simpler visualization, showing histograms of each of the fea‐
tures for the two classes, benign and malignant cancer:
fig, axes = plt.subplots(15, 2, figsize=(10, 20))
malignant = cancer.data[cancer.target == 0]
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benign = cancer.data[cancer.target == 1]
ax = axes.ravel()
for i in range(30):
_, bins = np.histogram(cancer.data[:, i], bins=50)
ax[i].hist(malignant[:, i], bins=bins, color='b', alpha=.5)
ax[i].hist(benign[:, i], bins=bins, color='r', alpha=.5)
ax[i].set_title(cancer.feature_names[i])
ax[i].set_yticks(())
fig.tight_layout()
plt.suptitle("cancer_histograms")
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Here we create a histogram for each of the features, counting how often a data point
appears with a feature in a certain range (called a bin).
Each plot overlays two histograms, one for all of the points of the benign class (blue)
and one for all the points in the malignant class (red). This gives us some idea of how
each feature is distributed across the two classes, and allows us to venture a guess as
to which features are better at distinguishing malignant and benign samples. For
example, the feature “smoothness error” seems quite uninformative, because the two
histograms mostly overlap, while the feature “worst concave points” seems quite
informative, because the histograms are quite disjoint.
However, this plot doesn’t show us anything about the, which indicate variables that
are varying together . We can find the first two principal components, and visualize
the data in this new, two-dimensional space, with a single scatter-plot.
Before we apply PCA, we scale our data so that each feature has unit variance using
StandardScaler:
from sklearn.datasets import load_breast_cancer
cancer = load_breast_cancer()
scaler = StandardScaler()
scaler.fit(cancer.data)
X_scaled = scaler.transform(cancer.data)
Learning the PCA transformation and applying it is as simple as applying a prepro‐
cessing transformation. We instantiate the PCA object, find the principal components
by calling the fit method, and then apply the rotation and dimensionality reduction
by calling transform.
By default, PCA only rotates (and shifts) the data, but keeps all principal components.
To reduce the dimensionality of the data, we need to specify how many components
we want to keep when creating the PCA object:
from sklearn.decomposition import PCA
# keep the first two principal components of the data
pca = PCA(n_components=2)
# fit PCA model to beast cancer data
pca.fit(X_scaled)
# transform data onto the first two principal components
X_pca = pca.transform(X_scaled)
print("Original shape: %s" % str(X_scaled.shape))
print("Reduced shape: %s" % str(X_pca.shape))
Original shape: (569, 30)
Reduced shape: (569, 2)
We can now plot the first two principal components:
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# plot fist vs second principal component, color by class
plt.figure(figsize=(8, 8))
plt.scatter(X_pca[:, 0], X_pca[:, 1], c=cancer.target, cmap=mglearn.tools.cm, s=60)
plt.gca().set_aspect("equal")
plt.xlabel("First principal component")
plt.ylabel("Second principal component")
It is important to note is that PCA is an unsupervised method, and does not use any
class information when finding the rotation. It simply looks at the correlations in the
data. For the scatter plot above, we plotted the first principal component against the
second principal component, and then used the class information to color the points.
You can see that the two classes separate quite well in this two-dimensional space.
This can lead us to believe that even a linear classifier (that would learn a line in this
space) could do a reasonably good job at distinguishing the two classes. We can also
see that the malignant (red) points are more spread-out than the benign (blue)
points, something that we could already see a bit from the histograms in Figure can‐
cer_histograms.
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A downside of PCA is that the two axes in the plot above are often not very easy to
interpret. The principal components correspond to directions in the original data, so
they are combinations of the original features. However, these combinations are usu‐
ally very complex, as well see below.
The principal components themselves are stored in the components_ attribute of the
PCA during fitting:
pca.components_.shape
(2, 30)
Each row in components_ corresponds to one principal component, sorted by their
importance (the first principal component comes first, etc). The columns correspond
to the original features, in this example “mean radius”, “mean texture” and so on.
Lets have a look at the content of components_:
print(pca.components_)
[[-0.22 -0.1 -0.23 -0.22 -0.14 -0.24 -0.26 -0.26 -0.14 -0.06 -0.21 -0.02
-0.21 -0.2 -0.01 -0.17 -0.15 -0.18 -0.04 -0.1 -0.23 -0.1 -0.24 -0.22
-0.13 -0.21 -0.23 -0.25 -0.12 -0.13]
[ 0.23 0.06 0.22 0.23 -0.19 -0.15 -0.06 0.03 -0.19 -0.37 0.11 -0.09
0.09 0.15 -0.2 -0.23 -0.2 -0.13 -0.18 -0.28 0.22 0.05 0.2 0.22
-0.17 -0.14 -0.1 0.01 -0.14 -0.28]]
We can also visualize the coefficients using a heatmap, which might be easier to
understand:
plt.matshow(pca.components_, cmap='viridis')
plt.yticks([0, 1], ["first component", "second component"])
plt.colorbar()
plt.xticks(range(len(cancer.feature_names)),
cancer.feature_names, rotation=60, ha='left');
plt.suptitle("pca_components_cancer")
You can see that in the first component, all feature have the same sign (its negative,
but as we mentioned above, it doesn’t matter in which direction you point the arrow).
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That means that there is a general correlation between all features. As one measure‐
ment is high, the others are likely to be high as well.
The second component has mixed signs, and both of the components involve all of
the 30 features. This mixing of all features is what makes explaining the axes in Figure
pca_components_cancer above so tricky.
Eigenfaces for feature extraction
Another application of PCA that we mentioned above is feature extraction. The idea
behind feature extraction is that it is possible to find a representation of your data
that is better suited to analysis than the raw representation you were given. A great
example of an application when feature extraction if helpful is with images. Images
are usually stored as red, green and blue intensities for each pixel. But images are
made up of many pixels, and only together are they meaningful; objects in images are
usually made up of thousands of pixels.
We will give a very simple application of feature extraction on images using PCA,
using face images from the “labeled faces in the wild” dataset. This dataset contains
face images of celebrities downloaded from the internet, and it includes faces of poli‐
ticians, singers, actors and athletes from the early 2000s. We use gray-scale versions of
these images, and scale them down for faster processing. You can see some of the
images below:
from sklearn.datasets import fetch_lfw_people
people = fetch_lfw_people(min_faces_per_person=20, resize=0.7)
image_shape = people.images[0].shape
fix, axes = plt.subplots(2, 5, figsize=(15, 8), subplot_kw={'xticks': (), 'yticks': ()})
for target, image, ax in zip(people.target, people.images, axes.ravel()):
ax.imshow(image)
ax.set_title(people.target_names[target])
plt.suptitle("some_faces")
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There are 3023 images, each 87 x 65 pixels large, belonging to 62 different people:
print(people.images.shape)
print(len(people.target_names))
(3023, 87, 65)
62
The dataset is a bit skewed, however, containing a lot of images of George W. Bush
and Colin Powell, as you can see here:
# count how often each target appears
counts = np.bincount(people.target)
# print counts next to target names:
for i, (count, name) in enumerate(zip(counts, people.target_names)):
print("{0:25} {1:3}".format(name, count), end=' ')
if (i + 1) % 3 == 0:
print()
Alejandro Toledo 39 Alvaro Uribe 35 Amelie Mauresmo 21
Andre Agassi 36 Angelina Jolie 20 Ariel Sharon 77
Arnold Schwarzenegger 42 Atal Bihari Vajpayee 24 Bill Clinton 29
Carlos Menem 21 Colin Powell 236 David Beckham 31
Donald Rumsfeld 121 George Robertson 22 George W Bush 530
Gerhard Schroeder 109 Gloria Macapagal Arroyo 44 Gray Davis 26
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Guillermo Coria 30 Hamid Karzai 22 Hans Blix 39
Hugo Chavez 71 Igor Ivanov 20 Jack Straw 28
Jacques Chirac 52 Jean Chretien 55 Jennifer Aniston 21
Jennifer Capriati 42 Jennifer Lopez 21 Jeremy Greenstock 24
Jiang Zemin 20 John Ashcroft 53 John Negroponte 31
Jose Maria Aznar 23 Juan Carlos Ferrero 28 Junichiro Koizumi 60
Kofi Annan 32 Laura Bush 41 Lindsay Davenport 22
Lleyton Hewitt 41 Luiz Inacio Lula da Silva 48 Mahmoud Abbas 29
Megawati Sukarnoputri 33 Michael Bloomberg 20 Naomi Watts 22
Nestor Kirchner 37 Paul Bremer 20 Pete Sampras 22
Recep Tayyip Erdogan 30 Ricardo Lagos 27 Roh Moo-hyun 32
Rudolph Giuliani 26 Saddam Hussein 23 Serena Williams 52
Silvio Berlusconi 33 Tiger Woods 23 Tom Daschle 25
Tom Ridge 33 Tony Blair 144 Vicente Fox 32
Vladimir Putin 49 Winona Ryder 24
To make the data less skewed, we will only take up to 50 images of each person.
Otherwise the feature extraction would be overwhelmed by the likelihood of George
W Bush.
mask = np.zeros(people.target.shape, dtype=np.bool)
for target in np.unique(people.target):
mask[np.where(people.target == target)[0][:50]] = 1
X_people = people.data[mask]
y_people = people.target[mask]
# scale the grey-scale values to be between 0 and 1
# instead of 0 and 255 for better numeric stability:
X_people = X_people / 255.
A common task in face recognition is to ask if a previously unseen face belongs to a
known person from a database. This has applications in photo collection, social
media and security. One way to solve this problem would be to build a classifier
where each person is a separate class. However, there are usually many different peo‐
ple in face databases, and very few images of the same person (i.e. very few training
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examples per class). That makes it hard to train most classifiers. Additionally, you
often want to easily add new people, without retraining a large model.
