Machine Learning Introduction PDF

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 and is also known as predictive analytics or statis...

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 and 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 a complex website like Facebook, Amazon, or Netflix, it is very likely that every part of the site 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 problems such as understanding stars, finding distant planets, discovering new particles, analyzing DNA sequences, and providing personalized cancer treatments. Your application doesn’t need to be as large-scale or world-changing as these exam‐ ples in order to benefit from machine learning, though. In this chapter, we will explain why machine learning has become so popular and discuss what kinds of problems can be solved using machine learning. Then, we will show you how to build your first machine learning model, introducing important concepts along the way. Why Machine Learning? In the early days of “intelligent” applications, many systems used handcoded rules of “if ” and “else” decisions to process data or adjust to user input. Think of a spam filter whose job is to move the appropriate incoming email messages to a spam folder. You could make up a blacklist of words that would result in an email being marked as 1 spam. This would be an example of using an expert-designed rule system to design an “intelligent” application. Manually crafting decision rules is feasible for some applica‐ tions, particularly those in which humans have a good understanding of the process to model. However, using handcoded rules to make decisions has two major disad‐ vantages: 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. Designing rules requires a deep understanding of how a decision should be made by a human expert. One example of where this handcoded approach will fail is in detecting faces in images. Today, every smartphone can detect a face in an image. However, face detec‐ tion was an unsolved problem until as recently 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 Machine Learning Can Solve The most successful kinds of machine learning algorithms are those that automate decision-making processes by generalizing from known examples. In this setting, which is known as supervised learning, the user provides the algorithm with pairs of inputs and desired outputs, and the algorithm finds a way to produce the desired out‐ put 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 with a large number of emails (which are the input), together with 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 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 algorithms in the form of the desired outputs for each example that they learn from. 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 2 | Chapter 1: Introduction 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 dataset 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 a tumor is benign based on a medical image Here the input is the image, and the output is whether the tumor is benign. To create a dataset 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. It might even be necessary to do additional diagnosis beyond the content of the image to determine whether the tumor in the image is cancerous or 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 dis‐ tributing 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 these examples is that although the inputs and out‐ puts look fairly straightforward, 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 diagnoses, on the other hand, requires not only expensive machinery but also rare and expensive expert knowledge, not to mention the ethical concerns and privacy issues. In the example of detecting credit card fraud, data col‐ lection 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 fraudu‐ lent and nonfraudulent activity is wait. Unsupervised algorithms are the other type of algorithm that we will cover in this book. In unsupervised learning, only the input data is known, and no known output data is given to the algorithm. While there are many successful applications of these methods, they are usually harder to understand and evaluate. Examples of unsupervised learning include: 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. Why Machine Learning? | 3 Segmenting customers into groups with similar preferences Given a set of customer 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.” Because you don’t know in advance 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 differ‐ ent from the norm. Each abnormal pattern might be very different, and you might not have any recorded instances of abnormal behavior. Because in this example you only observe traffic, and you don’t 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 or 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 have bought from your online shop. You might describe the image of a tumor by the grayscale values of each pixel, or maybe by using the size, shape, and color of the tumor. Each entity or row here is known as a sample (or data point) in machine learning, while the columns—the properties that describe these entities—are called features. Later in this book we will go into more detail on the topic of building a good repre‐ sentation 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 prediction 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 pre‐ dict their gender. This information is simply not contained in your data. If you add another feature that contains the patient’s first name, you will have much better luck, as it is often possible to tell the gender by a person’s first name. Knowing Your Task and Knowing Your Data Quite possibly the most important part in the machine learning process is under‐ standing the data you are working with and how it relates to the task you want to solve. 