A simple solution is to use a one-nearest-neighbor classifier which looks for the most
similar face image to the face you are classifying. A one-nearest-neighbor could in
principle work with only a single training example per class. Lets see how well KNeigh
borsClassifier does here:
from sklearn.neighbors import KNeighborsClassifier
# split the data in training and test set
X_train, X_test, y_train, y_test = train_test_split(
X_people, y_people, stratify=y_people, random_state=0)
# build a KNeighborsClassifier with using one neighbor:
knn = KNeighborsClassifier(n_neighbors=1)
knn.fit(X_train, y_train)
knn.score(X_test, y_test)
0.26615969581749049
We obtain an accuracy of 26.6%, which is not actually that bad for a 62 class classifi‐
cation problem (random guessing would give you around 1/62 = 1.5% accuracy), but
is also not great. We only correctly identify a person a every fourth time.
This is where PCA comes in. Computing distances in the original pixel space is quite
a bad way to measure similarity between faces [add a sentence saying why]. We hope
that using distances along principal components can improve our accuracy. Here we
enable the whitening option of PCA, which rescales the principal components to have
the same scale. This is the same as using StandardScaler after the transformation.
Reusing the data from Figure pca_illustration again, whitening corresponds to not
only rotating the data, but also rescaling it so that the center panel is a circle instead
of an ellipse:
mglearn.plots.plot_pca_whitening()
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We fit the PCA object to the training data and extract the first 100 principal compo‐
nents. Then we transform the training and test data:
pca = PCA(n_components=100, whiten=True).fit(X_train)
X_train_pca = pca.transform(X_train)
X_test_pca = pca.transform(X_test)
print(X_train_pca.shape)
(1537, 100)
The new data has 100 features, the first 100 principal components. Now, we can use
the new representation to classify our images using one-nearest-neighbors:
knn = KNeighborsClassifier(n_neighbors=1)
knn.fit(X_train_pca, y_train)
knn.score(X_test_pca, y_test)
0.36882129277566539
Our accuracy improved quite significantly, from 26.6% to 36.8%, confirming our
intuition that the principal components might provide a better representation of the
data.
For image data, we can also easily visualize the principal components that are found.
Remember that components correspond to directions in the input space. The input
space here is 50x37 gray-scale images, and so directions within this space are also
50x37 gray-scale images. Lets look at the first couple of principal components:
pca.components_.shape
(100, 5655)
fix, axes = plt.subplots(3, 5, figsize=(15, 12),
subplot_kw={'xticks': (), 'yticks': ()})
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fig.suptitle("pca_face_components")
for i, (component, ax) in enumerate(zip(pca.components_, axes.ravel())):
ax.imshow(component.reshape(image_shape),
cmap='viridis')
ax.set_title("%d. component" % (i + 1))
While we certainly can not understand all aspects of these components, we can guess
which aspects of the face images some of the components are capturing. The first
component seems to mostly encode the contrast between the face and the back‐
ground, the second component encodes differences in lighting between the right and
the left half of the face, and so on. While this representation is slightly more semantic
than the raw pixel values, it is still quite far from how a human might perceive a face.
As the PCA is based on pixels, the alignment of the face (the position of eyes, chin
and nose), as well as the lighting, both have a strong influence on how similar to
images are in their pixel representation. Alignment and lighting are probably not
what a human would perceive first. When asking a person to rate similarity of faces,
they are more likely to use attributes like age, gender, facial expression and hair style,
which are attributes that are hard to infer from the pixel intensities.
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Its important to keep in mind that algorithms often interpret data, in particular data
that humans are used to understand, like images, quite differently from how a human
would.
Lets come back to the specific case of PCA, though.
Above we introduced the PCA transformation as rotating the data, and then drop‐
ping the components with low variance.
Another useful interpretation is that we try to find some numbers (the new feature
values after the PCA rotation), so that we can express the test points as a weighted
sum of the principal components:
from matplotlib.offsetbox import OffsetImage, AnnotationBbox
image_shape = people.images[0].shape
plt.figure(figsize=(20, 3))
ax = plt.gca()
imagebox = OffsetImage(people.images[0], zoom=7, cmap="gray")
ab = AnnotationBbox(imagebox, (.05, 0.4), pad=0.0, xycoords='data')
ax.add_artist(ab)
for i in range(4):
imagebox = OffsetImage(pca.components_[i].reshape(image_shape), zoom=7, cmap="viridis")
ab = AnnotationBbox(imagebox, (.3 + .2 * i, 0.4),
pad=0.0,
xycoords='data'
)
ax.add_artist(ab)
if i == 0:
plt.text(.18, .25, 'x_%d *' % i, fontdict={'fontsize': 50})
else:
plt.text(.15 + .2 * i, .25, '+ x_%d *' % i, fontdict={'fontsize': 50})
plt.text(.95, .25, '+ ...', fontdict={'fontsize': 50})
plt.rc('text', usetex=True)
plt.text(.13, .3, r'\approx', fontdict={'fontsize': 50})
plt.axis("off")
plt.title("decomposition")
plt.rc('text', usetex=False) # THIS SHOULD NOT SHOW IN THE BOOK! it's needed for the figure above
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Here, $x_0$, $x_1$ and so on are the coefficients of the principal components for this
data point; in other words, they are the representation [of what? of the image? this is
not clear] in the rotated space.
Another way we can try to understand what a PCA model is doing is by looking at
the reconstructions of the original data using only some components. In Figure
pca_illustration, after dropping the second component and arriving at the third
panel , we undid the rotation and added the mean back to obtain new points in the
original space with the second component removed, as shown in the last panel.
We can do a similar transformation for the faces by reducing the data to only some
principal components and then rotating back into the original space. This return to
the original feature space can be done using the inverse_transform method. Here
we visualize the reconstruction of some faces using 10, 50, 100, 500 or 2000 compo‐
nents:
mglearn.plots.plot_pca_faces(X_train, X_test, image_shape)
plt.suptitle("pca_reconstructions");
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/home/andy/checkout/scikit-learn/sklearn/externals/joblib/logger.py:77: DeprecationWarning: The 'warn' function is deprecated, use 'warning' instead
logging.warn("[%s]: %s" % (self, msg))
________________________________________________________________________________
[Memory] Calling mglearn.plot_pca.pca_faces...
pca_faces(array([[ 0.036601, ..., 0.742484],
...,
[ 0.105882, ..., 0.393464]], dtype=float32),
array([[ 0.162091, ..., 0.677124],
...,
[ 0.109804, ..., 0.07451 ]], dtype=float32))
_______________________________________________________pca_faces - 12.9s, 0.2min
You can see that with using only the first 10 principal components, only the essence
of the picture, like the face orientation and lighting, is captured. Using more and
more principal components, more and more details in the image are preserved. This
corresponds to extending the sum in Figure decomposition to include more and
more terms. Using as many components as there are pixels would mean that we
would not discard any information after the rotation, and we would reconstruct the
image perfectly.
We can also try to use PCA to visualize all the faces in the dataset in a scatter plot
using the first two principal components, with classes given by who is shown in the
image, similarly to what we did for the cancer dataset:
plt.scatter(X_train_pca[:, 0], X_train_pca[:, 1], c=y_train, cmap='Paired', s=60)
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As you can see, when using just the first two principal components, the whole data is
just a big blob, with no separation of classes visible. This is not very surprising, given
that even with 10 components, as shown in Figure faces_pca_reconstruction above,
PCA only captures very rough characteristics of the faces.
Non-Negative Matrix Factorization (NMF)
Non-negative matrix factorization is another unsupervised learning algorithm that
aims to extract useful features. It works similarly to PCA and can also be used for
dimensionality reduction. As in PCA we are trying to write each data point as a
weighted sum of some components as illustrated in Figure decomposition. In PCA,
we wanted components that are orthogonal, and that explain as much variance of the
data as possible. In NMF, we want the components and the coefficients to be non-
negative; that is, we want both the components and the coefficients to be greater or
equal then zero.
Consequently, this method can only applied to data where each feature is non-
negative, as a non-negative sum of non-negative components can not become nega‐
tive. The process of decomposing data into a non-negative weighted sum is
particularly helpful for data that is created as the addition of several independent
sources, such as an audio track of multiple speakers, or music with many instruments.
In these situations, NMF can identify the original components that make up the com‐
bined data. Overall, NMF leads to more interpretable components than PCA, as neg‐
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ative components and coefficients can lead to hard-interpret cancellation effects. The
eigenfaces in Figure pca_face_components for example contain both positive and
negative parts, and as we mentioned in the description of PCA, the sign is actually
arbitrary.
Before we apply NMF to the face dataset, lets briefly revisit the synthetic data.
Applying NMF to synthetic data
In contrast to PCA, we need to ensure that our data is positive for NMF to be able to
operate on the data.
This means where the data lies relative to the origin (0, 0) actually matters for NMF.
Therefore, you can think of the non-negative components that are extracted as direc‐
tions from (0, 0) towards the data.
mglearn.plots.plot_nmf_illustration()
plt.suptitle("nmf_illustration")
The figure above shows the results of NMF on the two-dimensional toy data. For
NMF with two components, as shown on the left, it is clear that all points in the data
can be written as a positive combination of the two components. If there are enough
components to perfectly reconstruct the data (as many components are there are fea‐
tures), the algorithm will choose directions that point towards the extremes of the
data.
If we only use a single component, NMF creates a component that points towards the
mean, as pointing there best explains the data. You see that in contrast to PCA, reduc‐
ing the number of components not only removes directions, it actually changes all
directions! Components in NMF are also not ordered in any specific way, so there is
no “first non-negative component”: all components play an equal part.
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NMF uses a random initialization, which might lead to different results depending on
the random seed. In relatively easy cases as the synthetic data with two components,
where all the data can be explained perfectly, the randomness has little effect (though
it might change the order or scale of the components). In more complex situations,
there might be more drastic changes.
Applying NMF to face images
Now, let’s apply NMF to the “labeled faces in the wild” dataset we used above.
The main parameter of NMF is how many components we want to extract. Usually
this is lower than the number of input features (otherwise the data could be explained
by making each pixel a separate component).
First, lets inspect how the number of components impacts how well the data can be
reconstructed using NMF:
mglearn.plots.plot_nmf_faces(X_train, X_test, image_shape)
________________________________________________________________________________
[Memory] Calling mglearn.plot_nmf.nmf_faces...
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nmf_faces(array([[ 0.036601, ..., 0.742484],
...,
[ 0.105882, ..., 0.393464]], dtype=float32),
array([[ 0.162091, ..., 0.677124],
...,
[ 0.109804, ..., 0.07451 ]], dtype=float32))
_____________________________________________________nmf_faces - 763.1s, 12.7min
The quality of the back-transformed data is similar to PCA, but slightly worse. This is
expected, as PCA finds the optimum directions in terms of reconstruction. NMF is
usually not used for the ability to reconstruct or encode data, but rather for finding
interesting patterns within the data.
As a first look into the data, lets try extracting only a few components, say 15, on the
faced data:
from sklearn.decomposition import NMF
nmf = NMF(n_components=15, random_state=0)
nmf.fit(X_train)
X_train_nmf = nmf.transform(X_train)
X_test_nmf = nmf.transform(X_test)
fix, axes = plt.subplots(3, 5, figsize=(15, 12),
subplot_kw={'xticks': (), 'yticks': ()})
for i, (component, ax) in enumerate(zip(nmf.components_, axes.ravel())):
ax.imshow(component.reshape(image_shape))
ax.set_title("%d. component" % i)
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These components are all positive, and so resemble prototypes of faces much more so
than the components shown for PCA in Figure Eigenfaces. For example, one can
clearly see that component 3 shows a face rotated somewhat to the right, while com‐
ponent 7 shows a face somewhat rotated to the left. Lets look at the images for which
these components are particularly strong:
compn = 3
# sort by 3rd component, plot first 10 images
inds = np.argsort(X_train_nmf[:, compn])[::-1]
fig, axes = plt.subplots(2, 5, figsize=(15, 8),
subplot_kw={'xticks': (), 'yticks': ()})
fig.suptitle("Large component 3")
for i, (ind, ax) in enumerate(zip(inds, axes.ravel())):
ax.imshow(X_train[ind].reshape(image_shape))
compn = 7
# sort by 7th component, plot first 10 images
inds = np.argsort(X_train_nmf[:, compn])[::-1]
fig.suptitle("Large component 7")
fig, axes = plt.subplots(2, 5, figsize=(15, 8),
subplot_kw={'xticks': (), 'yticks': ()})
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for i, (ind, ax) in enumerate(zip(inds, axes.ravel())):
ax.imshow(X_train[ind].reshape(image_shape))
As expected, faces that have a high coefficient for component 3 are faces looking to
the right, while faces with a high component 7 are looking to the left. As mentioned
above, extracting patterns like these works best for data with additive structure,
including audio, gene expression data, and text data. We will see applications of NMF
to text data in Chapter 7 (Text Data).
There are many other algorithms that can be used to decompose each data point into
a weighted sum of a fixed set of components, as PCA and NMF do. Discussing all of
them is beyond the scope of this book, and describing the constraints made on the
components and coefficients often involves probability theory. If you are interested in
these kinds of pattern extraction, we recommend to study the user guide of Inde‐
pendent Component Analysis (ICA), Factor Analysis (FA) and Sparse Coding (dictio‐
nary learning), which are widely used decomposition methods.
Manifold learning with t-SNE
While PCA is often a good first approach for transforming your data so that you
might be able to visualize it using a scatter plot, the nature of the method (applying a
rotation and then dropping directions) limits its usefulness, as we saw with the scatter
plot of the labeled faces in the wild. There is a class of algorithms for visualization
called manifold learning algorithms which allows for much more complex mappings,
and often provides better visualizations. A particular useful one is the t-SNE algo‐
rithm.
Manifold learning algorithms are mainly aimed at visualization, and so are rarely
used to to generate more than two new features. Some of them, including t-SNE,
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compute a new representation of the training data, but don’t allow transformations of
new data. This means these algorithms can not be applied to a test set: rather, they
can only transform the data they were trained for. Manifold learning can be useful for
exploratory data analysis, but is rarely used if the final goal is supervised learning.
The idea behind t-SNE is to find a two-dimensional representation of the data that
preserve the distances between points as best as possible. t-SNE tarts with a random
two-dimensional representation for each data point, and then tries to make points
closer that are close in the original feature space, and points far apart that are far apart
in the original feature space. t-SNE puts more emphasis on points that are close by,
rather than preserving distances between far apart points. In other words, it tries to
preserve the information of which points are neighbors to each other.
We will apply the t-SNE manifold learning algorithm on a dataset of handwritten dig‐
its that is included in scikit-learn [Footnote: not to be confused with the much larger
MNIST dataset].
Each data point in this dataset is a 8x8 grey-scale image of a handwritten digit
between 0 and 1. Here is an example image for each class:
from sklearn.datasets import load_digits
digits = load_digits()
fig, axes = plt.subplots(2, 5, figsize=(10, 5),
subplot_kw={'xticks':(), 'yticks': ()})
for ax, img in zip(axes.ravel(), digits.images):
ax.imshow(img)
Lets use PCA to visualize the data reduced to two dimensions. We plot the first two
principal components, and color each dot by its class:
# build a PCA model
pca = PCA(n_components=2)
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pca.fit(digits.data)
# transform the digits data onto the first two principal components
digits_pca = pca.transform(digits.data)
colors = ["#476A2A", "#7851B8", "#BD3430", "#4A2D4E", "#875525",
"#A83683", "#4E655E", "#853541", "#3A3120","#535D8E"]
plt.figure(figsize=(10, 10))
plt.xlim(digits_pca[:, 0].min(), digits_pca[:, 0].max())
plt.ylim(digits_pca[:, 1].min(), digits_pca[:, 1].max())
for i in range(len(digits.data)):
# actually plot the digits as text instead of using scatter
plt.text(digits_pca[i, 0], digits_pca[i, 1], str(digits.target[i]),
color = colors[digits.target[i]],
fontdict={'weight': 'bold', 'size': 9})
plt.xlabel("first principal component")
plt.ylabel("second principal component")
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Here, we actually used the true digit classes as glyphs, to show which class is where.
The digits zero, six and four are relatively well-separated using the first two principal
components, though they still overlap. Most of the other digits overlap significantly.
Lets apply t-SNE to the same dataset, and compare results. As t-SNE does not sup‐
port transforming new data, the TSNE class has no transform method. Instead, we can
call the fit_transform method, which will build the model, and immediately return
the transformed data:
from sklearn.manifold import TSNE
tsne = TSNE(random_state=42)
# use fit_transform instead of fit, as TSNE has no transform method:
digits_tsne = tsne.fit_transform(digits.data)
plt.figure(figsize=(10, 10))
plt.xlim(digits_tsne[:, 0].min(), digits_tsne[:, 0].max() + 1)
plt.ylim(digits_tsne[:, 1].min(), digits_tsne[:, 1].max() + 1)
for i in range(len(digits.data)):
# actually plot the digits as text instead of using scatter
plt.text(digits_tsne[i, 0], digits_tsne[i, 1], str(digits.target[i]),
color = colors[digits.target[i]],
fontdict={'weight': 'bold', 'size': 9})
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The result of t-SNE is quite remarkable. All the classes are quite clearly separated.
The ones and nines are somewhat split up, but most of the classes form a single dense
group. Keep in mind that this method has no knowledge of the class labels: it is com‐
pletely unsupervised. Still, it can find a representation of the data in two dimensions
that clearly separates the classes, based solely on how close points are in the original
space.
t-SNE has some tuning parameters, though it often works well with the default set‐
tings. You can try playing with perplexity and early_exaggeration, though the
effects are usually minor.
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Clustering
As we described above, clustering is the task of partitioning the dataset into groups,
called clusters. The goal is to split up the data in such a way that points within a single
cluster are very similar and points in different clusters are different. Similarly to clas‐
sification algorithms, clustering algorithms assign (or predict) a number to each data
point, indicating which cluster a particular point belongs to.
k-Means clustering
k-Means clustering is one of the simplest and most commonly used clustering algo‐
rithms. It tries to find cluster centers that are representative of certain regions of the
data.
The algorithm alternates between two steps: assigning each data point to the closest
cluster center, and then setting each cluster center as the mean of the data points that
are assigned to it.
The algorithm is finished when the assignment of instances to clusters no longer
changes.
Figure kmeans_algorithm illustrates the algorithm on a synthetic dataset:
mglearn.plots.plot_kmeans_algorithm()
plt.suptitle("kmeans_algorithm");
We specified that we are looking for three clusters, so the algorithm was initialized by
declaring three data points as cluster centers (see “Initialization”). Then the iterative
algorithm starts: Each data point is assigned to the cluster center it is closest to (see
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Assign Points (1)”). Next, the cluster centers are updated to be the mean of the
assigned points (see “Recompute Centers (1)”). Then the process is repeated. After
the second iteration, the assignment of points to cluster centers remained unchanged,
so the algorithm stops.