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 kind of data and what problem setting it works best for. While you are building a machine learning solution, you should answer, or at least keep in mind, the following questions: 4 | Chapter 1: Introduction  What question(s) am I trying to answer? Do I think the data collected can answer that question? What is the best way to phrase my question(s) as a machine learning problem? Have I collected enough data to represent the problem I want to solve? What features of the data did I extract, and will these enable the right predictions? How will I measure success in my application? How will the machine learning solution interact with other parts of my research or business product? In a larger context, the algorithms and methods in machine learning are only one part of a greater process to solve a particular problem, and it is good to keep the big picture in mind at all times. Many people spend a lot of time building complex machine learning solutions, only to find out they don’t solve the right problem. When going deep into the technical aspects of machine learning (as we will in this book), it is easy to lose sight of the ultimate goals. While we will not discuss the ques‐ tions listed here in detail, we still encourage you to keep in mind all the assumptions that you might be making, explicitly or implicitly, when you start building machine learning models. Why Python? Python has become the lingua franca for many data science applications. It combines the power 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 processing, image processing, and more. This vast toolbox provides data scientists with a large array of general- and special-purpose functionality. One of the main advantages of using Python is the abil‐ ity to interact directly with the code, using a terminal or other tools like the Jupyter Notebook, which we’ll look at shortly. Machine learning and data analysis are funda‐ mentally iterative processes, in which the data drives the analysis. It is essential for these processes to have tools that allow quick iteration and easy interaction. As a general-purpose programming language, Python also allows for the creation of complex graphical user interfaces (GUIs) and web services, and for integration into existing systems. scikit-learn scikit-learn is an open source project, meaning that it is free to use and distribute, and anyone can easily obtain the source code to see what is going on behind the Why Python? | 5 scenes. The scikit-learn project is constantly being developed and improved, and it 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. scikit-learn is a very popular tool, and the most prominent Python library for machine learning. It is widely used in industry and academia, and a wealth of tutori‐ als and code snippets are available online. scikit-learn works well with a number of other scientific Python tools, which we will discuss later in this chapter. While reading this, we recommend that you also browse the scikit-learn user guide and API documentation for additional details on and many more options for 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 plot‐ ting and interactive development, you should also install matplotlib, IPython, and the Jupyter Notebook. We recommend using one of the following prepackaged Python distributions, which will provide the necessary packages: Anaconda A Python distribution made for large-scale data processing, predictive analytics, and scientific computing. Anaconda comes with NumPy, SciPy, matplotlib, pandas, IPython, Jupyter Notebook, and scikit-learn. Available on Mac OS, Windows, and Linux, it is a very convenient solution and is the one we suggest for people without an existing installation of the scientific Python packages. Ana‐ conda now also includes the commercial Intel MKL library for free. Using MKL (which is done automatically when Anaconda is installed) can give significant speed improvements for many algorithms in scikit-learn. Enthought Canopy Another Python distribution for scientific computing. This comes with NumPy, SciPy, matplotlib, pandas, 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 academic license and get free access to the paid subscription ver‐ sion of Enthought Canopy. Enthought Canopy is available for Python 2.7.x, and works on Mac OS, Windows, and Linux. Python(x,y) A free Python distribution for scientific computing, specifically for Windows. Python(x,y) comes with NumPy, SciPy, matplotlib, pandas, IPython, and scikit-learn. 6 | Chapter 1: Introduction If you already have a Python installation set up, you can use pip to install all of these packages: $ pip install numpy scipy matplotlib ipython scikit-learn pandas 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 NumPy and SciPy, we will be using pandas and matplotlib. We will also introduce the Jupyter Notebook, which is a 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.1 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 the Jupyter Notebook supports many programming languages, we only need the Python support. The Jupyter Notebook makes it easy to incorporate code, text, and images, and all of this book was in fact written as a Jupyter Notebook. All of the code examples we include can be downloaded from GitHub. 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 pseudorandom number generators. In scikit-learn, the NumPy array is the fundamental data structure. scikit-learn takes in data in the form of NumPy arrays. Any data you’re using will have to be con‐ verted to a NumPy array. The core functionality of NumPy is the ndarray class, a multidimensional (n-dimensional) array. All elements of the array must be of the same type. A NumPy array looks like this: In: import numpy as np x = np.array([[1, 2, 3], [4, 5, 6]]) print("x:\n{}".format(x)) 1 If you are unfamiliar with NumPy or matplotlib, we recommend reading the first chapter of the SciPy Lec‐ ture Notes. Essential Libraries and Tools | 7 Out: x: [[1 2 3] [4 5 6]] We will be using NumPy a lot in this book, and we will refer to objects of the NumPy ndarray class as “NumPy arrays” or just “arrays.” SciPy SciPy is 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: this provides sparse matrices, which are another representation that is used for data in scikit- learn. Sparse matrices are used whenever we want to store a 2D array that contains mostly zeros: In: 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{}".format(eye)) Out: NumPy array: [[ 1. 