Given new data points, k-Means will assign them to the closest cluster center. Here
are the boundaries of the cluster centers that were learned in the diagram above:
mglearn.plots.plot_kmeans_boundaries()
Applying k-Means with scikit-learn is quite straight-forward. Here we apply it to the
synthetic data that we used for the plots above. We instantiate the KMeans class, and
set the number of clusters we are looking for [footnote: If you don’t provide n_clus
ters it is set to eight by default. There is no particular reason why you should use
eight.]. Then we call the fit method with the data:
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
# generate synthetic two-dimensional data
X, y = make_blobs(random_state=1)
# build the clustering model:
kmeans = KMeans(n_clusters=3)
kmeans.fit(X)
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KMeans(copy_x=True, init='k-means++', max_iter=300, n_clusters=3, n_init=10,
n_jobs=1, precompute_distances='auto', random_state=None, tol=0.0001,
verbose=0)
During the algorithm, each training data point in X is assigned a cluster label. You can
find these labels in the kmeans.labels_ attribute:
print(kmeans.labels_)
[1 9 3 6 9 4 1 2 1 2 7 8 9 2 0 1 5 8 5 0 4 2 5 2 7 3 9 4 3 5 1 7 2 3 1 4 8
1 0 3 8 7 5 3 9 7 1 0 3 2 0 8 4 6 2 1 5 2 5 7 8 6 9 1 2 6 7 9 7 6 8 6 8 3
2 6 3 1 5 8 4 7 8 4 3 7 1 7 8 4 5 1 4 0 4 9 9 8 6 3 2 4 7 1 6 3 6 7 9 5 0
7 7 6 1 9 5 4 8 1 1 0 3 7 3 0 1 3 2 4 1 0 6 8 2 9 2 6 4 1 3 3 5 7 7 1 3 2
5 3 8 9 1 5 8 7 2 8 5 5 3 2 5 9 8 9 5 8 9 2 5 6 6 0 3 2 4 0 0 5 9 6 4 9 4
5 7 2 6 0 6 5 1 1 3 0 4 3 5 9]
As we asked for three clusters, the clusters are numbered 0 to 2.
You can also assign cluster labels to new points, using the predict method. Each new
point is assigned to the closest cluster center when predicting, but the existing model
is not changed. Running predict on the training set returns the same as labels_:
print(kmeans.predict(X))
[1 2 2 2 0 0 0 2 1 1 2 2 0 1 0 0 0 1 2 2 0 2 0 1 2 0 0 1 1 0 1 1 0 1 2 0 2
2 2 0 0 2 1 2 2 0 1 1 1 1 2 0 0 0 1 0 2 2 1 1 2 0 0 2 2 0 1 0 1 2 2 2 0 1
1 2 0 0 1 2 1 2 2 0 1 1 1 1 2 1 0 1 1 2 2 0 0 1 0 1]
You can see thgat clustering is somewhat similar to classification, in that each item
gets a label. However, there is no ground truth, and consequently the labels them‐
selves have no a priori meaning. Lets go back to the example of clustering face images
that we discusse before. It might be that the cluster 3 found by the algorithm contains
only faces of your friend Bela. You can only know that after you looked at the pic‐
tures, though, and the number 3 is arbitrary. The only information the algorithm
gives you is that all faces labeled as 3 are similar.
For the clustering we just computed on the two dimensional toy dataset, that means
that
Here is a plot of this data again. The cluster centers are stored in the cluster_cen
ters_ attribute, and we plot them as triangles:
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plt.scatter(X[:, 0], X[:, 1], c=kmeans.labels_, cmap=mglearn.cm3, s=60)
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1],
marker='^', s=100, linewidth=2, c=[0, 1, 2], cmap=mglearn.cm3)
We can also use more or less cluster centers:
fig, axes = plt.subplots(1, 2)
# using two cluster centers:
kmeans = KMeans(n_clusters=2)
kmeans.fit(X)
assignments = kmeans.labels_
axes[0].scatter(X[:, 0], X[:, 1], c=assignments, cmap=mglearn.cm2, s=60)
# using five cluster centers:
kmeans = KMeans(n_clusters=5)
kmeans.fit(X)
assignments = kmeans.labels_
axes[1].scatter(X[:, 0], X[:, 1], c=assignments, cmap='jet', s=60);
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Failure cases of k-Means
Even if you know the “right” number of clusters for a given dataset, k-Means might
not always be able to recover them. Each cluster is defined solely by its center, which
means that each cluster is a convex shape. As a result of this is that k-Means can only
capture relatively simple shapes.
k-Means also assumes that all clusters have the same “diameter” in some sense; it
always draws the boundary between clusters to be exactly in the middle between the
cluster centers. That can sometimes lead to surprising results, as shown below:
X, y = make_blobs(random_state=0)
plt.scatter(X[:, 0], X[:, 1]);
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k-Means also assumes that all directions are equally important for each cluster. The
plot below shows a two-dimensional dataset where there are three clearly separated
parts in the data. However, these groups are stretched towards the diagonal. As k-
Means only considers the distance to the nearest cluster center, it can’t handle this
kind of data.
# generate some random cluster data
X, y = make_blobs(random_state=170, n_samples=600)
rng = np.random.RandomState(74)
# transform the data to be streched
transformation = rng.normal(size=(2, 2))
X = np.dot(X, transformation)
# cluster the data into three clusters
kmeans = KMeans(n_clusters=3)
kmeans.fit(X)
y_pred = kmeans.predict(X)
# plot the cluster assignments and cluster centers
plt.scatter(X[:, 0], X[:, 1], c=y_pred, cmap=mglearn.cm3)
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1],
marker='^', c=['b', 'r', 'g'], s=60, linewidth=2);
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k-Means also performs poorly if the clusters have more complex shapes, like the
two_moons data we encountered in Chapter 2:
# generate synthetic two_moons data (with less noise this time)
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
# cluster the data into two clusters
kmeans = KMeans(n_clusters=2)
kmeans.fit(X)
y_pred = kmeans.predict(X)
# plot the cluster assignments and cluster centers
plt.scatter(X[:, 0], X[:, 1], c=y_pred, cmap=mglearn.cm3, s=60)
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1],
marker='^', c=['b', 'g'], s=60, linewidth=2);
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Here, we would hope that the clustering algorithm can discover the two half-moon
shapes. However, this is not possible using the k-Means algorithm.
Vector Quantization - Or Seeing k-Means as Decomposition
Even though k-Means is a clustering algorithm, there are interesting parallels
between k-Means and decomposition methods like PCA and NMF that we discussed
above. You might remember that PCA tries to find directions of maximum variance
in the data, while NMF tries to find additive components, which often correspond to
extremes” or “parts” of the data (see Figure nmf_illustration). Both methods tried to
express data points as a sum over some components.
k-Means on the other hand tries to represent each data point using a cluster center.
You can think of that as each point being represented using only a single component,
which is given by the cluster center. This view of k-Means as a decomposition
method, where each point is represented using a single component, is called vector
quantization.
Here is a side-by-side comparison of PCA, NMF and k-Means, showing the compo‐
nents extracted, as well as reconstructions of faces from the test set using 100 compo‐
nents. For k-Means, the reconstruction is the closest cluster center found on the
training set:
X_train, X_test, y_train, y_test = train_test_split(X_people, y_people, stratify=y_people, random_state=0)
nmf = NMF(n_components=100)
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nmf.fit(X_train)
pca = PCA(n_components=100)
pca.fit(X_train)
kmeans = KMeans(n_clusters=100)
kmeans.fit(X_train)
X_reconstructed_pca = pca.inverse_transform(pca.transform(X_test))
X_reconstructed_kmeans = kmeans.cluster_centers_[kmeans.predict(X_test)]
X_reconstructed_nmf = np.dot(nmf.transform(X_test), nmf.components_)
fig, axes = plt.subplots(3, 5, figsize=(8, 8)) #, subplot_kw={'xticks': (), 'yticks': ()}
fig.suptitle("Extracted Components")
for ax, comp_kmeans, comp_pca, comp_nmf in zip(axes.T, kmeans.cluster_centers_, pca.components_, nmf.components_):
ax[0].imshow(comp_kmeans.reshape(image_shape))
ax[1].imshow(comp_pca.reshape(image_shape), cmap='viridis')
ax[2].imshow(comp_nmf.reshape(image_shape))
axes[0, 0].set_ylabel("kmeans")
axes[1, 0].set_ylabel("pca")
axes[2, 0].set_ylabel("nmf")
fig, axes = plt.subplots(4, 5, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(8, 8))
fig.suptitle("Reconstructions")
for ax, orig, rec_kmeans, rec_pca, rec_nmf in zip(axes.T, X_test, X_reconstructed_kmeans,
X_reconstructed_pca, X_reconstructed_nmf):
ax[0].imshow(orig.reshape(image_shape))
ax[1].imshow(rec_kmeans.reshape(image_shape))
ax[2].imshow(rec_pca.reshape(image_shape))
ax[3].imshow(rec_nmf.reshape(image_shape))
axes[0, 0].set_ylabel("original")
axes[1, 0].set_ylabel("kmeans")
axes[2, 0].set_ylabel("pca")
axes[3, 0].set_ylabel("nmf")
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An interesting aspect of vector quantization using k-Means is that we can use many
more clusters than input dimensions to encode our data. Lets go back to the
two_moons data. Using PCA or NMF, there is nothing much we can do to this data, as
it lives in only two dimensions. Reducing it to one dimension with PCA or NMF
would completely destroy the structure of the data. But we can find a more expressive
representation using k-Means, by using more cluster centers:
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
kmeans = KMeans(n_clusters=10)
kmeans.fit(X)
y_pred = kmeans.predict(X)
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plt.scatter(X[:, 0], X[:, 1], c=y_pred, s=60, cmap='Paired')
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1],
marker='^', c=range(kmeans.n_clusters), s=60, linewidth=2, cmap='Paired')
print(y_pred)
[1 9 3 6 9 4 1 2 1 2 7 8 9 2 0 1 5 8 5 0 4 2 5 2 7 3 9 4 3 5 1 7 2 3 1 4 8
1 0 3 8 7 5 3 9 7 1 0 3 2 0 8 4 6 2 1 5 2 5 7 8 6 9 1 2 6 7 9 7 6 8 6 8 3
2 6 3 1 5 8 4 7 8 4 3 7 1 7 8 4 5 1 4 0 4 9 9 8 6 3 2 4 7 1 6 3 6 7 9 5 0
7 7 6 1 9 5 4 8 1 1 0 3 7 3 0 1 3 2 4 1 0 6 8 2 9 2 6 4 1 3 3 5 7 7 1 3 2
5 3 8 9 1 5 8 7 2 8 5 5 3 2 5 9 8 9 5 8 9 2 5 6 6 0 3 2 4 0 0 5 9 6 4 9 4
5 7 2 6 0 6 5 1 1 3 0 4 3 5 9]
We used 10 cluster centers, which means each point is now assigned a number
between 0 and 9. We can see this as the data being represented using 10 components
(that is, we have ten new features), with all features being zero, apart from the one
that represents the cluster center the point is assigned to. Using this 10-dimensional
representation, it would now be possible to separate the two half-moon shapes using
a linear model, which would not have been possible using the original two features.
[really? can you show what that would look like, maybe?]
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It is also possible to get an even more expressive representation of the data by using
the distances to each of the cluster centers as features. This can be done using the
transform method of kmeans:
distance_features = kmeans.transform(X)
print(distance_features.shape)
print(distance_features)
(200, 10)
[[ 1.53 0.2 1.03 ..., 1.12 0.92 1.14]
[ 2.56 1.01 0.54 ..., 2.28 1.14 0.12]
[ 0.8 0.93 1.33 ..., 0.72 0.79 1.75]
...,
[ 1.12 0.81 1.02 ..., 1.05 0.45 1.49]
[ 0.88 1.03 1.76 ..., 0.34 1.39 1.98]
[ 2.5 0.91 0.59 ..., 2.19 1.15 0.05]]
k-Means is a very popular algorithm for clustering, not only because it is relatively
easy to understand and implement, but also because it runs relatively quickly. k-
Means scales easily to large datasets, and scikit-learn even includes a more scalable
variant in the MiniBatchKMeans class, which can handle very large datasets.
One of the drawbacks of k-Means is that it relies on a random initialization, which
means the outcome of the algorithm depends on a random seed. By default, scikit-
learn runs the algorithm 10 times with 10 different random initializations, and
returns the best result [Footnote: best here meaning that the sum of variances of the
clusters is small].
Further downsides of k-Means are the relatively restrictive assumptions made on the
shape of clusters, and the requirement to specify the number of clusters you are look‐
ing for (which might not be known in a real-world application).
Next, we will look at two more clustering algorithms that improve upon these proper‐
ties in some ways.
Agglomerative Clustering
Agglomerative clustering refers to a collection of clustering algorithms that all build
upon the same principles: The algorithm starts by declaring each point its own clus‐
ter, and then merges the two most similar clusters until some stopping criterion is
satisfied.
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The stopping criterion implemented in scikit-learn is the number of clusters, so simi‐
lar cluster are merged until only the specified number of clusters is left.
There are several linkage criteria that specify how exactly “most similar cluster” is
measured. [this sentence looks a lot like the next sentence - combine into one]
The following three choices are implemented in scikit-learn:
“ward, which is the default choice. Ward picks the two clusters to merge such
that the variance within all clusters increases the least. This often leads to clusters
that are relatively equally sized.
average” linkage merges the two clusters that have the smallest average distance
between all their points.
complete” linkage (also known as maximum linkage) merges the two clusters
that have the smallest maximum distance between their points.
[you keep writing ‘the two clusters', but you did not specify that there would be two
clusters - rahter said that there are a ‘number of clusters']
Ward is generally a good default; all our examples below will use ward.
The plot below illustrates the progression of agglomerative clustering on a two-
dimensional dataset, looking for three clusters.
mglearn.plots.plot_agglomerative_algorithm()
plt.suptitle("agglomerative_algorithm");
In the beginning, each point is its own cluster. Then, in each step, the two clusters
that are closest are merged. In the first four steps, two single point clusters are picked
and these are joined into two-point clusters. In step four, one of the two-point clus‐
ters is extended to a third point, and so on. In step 9, there are only three clusters
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remaining. As we specified that we are looking for three clusters, the algorithm then
stops.
Lets have a look at how agglomerative clustering performs on the simple three-
cluster data we used above.
Because of the way the algorithm works, agglomerative clustering can not make pre‐
dictions for new data points. Therefore, agglomerative clustering has no predict
method. To build the model, and get the cluster memberships on the training set, use
the fit_predict method instead. [footnote: we could also use the labels_ attribute
as we did for k-Means]
from sklearn.cluster import AgglomerativeClustering
X, y = make_blobs(random_state=1)
agg = AgglomerativeClustering(n_clusters=3)
assignment = agg.fit_predict(X)
plt.scatter(X[:, 0], X[:, 1], c=assignment, cmap=mglearn.cm3, s=60)
As expected, the algorithm recovers the clustering perfectly. While the scikit-learn
implementation of agglomerative clustering requires you to specify a number of clus‐
ters you want the algorithm to find, agglomerative clustering methods provide some
help with choosing the right number, which we will now discuss next.
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Hierarchical Clustering and Dendrograms
Agglomerative clustering produces what is known as a hierarchical clustering. The
clustering proceeds iteratively, and every point makes a journey from being a single
point cluster to belonging to some final cluster. Each intermediate step provides a
clustering of the data (with a different number of clusters). It is sometimes helpful to
look at all possible clusterings jointly.
The figure below shows an overlay of all possible clusterings shown in Figure agglom‐
erative_algorithm, providing some insight into how each cluster breaks up into
smaller clusters.
mglearn.plots.plot_agglomerative()
While this visualization provides a very detailed view of the hierarchical clustering, it
relies on the two-dimensional nature of the data, and can therefore not be used on
datasets that have more than two features. There is, however, another tool to visualize
hierarchical clustering, called a dendrogram (as shown in Figure dendrogram below).
Unfortunately, scikit-learn currently does not have the functionality to draw dendro‐
grams. However, you can generate them easily using scipy.
The scipy clustering algorithms have a slightly different interface from the scikit-learn
clustering algorithms.
scipy provides function that take data arrays X linkage array encoding cluster similari‐
ties.
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We can then feed this linkage array into the scipy dendrogram function to plot the
dendrogram:
# import the dendrogram function and the ward clustering function from scipy
from scipy.cluster.hierarchy import dendrogram, ward
X, y = make_blobs(random_state=0, n_samples=12)
# apply the ward clustering to the data array X
# The scipy ward function returns an array that specifies the distances bridged when performing agglomerative clustering
linkage_array = ward(X)
# now we plot the dendrogram for the linkage_array containing the distances between clusters
dendrogram(linkage_array);
# mark the cuts in the tree that signify two or three clusters
ax = plt.gca()
bounds = ax.get_xbound()
ax.plot(bounds, [7.25, 7.25], '--', c='k')
ax.plot(bounds, [4, 4], '--', c='k')
ax.text(bounds[1], 7.25, ' two clusters', verticalalignment='center', fontdict={'size': 15})
ax.text(bounds[1], 4, ' three clusters', verticalalignment='center', fontdict={'size': 15})
plt.title("dendrogram")
The dendrogram shows data points as points on the bottom (numbered from zero to
eleven). Then, a tree is plotted with these points (representing single-point clusters)
as the leafs, and a new node parent is added for each two clusters that are joined.
Reading from bottom to top, the data points 1 and 4 are joined first (as you could see
in Figure agglomerative_algorithm). Next, points 6 and 9 are joined into a cluster,
and so on. The top level, there are two branches, one consisting of point 11, 0, 5, 10,
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7, 6 and 9, and the other one consisting of points 1, 4, 3, 2 and 8. These correspond to
the two largest clusters in in the left hand side of the plot.
The y axis in the dendrogram not only specifies when in the agglomerative algorithm
two clusters get merged. The length of each branch also shows how far apart the
merged clusters are. The longest branches in this dendogram are the three lines that
are around the “10” mark on the y-axis. That these are the longest branches indicates
that going from three to two clusters meant merging some very far-apart points. We
see this again at the top of the chart, where merging the two remaining clusters into a
single cluster again bridges a large distance.
Unfortunately, agglomerative clustering still fails at separating complex shapes like
the two_moons dataset. The same is not true for the next algorithm we will look at,
DBSCAN.
DBSCAN
Another very useful clustering algorithm is DBSCAN (which stands for “Density-
based spatial clustering of applications with noise”). The main benefits of DBSCAN
are that a) it does not require the user to set the number of clusters a priori, b) it can
capture clusters of complex shapes, and c) it can identify point that are not part of any
cluster.
DBSCAN is somewhat slower than agglomerative clustering and k-Means, but still
scales to relatively large datasets.
The way DBSCAN works is by identifying points that are in “crowded” regions of the
feature space, where many data points are close together. These regions are referred
to as dense regions in feature space. The idea behind DBSCAN is that clusters form
dense regions of data, separated by regions that are relatively empty.