0. 0. 0.] [ 0. 1. 0. 0.] [ 0. 0. 1. 0.] [ 0. 0. 0. 1.]] In: # Convert the NumPy array to a SciPy sparse matrix in CSR format # Only the nonzero entries are stored sparse_matrix = sparse.csr_matrix(eye) print("\nSciPy sparse CSR matrix:\n{}".format(sparse_matrix)) Out: SciPy sparse CSR matrix: (0, 0) 1.0 (1, 1) 1.0 (2, 2) 1.0 (3, 3) 1.0 8 | Chapter 1: Introduction Usually it is not possible to create dense representations of sparse data (as they would not fit into memory), so we need to create sparse representations directly. Here is a way to create the same sparse matrix as before, using the COO format: In: data = np.ones(4) row_indices = np.arange(4) col_indices = np.arange(4) eye_coo = sparse.coo_matrix((data, (row_indices, col_indices))) print("COO representation:\n{}".format(eye_coo)) Out: COO representation: (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 functions for making publication-quality visualizations such as line charts, histograms, scatter plots, and so on. Visualizing your data and different aspects of your analysis can give you important insights, and we will be using matplotlib for all our visualizations. When working inside the Jupyter Notebook, you can show figures directly in the browser by using the %matplotlib notebook and %matplotlib inline commands. We recommend using %matplotlib notebook, which provides an interactive envi‐ ronment (though we are using %matplotlib inline to produce this book). For example, this code produces the plot in Figure 1-1: In: %matplotlib inline import matplotlib.pyplot as plt # Generate a sequence of numbers from -10 to 10 with 100 steps in between x = np.linspace(-10, 10, 100) # Create a second array using sine y = np.sin(x) # The plot function makes a line chart of one array against another plt.plot(x, y, marker="x") Essential Libraries and Tools | 9 Figure 1-1. Simple line plot of the sine function using matplotlib pandas pandas is a Python library for data wrangling and analysis. It is built around a data structure called the DataFrame that is modeled after the R DataFrame. Simply put, a 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. In contrast to NumPy, which requires that all entries in an array be of the same type, pandas allows each column to have a separate type (for example, integers, dates, floating-point numbers, and strings). Another valuable tool provided by pandas is its ability to ingest from a great variety of file formats and data‐ bases, like SQL, Excel files, and comma-separated values (CSV) files. Going into detail about the functionality of pandas is out of the scope of this book. However, Python for Data Analysis by Wes McKinney (O’Reilly, 2012) provides a great guide. Here is a small example of creating a DataFrame using a dictionary: In: 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) # IPython.display allows "pretty printing" of dataframes # in the Jupyter notebook display(data_pandas) 10 | Chapter 1: Introduction This produces the following output: Age Location Name 0 24 New York John 1 13 Paris Anna 2 53 Berlin Peter 3 33 London Linda There are several possible ways to query this table. For example: In: # Select all rows that have an age column greater than 30 display(data_pandas[data_pandas.Age > 30]) This produces the following result: Age Location Name 2 53 Berlin Peter 3 33 London Linda mglearn This book comes with accompanying code, which you can find on GitHub. The accompanying code includes not only all the examples shown in this book, but also the mglearn library. This is a library of utility functions we wrote for this book, so that we don’t clutter up our code listings with details of plotting and data loading. If you’re interested, you can look up all the functions in the repository, but the details of the mglearn module are not really important to the material in this book. If you see a call to mglearn in the code, it is usually a way to make a pretty picture quickly, or to get our hands on some interesting data. Throughout the book we make ample use of NumPy, matplotlib and pandas. All the code will assume the following imports: import numpy as np import matplotlib.pyplot as plt import pandas as pd import mglearn We also assume that you will run the code in a Jupyter Notebook with the %matplotlib notebook or %matplotlib inline magic enabled to show plots. If you are not using the notebook or these magic commands, you will have to call plt.show to actually show any of the figures. Essential Libraries and Tools | 11 Python 2 Versus Python 3 There are two major versions of Python that are widely used at the moment: Python 2 (more precisely, 2.7) and Python 3 (with the latest release being 3.5 at the time of writing). This sometimes leads to some confusion. Python 2 is no longer actively developed, but because Python 3 contains major changes, Python 2 code usually does not run on Python 3. If you are new to Python, or are starting a new project from scratch, we highly recommend using the latest version of Python 3 without changes. If you have a large codebase that you rely on that is written for Python 2, you are excused from upgrading for now. However, you should try to migrate to Python 3 as soon as possible. When writing any new code, it is for the most part quite easy to write code that runs under Python 2 and Python 3. 2 If you don’t have to interface with legacy software, you should definitely use Python 3. All the code in this book is writ‐ ten in a way that works for both versions. However, the exact output might differ slightly under Python 2. Versions Used in this Book We are using the following versions of the previously mentioned libraries in this book: In: import sys print("Python version: {}".format(sys.version)) import pandas as pd print("pandas version: {}".format(pd.