Points that are within a dense region are called core samples, and they are defined as
follows.
There are two parameters in DBSCAN, min_samples and eps. If there are at least
min_samples many data points within a distance of eps to a given data point, its
called a core sample. Core samples that are closer than the distance eps are put into
the same cluster by DBSCAN.
The algorithm works by picking a point to start with.
It then finds all points with distance eps or less. If there are less than min_samples
points within distance eps or less, this point is labeled as noise, meaning that this
point doesn’t belong to any cluster.
If there are more than min_samples points within a distance of eps, the point is
labeled a core sample and assigned a new cluster label. Then, all neighbors (withing
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eps) of the point are visited. If they have not been assigned a cluster yet, they are
assigned the new cluster label we just created. If they are core samples, their neigh‐
bors are visited in turn, and so on.
The cluster grows, until there are no more core-samples within distance eps of the
cluster.
Then another point, which hasn’t yet been visited, is picked, and the same procedure
is repeated.
In the end, there are three kinds of points: core points, points that are within distance
eps of core points (called boundary points), and noise. When running the DBSCAN
algorithm on a particular dataset multiple times, the clustering of the core points is
always the same, and the same points will always be labeled as noise. However, a
boundary point might be neighbor to core samples of more than one cluster. There‐
fore, the cluster membership of boundary points depends on the order in which
points are visited. Usually there are only few boundary points, and this slight depend‐
ence on the order of points is not important.
Lets apply DBSCAN on the synthetic data from above. As in agglomerative cluster‐
ing, DBSCAN does not allow predictions on new test data, so we will use the fit_pre
dict method to perform clustering and return the cluster labels in one step:
from sklearn.cluster import DBSCAN
X, y = make_blobs(random_state=0, n_samples=12)
dbscan = DBSCAN()
clusters = dbscan.fit_predict(X)
clusters
array([-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1])
As you can see, all data points were assigned the label -1, which stands for noise. This
is a consequence of the default parameter settings for eps and min_samples, which
are not attuned to small toy datasets.
Lets investigate the effect of changing eps and min_samples.
fig, axes = plt.subplots(3, 4, figsize=(11, 8), subplot_kw={'xticks': (), 'yticks': ()})
# Plot clusters as red, green and blue, and outliers (-1) as white
colors = np.array(['r', 'g', 'b', 'w'])
# iterate over settings of min_samples and eps
for i, min_samples in enumerate([2, 3, 5]):
for j, eps in enumerate([1, 1.5, 2, 3]):
# instantiate DBSCAN with a particular setting
dbscan = DBSCAN(min_samples=min_samples, eps=eps)
# get cluster assignments
clusters = dbscan.fit_predict(X)
print("min_samples: %d eps: %f cluster: %s" % (min_samples, eps, clusters))
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# vizualize core samples and clusters.
sizes = 60 * np.ones(X.shape[0])
# size is given by whether something is a core sample
sizes[dbscan.core_sample_indices_] *= 4
axes[i, j].scatter(X[:, 0], X[:, 1], c=colors[clusters], s=sizes)
axes[i, j].set_title("min_samples: %d eps: %.1f" % (min_samples, eps))
fig.tight_layout()
min_samples: 2 eps: 1.000000 cluster: [-1 0 0 -1 0 -1 1 1 0 1 -1 -1]
min_samples: 2 eps: 1.500000 cluster: [0 1 1 1 1 0 2 2 1 2 2 0]
min_samples: 2 eps: 2.000000 cluster: [0 1 1 1 1 0 0 0 1 0 0 0]
min_samples: 2 eps: 3.000000 cluster: [0 0 0 0 0 0 0 0 0 0 0 0]
min_samples: 3 eps: 1.000000 cluster: [-1 0 0 -1 0 -1 1 1 0 1 -1 -1]
min_samples: 3 eps: 1.500000 cluster: [0 1 1 1 1 0 2 2 1 2 2 0]
min_samples: 3 eps: 2.000000 cluster: [0 1 1 1 1 0 0 0 1 0 0 0]
min_samples: 3 eps: 3.000000 cluster: [0 0 0 0 0 0 0 0 0 0 0 0]
min_samples: 5 eps: 1.000000 cluster: [-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1]
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min_samples: 5 eps: 1.500000 cluster: [-1 0 0 0 0 -1 -1 -1 0 -1 -1 -1]
min_samples: 5 eps: 2.000000 cluster: [-1 0 0 0 0 -1 -1 -1 0 -1 -1 -1]
min_samples: 5 eps: 3.000000 cluster: [0 0 0 0 0 0 0 0 0 0 0 0]
In this plot, points that belong to clusters are colored, while the noise points are
shown in white.
Core samples are shown as large points, while border points are displayed as smaller
points.
Increasing eps (going from left to left to right in the figure) means that more points
will be included in a cluster. This makes clusters grow, but might also lead to multiple
clusters joining into one. Increasing min_samples (going from top to bottom in the
figure) means that fewer points will be core points, and more points will be labeled as
noise.
The parameter eps is somewhat more important, as it determines what it means for
points to be “close.
Setting eps to be very small will mean that no points are core samples, and may lead
to all points being labeled as noise. Setting eps to be very large will result in all points
forming a single cluster.
The setting of min_samples mostly determines whether points in less dense regions
will be labeled as outliers, or as their own cluster. If you decrease min_samples, any‐
thing that would have been a cluster with less than min_samples many samples will
now be labeled as noise. The min_samples therefore determines the minimum cluster
size. You can see this very clearly in the plot above, when going from min_samples=3
to min_samples=5 with eps=1.5. With min_samples=3, there are three clusters: one of
four points, one of five points and one of three points. Using min_samples=5, the two
smaller clusters (with three or four points) are now labeled as noise, and only the
cluster with 5 samples remains.
While DBSCAN doesnt require setting the number of clusters explicitly, setting eps
implicitly controls how many clusters will be found.
Finding a good setting for eps is sometimes easier after scaling the data using Stand
ardScaler or MinMaxScaler, as using these scaling techniques will ensure that all fea‐
tures have similar ranges.
Below is the result of DBSCAN running on the two_moons dataset. The algorithm
actually finds the two half-circles and separates them using the default settings.
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
# Rescale the data to zero mean and unit variance
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scaler = StandardScaler()
scaler.fit(X)
X_scaled = scaler.transform(X)
dbscan = DBSCAN()
clusters = dbscan.fit_predict(X_scaled)
# plot the cluster assignments
plt.scatter(X_scaled[:, 0], X_scaled[:, 1], c=clusters, cmap=mglearn.cm2, s=60)
As the algorithm produced the desired number of clusters (two), the parameter set‐
tings seem to work well.
If we decrease eps to 0.2 (from the default of 0.5), we will get 8 clusters, which are
clearly too many. Increasing eps to 0.7 results in a single cluster.
When using DBSCAN, you need to be careful about handling the returned cluster
assignments. The use of -1 to indicate noise might result in unexpected effects when
using the cluster labels to index another array. [an example here could be useful - sort
of know what youre talking about, but sort of not]
Comparing and evaluating clustering algorithms
After talking about the algorithms behind k-Means, agglomerative clustering and
DBSCAN, we will now compare them on some real world datasets. One of the chal‐
lenges in applying clustering algorithms is that it is very hard to access how well a
clustering algorithm worked, and to compare outcomes between different algorithms.
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Evaluating clustering with ground truth
There are metrics that can be used to assess the outcome of a clustering algorithm
relative to a ground truth clustering, the most important ones being the adjusted rand
index (ARI) and normalized mutual information (NMI), which both provide a quanti‐
tative measure between 0 and 1.
Below we compare the k-Means, agglomerative clustering and DBSCAN algorithms
using ARI. We also include what it looks like when we randomly assign points to two
clusters for comparison.
from sklearn.metrics.cluster import adjusted_rand_score
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
# Rescale the data to zero mean and unit variance
scaler = StandardScaler()
scaler.fit(X)
X_scaled = scaler.transform(X)
fig, axes = plt.subplots(1, 4, figsize=(15, 3), subplot_kw={'xticks': (), 'yticks': ()})
# make a list of algorithms to use
algorithms = [KMeans(n_clusters=2), AgglomerativeClustering(n_clusters=2), DBSCAN()]
# create a random cluster assignment for reference:
random_state = np.random.RandomState(seed=0)
random_clusters = random_state.randint(low=0, high=2, size=len(X))
# plot random assignment:
axes[0].scatter(X_scaled[:, 0], X_scaled[:, 1], c=random_clusters, cmap=mglearn.cm3, s=60)
axes[0].set_title("Random assignment - ARI: %.2f" % adjusted_rand_score(y, random_clusters))
for ax, algorithm in zip(axes[1:], algorithms):
# plot the cluster assignments and cluster centers
clusters = algorithm.fit_predict(X_scaled)
ax.scatter(X_scaled[:, 0], X_scaled[:, 1], c=clusters, cmap=mglearn.cm3, s=60)
ax.set_title("%s - ARI: %.2f" % (algorithm.__class__.__name__, adjusted_rand_score(y, clusters)))
The adjusted rand index provides intuitive results, with a random cluster assignment
having a score of 0, and DBSCAN (which recovers the desired clustering perfectly)
having a score of 1. A common mistake when evaluating clustering in this way is to
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use accuracy_score instead of a clustering metric like adjusted_rand_score and
normalized_mutual_info_score.