__version__)) import matplotlib print("matplotlib version: {}".format(matplotlib.__version__)) import numpy as np print("NumPy version: {}".format(np.__version__)) import scipy as sp print("SciPy version: {}".format(sp.__version__)) import IPython print("IPython version: {}".format(IPython.__version__)) import sklearn print("scikit-learn version: {}".format(sklearn.__version__)) 2 The six package can be very handy for that. 12 | Chapter 1: Introduction Out: Python version: 3.5.2 |Anaconda 4.1.1 (64-bit)| (default, Jul 2 2016, 17:53:06) [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] pandas version: 0.18.1 matplotlib version: 1.5.1 NumPy version: 1.11.1 SciPy version: 0.17.1 IPython version: 5.1.0 scikit-learn version: 0.18 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, let’s dive into our first application of machine learning. This book assumes that you have version 0.18 or later of scikit- learn. The model_selection module was added in 0.18, and if you use an earlier version of scikit-learn, you will need to adjust the imports from this module. 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 terms. Let’s assume that a hobby botanist is interested in distinguishing the species of some iris flowers that she has found. She has collected some measurements associated with each iris: the length and width of the petals and the length and width of the sepals, all measured in centimeters (see Figure 1-2). 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. A First Application: Classifying Iris Species | 13 Figure 1-2. Parts of the iris flower Because 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 classification problem. The possi‐ ble outputs (different species of irises) are called classes. Every iris in the dataset belongs to one of three classes, so 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. Meet the Data The data we will use for this example is the Iris dataset, a classical dataset in machine learning and statistics. It is included in scikit-learn in the datasets module. We can load it by calling the load_iris function: In: from sklearn.datasets import load_iris iris_dataset = 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: 14 | Chapter 1: Introduction In: print("Keys of iris_dataset: \n{}".format(iris_dataset.keys())) Out: Keys of iris_dataset: dict_keys(['target_names', 'feature_names', 'DESCR', 'data', 'target']) The value of 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): In: print(iris_dataset['DESCR'][:193] + "\n...") Out: 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 of the key target_names is an array of strings, containing the species of flower that we want to predict: In: print("Target names: {}".format(iris_dataset['target_names'])) Out: Target names: ['setosa' 'versicolor' 'virginica'] The value of feature_names is a list of strings, giving the description of each feature: In: print("Feature names: \n{}".format(iris_dataset['feature_names'])) Out: Feature names: ['sepal length (cm)', 'sepal width (cm)', 'petal length (cm)', 'petal width (cm)'] The data itself is contained in the target and data fields. data contains the numeric measurements of sepal length, sepal width, petal length, and petal width in a NumPy array: A First Application: Classifying Iris Species | 15 In: print("Type of data: {}".format(type(iris_dataset['data']))) Out: Type of data: The rows in the data array correspond to flowers, while the columns represent the four measurements that were taken for each flower: In: print("Shape of data: {}".format(iris_dataset['data'].shape)) Out: Shape of data: (150, 4) We see that the array 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 multiplied by 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: In: print("First five columns of data:\n{}".format(iris_dataset['data'][:5])) Out: First five columns of data: [[ 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]] From this data, we can see that all of the first five flowers have a petal width of 0.2 cm and that the first flower has the longest sepal, at 5.1 cm. The target array contains the species of each of the flowers that were measured, also as a NumPy array: In: print("Type of target: {}".format(type(iris_dataset['target']))) Out: Type of target: target is a one-dimensional array, with one entry per flower: 16 | Chapter 1: Introduction In: print("Shape of target: {}".format(iris_dataset['target'].shape)) Out: Shape of target: (150,) The species are encoded as integers from 0 to 2: In: print("Target:\n{}".format(iris_dataset['target'])) Out: Target: [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 meanings 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. But before we can apply our model to new measurements, we need to know whether it actually works—that is, whether we should trust its predictions. Unfortunately, we cannot use the data we used 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). To assess the model’s performance, we show it new data (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. One 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 assess how well the model works; this is called the 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. 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, is declared as the test set. Deciding A First Application: Classifying Iris Species | 17 how much data you want to put into the training and the test set respectively is some‐ what 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 lowercase y. This is inspired by the standard formulation f(x)=y in mathematics, where x is the input to a function and y is the output. Following more conventions from mathematics, we use a capital X because the data is a two-dimensional array (a matrix) and a lowercase y because the target is a one-dimensional array (a vector). Let’s call train_test_split on our data and assign the outputs using this nomencla‐ ture: In: from sklearn.model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split( iris_dataset['data'], iris_dataset['target'], random_state=0) Before making the split, the train_test_split function shuffles the dataset using a pseudorandom number generator. If we just took the last 25% of the data as a test set, all the data points would have the label 2, as the data points are sorted by the label (see the output for iris['target'] shown earlier). Using a test set containing only one of the three classes would not tell us much about how well our model generalizes, 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 pseudorandom 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 is 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%: In: print("X_train shape: {}".format(X_train.shape)) print("y_train shape: {}".format(y_train.shape)) Out: X_train shape: (112, 4) y_train shape: (112,) 18 | Chapter 1: Introduction In: print("X_test shape: {}".format(X_test.shape)) print("y_test shape: {}".format(y_test.shape)) Out: X_test shape: (38, 4) y_test shape: (38,) First Things First: 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. 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 and another along the y-axis, and draws a dot for each data point. Unfortunately, computer screens have only two dimensions, which allows us to plot only 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 possible pairs of features. If you have a small number of features, such as the four we have here, this is quite reasonable. You should keep in mind, however, that a pair plot does not show the interaction of all of features at once, so some interesting aspects of the data may not be revealed when visualizing it this way. Figure 1-3 is a pair plot of the features in the training set. The data points are colored according to the species the iris belongs to. To create the plot, we first convert the NumPy array into a pandas DataFrame. pandas has a function to create pair plots called scatter_matrix. The diagonal of this matrix is filled with histograms of each feature: In: # create dataframe from data in X_train # label the columns using the strings in iris_dataset.feature_names iris_dataframe = pd.DataFrame(X_train, columns=iris_dataset.feature_names) # create a scatter matrix from the dataframe, color by y_train grr = pd.scatter_matrix(iris_dataframe, c=y_train, figsize=(15, 15), marker='o', hist_kwds={'bins': 20}, s=60, alpha=.8, cmap=mglearn.cm3) A First Application: Classifying Iris Species | 19 Figure 1-3. Pair plot of the Iris dataset, colored by class label 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 First 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. 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 assigns the label of this training point to the new data point. 20 | Chapter 1: Introduction The k in k-nearest neighbors signifies that instead of using only the closest neighbor to the new data point, we can consider any fixed number k of neighbors in the train‐ ing (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 detail about this in Chapter 2; for now, we’ll use only a single neighbor. All machine learning models in scikit-learn are implemented in their own classes, 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 most important parameter of KNeighbor sClassifier is the number of neighbors, which we will set to 1: In: from sklearn.neighbors import KNeighborsClassifier knn = KNeighborsClassifier(n_neighbors=1) The knn object encapsulates the algorithm that will be used 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 that 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: In: knn.fit(X_train, y_train) Out: KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski', metric_params=None, n_jobs=1, n_neighbors=1, p=2, weights='uniform') The fit method returns the knn object itself (and modifies it in place), so we get a string representation of our classifier. The representation shows us which parameters were used in creating the model. Nearly all of them are the default values, but you can also find n_neighbors=1, which is the parameter that we passed. Most models in scikit-learn have many parameters, but the majority of them are either speed opti‐ mizations or for very special use cases. You don’t have to worry about the other parameters shown in this representation. Printing a scikit-learn model can yield very long strings, but don’t be intimidated by these. We will cover all the important parameters in Chapter 2. In the remainder of this book, we will not show the output of fit because it doesn’t contain any new information. A First Application: Classifying Iris Species | 21 Making Predictions We can now make predictions using this model on new data for which we might not know the correct labels. Imagine we found an iris in the wild with a sepal length of 5 cm, a sepal width of 2.9 cm, a petal length of 1 cm, and a petal width of 0.2 cm. What species of iris would this be? We can put this data into a NumPy array, again by calculating the shape—that is, the number of samples (1) multiplied by the number of features (4): In: X_new = np.array([[5, 2.9, 1, 0.2]]) print("X_new.shape: {}".format(X_new.shape)) Out: X_new.shape: (1, 4) Note that we made the measurements of this single flower into a row in a two- dimensional NumPy array, as scikit-learn always expects two-dimensional arrays for the data. To make a prediction, we call the predict method of the knn object: In: prediction = knn.predict(X_new) print("Prediction: {}".format(prediction)) print("Predicted target name: {}".format( iris_dataset['target_names'][prediction])) Out: Prediction: Predicted target name: ['setosa'] 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 don’t know the cor‐ rect species of this sample, which is 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 is for each iris in the test set. Therefore, we can make a prediction for each 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: 22 | Chapter 1: Introduction In: y_pred = knn.predict(X_test) print("Test set predictions:\n {}".format(y_pred)) Out: Test set predictions: [2 1 0 2 0 2 0 1 1 1 2 1 1 1 1 0 1 1 0 0 2 1 0 0 2 0 0 1 1 0 2 1 0 2 2 1 0 2] In: print("Test set score: {:.2f}".format(np.mean(y_pred == y_test))) Out: Test set score: 0.97 We can also use the score method of the knn object, which will compute the test set accuracy for us: In: print("Test set score: {:.2f}".format(knn.score(X_test, y_test))) Out: Test set score: 0.97 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 model 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 and Outlook Let’s summarize what we learned in this chapter. We started with a brief introduction to machine learning and its applications, then discussed the distinction between supervised and unsupervised learning and gave an overview of the tools we’ll be using in this book. Then, we formulated the task of predicting which species of iris a particular flower belongs to by using physical measurements 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 classification problem. The possible species are called classes in the classification problem, and the species of a single iris is called its label. The Iris 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, Summary and Outlook | 23 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 contains one class label, an integer ranging 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, previously 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. This is implemented in the KNeighborsClassifier class, which contains the algorithm that builds the model as well as the algorithm that makes 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, which 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. Here is a summary of the code needed for the whole training and evaluation procedure: In: X_train, X_test, y_train, y_test = train_test_split( iris_dataset['data'], iris_dataset['target'], random_state=0) knn = KNeighborsClassifier(n_neighbors=1) knn.fit(X_train, y_train) print("Test set score: {:.2f}".format(knn.score(X_test, y_test))) Out: Test set score: 0.97 This snippet contains the core code for applying any machine learning algorithm using scikit-learn. The fit, predict, and score methods are the common inter‐ face 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 supervised models in scikit-learn and how to apply them successfully. 24 | Chapter 1: Introduction  CHAPTER 2 Supervised Learning As we mentioned earlier, supervised machine learning is one of the most commonly used and successful types of machine learning. In this chapter, we will describe super‐ vised learning in more detail and explain several popular supervised learning algo‐ rithms. We already saw an application of supervised machine learning in Chapter 1: classifying iris flowers into several species using physical measurements of the flowers. 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 for new, never-before-seen data. Super‐ vised learning often requires human effort to build the training set, but afterward automates and often speeds up an otherwise laborious or infeasible task. Classification and Regression There are two major types of supervised machine learning problems, called classifica‐ tion 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 we used the example of classifying irises into one of three possible species. Classification is sometimes separated into binary classification, which is the special case of distinguishing between exactly two classes, and multiclass classification, which is classification between more than two classes. You can think of binary classification as trying to answer a yes/no question. Classifying emails as either spam or not spam is an example of a binary classification problem. In this binary classification task, the yes/no question being asked would be “Is this email spam?” 25  In binary classification we often speak of one class being the posi‐ tive class and the other class being the negative class. Here, positive doesn’t represent having benefit or value, but rather what the object of the study is. So, when looking for spam, “positive” could mean the spam class. Which of the two classes is called positive is often a subjective matter, and specific to the domain. The iris example, on the other hand, is an example of a multiclass classification prob‐ lem. Another example 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 floating-point number in programming terms (or real number in mathematical terms). Predicting a person’s annual income from their education, their age, and where they live is an 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 continuity in the output. If there is continuity between possible outcomes, then the problem is a regression problem. Think about predicting annual income. There is a clear continuity in the output. Whether a person makes $40,000 or $40,001 a year does not make a tangible difference, even though these are different amounts of money; if our algorithm predicts $39,999 or $40,001 when it should have predicted $40,000, we don’t mind that much. By contrast, 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.1 Generalization, Overfitting, and Underfitting In supervised learning, we want to build 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 accurately as possible. 1 We ask linguists to excuse the simplified presentation of languages as distinct and fixed entities. 