The problem in using accuracy is that it requires the assigned cluster labels to exactly
match the ground truth. However, the cluster labels themselves are meaningless, and
only which points are in the same cluster matters:
from sklearn.metrics import accuracy_score
# These two labelings of points correspond to the same clustering:
clusters1 = [0, 0, 1, 1, 0]
clusters2 = [1, 1, 0, 0, 1]
# accuracy is zero, as none of the labels are the same:
print("Accuracy: %.2f" % accuracy_score(clusters1, clusters2))
# adjusted rand score is 1, as the clustering is exactly the same:
print("ARI: %.2f" % adjusted_rand_score(clusters1, clusters2))
Accuracy: 0.00
ARI: 1.00
Evaluating clustering without ground truth
Although we have just shown how to evaluate on clustering algorithms, in practice,
there is a big problem with the evaluation using measures like ARI.
When applying clustering algorithms, there is usually no ground truth to which to
compare. If we knew the right clustering of the data, we could use this information to
build a supervised model like a classifier.
Therefore, using metric like ARI and NMI usually only really helps in developing
algorithms, not in assessing success in an application.
There are scoring metrics for clustering that don’t require ground truth, like the sil‐
houette coecient. However, these often dont work well in practice. The silhouette
score computes the compactness of a cluster, where higher is better, with a perfect
score of 1. While compact clusters are good, compactness doesn’t allow for complex
shapes.
Here is an example comparing the outcome of k-Means, agglomerative clustering and
DBSCAN on the two moons using the silhouette score:
from sklearn.metrics.cluster import silhouette_score
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
# Rescale the data to zero mean and unit variance
scaler = StandardScaler()
scaler.fit(X)
X_scaled = scaler.transform(X)
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fig, axes = plt.subplots(1, 4, figsize=(15, 3), subplot_kw={'xticks': (), 'yticks': ()})
# create a random cluster assignment for reference:
random_state = np.random.RandomState(seed=0)
random_clusters = random_state.randint(low=0, high=2, size=len(X))
# plot random assignment:
axes[0].scatter(X_scaled[:, 0], X_scaled[:, 1], c=random_clusters, cmap=mglearn.cm3, s=60)
axes[0].set_title("Random assignment: %.2f" % silhouette_score(X_scaled, random_clusters))
algorithms = [KMeans(n_clusters=2), AgglomerativeClustering(n_clusters=2), DBSCAN()]
for ax, algorithm in zip(axes[1:], algorithms):
clusters = algorithm.fit_predict(X_scaled)
# plot the cluster assignments and cluster centers
ax.scatter(X_scaled[:, 0], X_scaled[:, 1], c=clusters, cmap=mglearn.cm3, s=60)
ax.set_title("%s : %.2f" % (algorithm.__class__.__name__, silhouette_score(X_scaled, clusters)))
As you can see, k-Means gets the highest silhouette score, even though we might pre‐
fer the result produced by DBSCAN.
A slightly better strategy for evaluating clusters are robustness-based clustering met‐
rics. These run an algorithm after adding some noise to the data, or using different
parameter settings, and compare the outcomes. The idea is that if many algorithm
parameters and many perturbations of the data return the same result, it is likely to
be trustworthy.
Unfortunately, this strategy is not implemented in scikit-learn at the time of writing.
Even if we get a very robust clustering, or a very high silhouette score, we still don’t
know if there is any semantic meaning in the clustering, or whether the clustering
reflects an aspect of the data that we are interested in.
Lets go back to the example of face images. We hope to find groups of similar faces,
say men and women, or old people and young people, or people with beards and
without. Lets say we cluster the data into two clusters, and all algorithms agree about
which points should be clustered together. We still don’t know if the clusters that are
found correspond in any way to the concepts we are interested in. It could be that
they found side-views versus front-views. Or pictures taken at night versus pictures
taken during the day. Or pictures taken with iPhones versus pictures taken with
Android phones.
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Only if we actually analyze the clusters can we know whether the clustering corre‐
sponds to anything we are interested in.
Comparing algorithms on the faces dataset
Lets apply the k-Means, DBSCAN and agglomerative clustering algorithms to the
labeled faces in the wild dataset, and see if any of them find interesting structure.
We will use the eigenface representation of the data, as produced by
PCA(whiten=True), with 100 components. We saw above that this is a more semantic
representation of the face images than the raw pixels. It will also make computation
faster.
Its a good exercise for you to run the experiments below on the original data and
compare results.
# extract eigenfaces from lfw data and transform data
from sklearn.decomposition import PCA
pca = PCA(n_components=100, whiten=True)
pca.fit_transform(X_people)
X_pca = pca.transform(X_people)
We will start by applying DBSCAN, which we just discussed.
Analyzing the faces dataset with DBSCAN
# apply DBSCAN with default parameters
dbscan = DBSCAN()
labels = dbscan.fit_predict(X_pca)
np.unique(labels)
array([-1])
We see that all returned labels are -1, so all of the data was labeled as “noise” by
DBSCAN. There are two things we can change to help this: we can make eps higher,
to expand the neighborhood of each point, and min_samples lower, to consider
smaller groups of points as clusters. Lets try changing min_samples first:
dbscan = DBSCAN(min_samples=3)
labels = dbscan.fit_predict(X_pca)
np.unique(labels)
array([-1])
Even when considering groups of three points, everything is labeled as noise. So we
need to increase eps.
dbscan = DBSCAN(min_samples=3, eps=15)
labels = dbscan.fit_predict(X_pca)
np.unique(labels)
array([-1, 0])
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Using a much larger eps=15 we get only a single clusters and noise points. We can use
this result and find out what the “noise” looks like compared to the rest of the data. To
understand better whats happening, lets look at how many points are noise, and how
many points are inside the cluster:
# count number of points in all clusters and noise.
# bincount doesn't allow negative numbers, so we need to add 1.
# the first number in the result corresponds to noise points
np.bincount(labels + 1)
array([ 27, 2036])
There are only very few noise points, 27, so we can look at all of them:
noise = X_people[labels==-1]
fig, axes = plt.subplots(3, 9, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(12, 4))
for image, ax in zip(noise, axes.ravel()):
ax.imshow(image.reshape(image_shape), vmin=0, vmax=1)
Comparing these images to the random sample of face images from Figure
some_faces, we can guess why they were labeled as noise: the image in the sixth image
in the first row one has a person drinking from a glass, there are images with hats,
and the second to last image has a hand in front of the face. The other images contain
odd angles or crops that are too close (see the first image in the first row) or too wide
(see the last image in the first row).
This kind of analysis, trying to find “the odd one out”, is called outlier detection. If this
was a real application, we might try to do a better job in cropping images, to get more
homogeneous data. There is little we can do about people sometimes wearing hats or
drinking from a glass, but its good to know that these are issues in the data that any
algorithm we might apply needs to handle.
If we want to find more interesting clusters than just one large one, we need to set eps
smaller, somewhere between 15 and 0.5 (the default). Lets have a look at what differ‐
ent values of eps result in:
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for eps in [1, 3, 5, 7, 9, 11, 13]:
print("\neps=%d" % eps)
dbscan = DBSCAN(eps=eps, min_samples=3)
labels = dbscan.fit_predict(X_pca)
print("Number of clusters: %s" % np.unique(labels))
print("Clusters: %s" % np.bincount(labels + 1))
eps=1
Number of clusters: [-1]
Clusters: [2063]
eps=3
Number of clusters: [-1]
Clusters: [2063]
eps=5
Number of clusters: [-1]
Clusters: [2063]
eps=7
Number of clusters: [-1 0 1 2 3 4 5 6 7 8 9 10 11 12]
Clusters: [2008 4 6 3 6 9 5 3 3 4 3 3 3 3]
eps=9
Number of clusters: [-1 0 1 2]
Clusters: [1273 784 3 3]
eps=11
Number of clusters: [-1 0]
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Clusters: [ 429 1634]
eps=13
Number of clusters: [-1 0]
Clusters: [ 115 1948]
For small numbers of eps, again all points are labeled as noise. For eps=7, we get
many noise points, and many smaller clusters. For eps=9 we still get many noise
points, one big cluster and some smaller clusters. Starting from eps=11 we get only
one large cluster and noise.
What is interesting to note is that there are never more than one large cluster. There
is at most one large cluster containing most of the points, and some smaller clusters.
This indicates that there are not two or three different kinds of face images in the data
that are very distinct, but that all images are more or less equally similar (or dissimi‐
lar) from the rest.
The results for eps=7 look most interesting, with many small clusters. We investigate
this clustering in more detail, by visualizing all of the points in each of the 13 small
clusters:
# dbscan = DBSCAN(min_samples=3, eps=7)
labels = dbscan.fit_predict(X_pca)
for cluster in range(max(labels)):
mask = labels == cluster
n_images = np.sum(mask)
fig, axes = plt.subplots(1, n_images, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(n_images * 1.5, 4))
for image, label, ax in zip(X_people[mask], y_people[mask], axes):
ax.imshow(image.reshape(image_shape), vmin=0, vmax=1)
ax.set_title(people.target_names[label].split()[-1])
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Some of the clusters correspond to people with very distinct faces (within this data‐
set), such as Tayyip or Koizumi. Within each cluster, the orientation of the face is also
quite fixed, as well as the facial expression. Some of the clusters contain faces of mul‐
tiple people, but they share a similar orientation of the face and expression.
This concludes our analysis of the DBSCAN algorithm applied to the faces dataset.
As you can see, we are doing a very manual analysis here, different from the much
more automatic search approach we could use for supervised learning, based on
$R^2$ or accuracy.
Lets move on to applying k-Means and Agglomerative Clustering.
Analyzing the faces dataset with k-Means
We saw that it was not possible to create more than one big cluster using DBSCAN.
Agglomerative clustering and k-Means are much more likely to create clusters of even
size, but we do need to set a number of clusters.