26 | Chapter 2: Supervised Learning Usually we build a model in such a way that it can make accurate predictions on the training set. If the training and test sets 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 ourselves to build very complex models, we can always be as accurate as we like on the training set. Let’s take a look at a made-up example to illustrate this point. Say a novice data scien‐ tist wants to predict whether a customer will buy a boat, given records of previous boat buyers and customers who we know are not interested in buying a boat.2 The goal is to send out promotional emails to people who are likely to actually make a purchase, but not bother those customers who won’t be interested. Suppose we have the customer records shown in Table 2-1. Table 2-1. Example data about customers Age Number of Owns house Number of children Marital status Owns a dog Bought a boat cars owned 66 1 yes 2 widowed no yes 52 2 yes 3 married no yes 22 0 no 0 married yes no 25 1 no 1 single no no 44 0 no 2 divorced yes no 39 1 yes 2 married yes no 26 1 no 2 single no no 40 3 yes 1 married yes no 53 2 yes 2 divorced no yes 64 2 yes 3 divorced no no 58 2 yes 2 married yes yes 33 1 no 1 single no no After looking at the data for a while, our novice data scientist comes up with the fol‐ lowing rule: “If the customer is older than 45, and has less than 3 children or is not divorced, then they want to buy a boat.” When asked how well this rule of his does, our data scientist answers, “It’s 100 percent accurate!” And indeed, on the data that is in the table, the rule is perfectly accurate. There are many possible rules we could come up with that would explain perfectly if someone in this dataset wants to buy a boat. No age appears twice in the data, so we could say people who are 66, 52, 53, or 2 In the real world, this is actually a tricky problem. While we know that the other customers haven’t bought a boat from us yet, they might have bought one from someone else, or they may still be saving and plan to buy one in the future. Generalization, Overfitting, and Underfitting | 27 58 years old want to buy a boat, while all others don’t. While we can make up many rules that work well on this data, remember that we are not interested in making pre‐ dictions for this dataset; we already know the answers for these customers. We want to know if new customers are likely to buy a boat. We therefore want to find a rule that will work well for new customers, and achieving 100 percent accuracy on the training set does not help us there. We might not expect that the rule our data scientist came up with will work very well on new customers. It seems too complex, and it is sup‐ ported by very little data. For example, the “or is not divorced” part of the rule hinges on a single customer. The only measure of whether an algorithm will perform well on new data is the eval‐ uation on the test set. However, intuitively3 we expect simple models to generalize better to new data. If the rule was “People older than 50 want to buy a boat,” and this would explain the behavior of all the customers, we would trust it more than the rule involving children and marital status in addition to age. Therefore, we always want to find the simplest model. Building a model that is too complex for the amount of information we have, as our novice data scientist did, is called overfitting. Overfitting occurs when you fit a model too closely to the particularities of the training set and obtain a model that works well on the training set but is not able to generalize to new data. On the other hand, if your model is too simple—say, “Everybody who owns a house buys a boat”—then you might not be able to capture all the aspects of and vari‐ ability in the data, and your model will do badly even on the training set. Choosing too simple a model is called underfitting. 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 each individual data point in our training set, and the model will not gener‐ alize well to new data. There is a sweet spot in between that will yield the best generalization performance. This is the model we want to find. The trade-off between overfitting and underfitting is illustrated in Figure 2-1. 3 And also provably, with the right math. 28 | Chapter 2: Supervised Learning Figure 2-1. Trade-off of model complexity against training and test accuracy Relation of Model Complexity to Dataset Size It’s important to note that model complexity is intimately tied to the variation of inputs contained in your training dataset: the larger variety of data points your data‐ set contains, the more complex a model you can use without overfitting. Usually, col‐ lecting more data points will yield more variety, so larger datasets allow building more complex models. However, simply duplicating the same data points or collect‐ ing very similar data will not help. Going back to the boat selling example, if we saw 10,000 more rows of customer data, and all of them complied with the rule “If the customer is older than 45, and has less than 3 children or is not divorced, then they want to buy a boat,” we would be much more likely to believe this to be a good rule than when it was developed using only the 12 rows in Table 2-1. Having more data and building appropriately more complex models can often work wonders for supervised learning tasks. In this book, we will focus on working with datasets of fixed sizes. In the real world, you often have the ability to decide how much data to collect, which might be more beneficial than tweaking and tuning your model. Never underestimate the power of more data. Supervised Machine Learning Algorithms We will now review 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, and provide an over‐ Supervised Machine Learning Algorithms | 29 view of how each algorithm builds a model. We will examine the strengths and weak‐ nesses of each algorithm, and what kind of data they can best be applied to. We will also explain the meaning of the most important parameters and options.4 Many algo‐ rithms have a classification and a regression variant, and we will describe both. It is not necessary to read through the descriptions 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. Some Sample Datasets We will use several datasets to illustrate the different algorithms. Some of the datasets will be small and synthetic (meaning made-up), designed to highlight particular aspects of the algorithms. Other datasets will be large, real-world examples. An example of a synthetic two-class classification dataset is the forge dataset, which has two features. The following code creates a scatter plot (Figure 2-2) visualizing all of the data points in this dataset. The plot has the first feature on the x-axis and the second feature on the y-axis. As is always the case in scatter plots, each data point is represented as one dot. The color and shape of the dot indicates its class: In: # generate dataset X, y = mglearn.datasets.make_forge() # plot dataset mglearn.discrete_scatter(X[:, 0], X[:, 1], y) plt.legend(["Class 0", "Class 1"], loc=4) plt.xlabel("First feature") plt.ylabel("Second feature") print("X.shape: {}".format(X.shape)) Out: X.shape: (26, 2) 4 Discussing all of them is beyond the scope of the book, and we refer you to the scikit-learn documentation for more details. 