We could set the number of clusters to the known number of people in the dataset,
though it is very unlikely that an unsupervised clustering algorithm will recover
them.
Instead, we can start with a low number of clusters, like 10, which might allow us to
analyze each of the clusters.
n_clusters = 10
# extract clusters with k-Means
km = KMeans(n_clusters=n_clusters, random_state=0)
labels_km = km.fit_predict(X_pca)
print("cluster sizes k-Means: %s" % np.bincount(labels_km))
cluster sizes k-Means: [185 146 168 190 153 284 263 133 223 318]
As you can see, k-Means clustering partitioned the data into relatively similarly sized
clusters from 133 to 318. This is quite different from the result of DBSCAN.
We can further analyze the outcome of k-Means by visualizing the cluster centers. As
we clustered in the representation produced by PCA, we need to rotate the cluster
centers back into the original space to visualize them, using pca.inverse_transform.
fig, axes = plt.subplots(2, 5, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(12, 4))
for center, ax in zip(km.cluster_centers_, axes.ravel()):
ax.imshow(pca.inverse_transform(center).reshape(image_shape), vmin=0, vmax=1)
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The cluster centers found by k-Means are very smooth version of faces. This is not
very surprising, given that each center is an average of 133 to 318 face images. Work‐
ing with a reduced PCA representation adds to the smoothness of the images (com‐
pared to faces reconstructed using 100 PCA dimensions in Figure
pca_reconstructions).
The clustering seems to pick up on different orientations of the face, different expres‐
sion (the third cluster center seems to show a smiling face), and presence of collars
(see the second to last cluster center).
For a more detailed view, we show for each cluster center the five most typical images
in the cluster (the images that are assigned to the cluster and closest to the cluster
center) and the five most atypical images in the cluster (the images that are assigned
to the cluster and furthest to the cluster center):
n_clusters = 10
for cluster in range(n_clusters):
center = km.cluster_centers_[cluster]
mask = km.labels_ == cluster
dists = np.sum((X_pca - center) ** 2, axis=1)
dists[~mask] = np.inf
inds = np.argsort(dists)[:5]
dists[~mask] = -np.inf
inds = np.r_[inds, np.argsort(dists)[-5:]]
fig, axes = plt.subplots(1, 11, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(10, 8))
axes[0].imshow(pca.inverse_transform(center).reshape(image_shape), vmin=0, vmax=1)
for image, label, asdf, ax in zip(X_people[inds], y_people[inds], labels_km[inds], axes[1:]):
ax.imshow(image.reshape(image_shape), vmin=0, vmax=1)
ax.set_title("%s" % (people.target_names[label].split()[-1]), fontdict={'fontsize': 9})
print("kmeans_face_clusters")
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kmeans_face_clusters
Figure kmeans_face_clusters confirms our intuition about smiling faces for the third
cluster, and also the importance of orientation for the other clusters. The “atypical”
points are not very similar to the cluster center, though, and the assignment seems
somewhat arbitrary. This can be attributed to the fact that k-Means partitions all the
data points, and doesn’t have a concept of “noise” points, as DBSCAN does.
Using a larger number of clusters, the algorithm could find finer distinctions. How‐
ever, adding more clusters makes manual inspection even harder.
Analyzing the faces dataset with agglomerative clustering
Now, let’s look at the results of agglomerative clustering:
# extract clusters with ward agglomerative clustering
agglomerative = AgglomerativeClustering(n_clusters=10)
labels_agg = agglomerative.fit_predict(X_pca)
print("cluster sizes agglomerative clustering: %s" % np.bincount(labels_agg))
cluster sizes agglomerative clustering: [167 50 252 367 160 80 50 67 521 349]
Agglomerative clustering produces relatively equally sized clusters, with cluster sizes
between 50 and 521. These are more uneven than k-Means, but much more even
than the ones produced by DBSCAN.
We can compute the ARI to measure if the two partitions of the data given by
agglomerative clustering and k-Means are similar:
adjusted_rand_score(labels_agg, labels_km)
0.09733057984938646
An ARI of only 0.1 means that the two clusterings labels_agg and labels_km have
quite little in common. This is not very surprising, given the fact that points further
away from the cluster centers seem to have little in common for k-Means.
Next, we might want to plot the dendrogram. We limit the depth of the tree in the
plot, as branching down to the individual 2063 data points would result in an unread‐
ably dense plot.
# import the dendogram function and the ward clustering function from scipy
from scipy.cluster.hierarchy import dendrogram, ward
# apply the ward clustering to the data array X
# The scipy ward function returns an array that specifies the distances bridged
# when performing agglomerative clustering
linkage_array = ward(X_pca)
# now we plot the dendogram for the linkage_array containing the distances between clusters
plt.figure(figsize=(20, 5))
dendrogram(linkage_array, p=7, truncate_mode='level', no_labels=True);
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Creating ten clusters, we cut across the tree at the very top, where there are 10 vertical
lines. In the dendrogram for the toy data shown in Figure dendrogram, you could see
by the length of the branches that two or three clusters might capture the data appro‐
priately. For the faces data, there doesn’t seem to be a very natural cut-off point. There
are some branches that represent more distinct groups, but there doesn’t seem to be a
particular number of clusters that is a good fit. This is not particularly surprising,
given the results of DBSCAN, which tried to cluster all points together.
Here is a visualization of the ten clusters, similarly to k-Means above.
Note that there is no notion of cluster center in agglomerative clustering (though we
could compute the mean), and we simply show the first couple of points in each clus‐
ter. We show the number of points in each cluster to the left of the first image.
n_clusters = 10
for cluster in range(n_clusters):
mask = labels_agg == cluster
fig, axes = plt.subplots(1, 10, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(15, 8))
axes[0].set_ylabel(np.sum(mask))
for image, label, asdf, ax in zip(X_people[mask], y_people[mask], labels_agg[mask], axes):
ax.imshow(image.reshape(image_shape), vmin=0, vmax=1)
ax.set_title("%s" % people.target_names[label].split()[-1], fontdict={'fontsize': 9})
While some of the clusters seem to have a semantic theme, many of them are too
large to be actually homogeneous. To get more homogeneous cluster, we run the
algorithm again, this time with 40 clusters, and pick out some of the clusters that are
particularly interesting:
# extract clusters with ward agglomerative clustering
agglomerative = AgglomerativeClustering(n_clusters=40)
labels_agg = agglomerative.fit_predict(X_pca)
print("cluster sizes agglomerative clustering: %s" % np.bincount(labels_agg))
n_clusters = 40
for cluster in [15, 7, 17, 20, 25, 29]: # hand-picked "interesting" clusters
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mask = labels_agg == cluster
fig, axes = plt.subplots(1, 15, subplot_kw={'xticks': (), 'yticks': ()}, figsize=(15, 8))
cluster_size = np.sum(mask)
axes[0].set_ylabel(cluster_size)
for image, label, asdf, ax in zip(X_people[mask], y_people[mask], labels_agg[mask], axes):
ax.imshow(image.reshape(image_shape), vmin=0, vmax=1)
ax.set_title("%s" % (people.target_names[label].split()[-1]), fontdict={'fontsize': 9})
for i in range(cluster_size, 15):
axes[i].set_visible(False)
cluster sizes agglomerative clustering: [ 50 111 103 120 60 169 50 53 10 33 187 30 50 85 53 28 35 6
111 18 3 78 81 36 10 16 22 5 57 27 75 26 92 5 13 35
8 23 20 69]
Here, the clustering seems to have picked up on “dark skinned and smiling”, “serious
eyebrows, “collared shirt”, “white hat and showing front teeth, “high forehead” and
Asian.
We could also] find these highly similar clusters using the dendrogram, if we did
more a detailed analysis.
Summary of Clustering Methods
We saw above that applying and evaluating clustering is a highly qualitative proce‐
dure, and often most helpful in the exploratory phase of data analysis. We looked at
three clustering algorithms, k-Means, DBSCAN and Agglomerative Clustering. All
three have a way of controlling the granularity of clustering. k-Means and Agglomer‐
ative Clustering allow you to directly specify the number of desired clusters, while
DBSCAN lets you define proximity using the eps parameter, which indirectly influ‐
ences cluster size.
All three methods can be used on large, real-world datasets, are relatively easy to
understand, and allow for clustering into many clusters.
Each of the algorithms has somewhat different strengths. k-Means allows for a char‐
acterization of the clusters using the cluster means. It can also be viewed as a decom‐
position method, where each data point is represented by its cluster center.
DBSCAN allows for the detection of “noise points” that are not assigned any cluster,
and it can help automatically determine the number of clusters. In contrast to the
other two methods, it allow for complex cluster shapes, as we saw in the two_moons
example. DBSCAN sometimes produces clusters of very differing size, which can be a
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strength or a weakness. Agglomerative clustering can provide a whole hierarchy of
possible partitions of the data, which can be easily inspected via dendrograms.
Summary and Outlook
This chapter introduced a range of unsupervised learning algorithms that can be
applied for exploratory data analysis and preprocessing. Having the right representa‐
tion of the data is often crucial for supervised or unsupervised learning to succeed,
and preprocessing and decomposition methods play an important part in data prepa‐
ration.
Decomposition, manifold learning and clustering are essential tools to further your
understanding of your data, and can be the only way to make sense of your data in
the absence of supervision information. Even in the supervised setting, exploratory
tools are important for a better understanding of the properties of the data.
Often it is hard to quantify the usefulness of an unsupervised algorithm, though this
shouldnt deter you from using them to create insights from your data.
With these methods under your belt, you are now equipped with all the essential
learning algorithms that machine learning practitioners use every day.
We encourage you to