30 | Chapter 2: Supervised Learning Figure 2-2. Scatter plot of the forge dataset 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. The wave dataset has a single input feature and a continuous target variable (or response) that we want to model. The plot created here (Figure 2-3) shows the single feature on the x-axis and the regression target (the output) on the y-axis: In: X, y = mglearn.datasets.make_wave(n_samples=40) plt.plot(X, y, 'o') plt.ylim(-3, 3) plt.xlabel("Feature") plt.ylabel("Target") Supervised Machine Learning Algorithms | 31 Figure 2-3. Plot of the wave dataset, with the x-axis showing the feature and the y-axis showing the regression target We are using these very simple, low-dimensional datasets because we can easily visu‐ alize them—a printed page has two dimensions, so data with more than two features 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 datasets with two real-world datasets that are included in scikit-learn. One is the Wisconsin Breast Cancer dataset (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 meas‐ urements of the tissue. The data can be loaded using the load_breast_cancer function from scikit-learn: In: from sklearn.datasets import load_breast_cancer cancer = load_breast_cancer() print("cancer.keys(): \n{}".format(cancer.keys())) 32 | Chapter 2: Supervised Learning Out: cancer.keys(): dict_keys(['feature_names', 'data', 'DESCR', 'target', 'target_names']) 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']). The dataset consists of 569 data points, with 30 features each: In: print("Shape of cancer data: {}".format(cancer.data.shape)) Out: Shape of cancer data: (569, 30) Of these 569 data points, 212 are labeled as malignant and 357 as benign: In: print("Sample counts per class:\n{}".format( {n: v for n, v in zip(cancer.target_names, np.bincount(cancer.target))})) Out: Sample counts per class: {'benign': 357, 'malignant': 212} To get a description of the semantic meaning of each feature, we can have a look at the feature_names attribute: In: print("Feature names:\n{}".format(cancer.feature_names)) Out: Feature names: ['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'] Supervised Machine Learning Algorithms | 33 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 such as crime rate, prox‐ imity to the Charles River, highway accessibility, and so on. The dataset contains 506 data points, described by 13 features: In: from sklearn.datasets import load_boston boston = load_boston() print("Data shape: {}".format(boston.data.shape)) Out: 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 features, but also the product of crime rate and highway accessibility. Including derived feature like these is called feature engi‐ neering, which we will discuss in more detail in Chapter 4. This derived dataset can be loaded using the load_extended_boston function: In: X, y = mglearn.datasets.load_extended_boston() print("X.shape: {}".format(X.shape)) Out: X.shape: (506, 104) The resulting 104 features are the 13 original features together with the 91 possible combinations of two features within those 13.5 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 neighbors (k-NN) algorithm that we saw in the pre‐ vious chapter. 5 This is called the binomial coefficient, which is the number of combinations of k elements that can be selected n from a set of n elements. Often this is written as and spoken as “n choose k”—in this case, “13 choose 2.” k 34 | Chapter 2: Supervised Learning k-Nearest Neighbors The k-NN algorithm is arguably the simplest machine learning algorithm. Building the model consists only 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—its “nearest neighbors.” k-Neighbors classification In its simplest version, the k-NN algorithm only considers exactly one nearest neigh‐ bor, 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 2-4 illustrates this for the case of classification on the forge dataset: In: mglearn.plots.plot_knn_classification(n_neighbors=1) Figure 2-4. Predictions made by the one-nearest-neighbor model on the forge dataset Here, we added three new data points, shown as stars. For each of them, we marked the closest point in the training set. The prediction of the one-nearest-neighbor algo‐ rithm is the label of that point (shown by the color of the cross). Supervised Machine Learning Algorithms | 35 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-nearest neighbors algorithm comes from. When considering more than one neighbor, we use voting to assign a label. This means that for each test point, we count how many neighbors belong to class 0 and how many neighbors belong to class 1. We then assign the class that is more frequent: in other words, the majority class among the k-nearest neighbors. The following example (Figure 2-5) uses the three closest neighbors: In: mglearn.plots.plot_knn_classification(n_neighbors=3) Figure 2-5. Predictions made by the three-nearest-neighbors model on the forge dataset Again, the prediction is shown as the color of the cross. You can see that the predic‐ tion for the new data point at the top left is not the same as the prediction when we used only one neighbor. While this illustration is for a binary classification problem, this method can be applied to datasets 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 let’s 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 general‐ ization performance, as discussed in Chapter 1: 36 | Chapter 2: Supervised Learning In: 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 3: In: 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: In: clf.fit(X_train, y_train) To make predictions on the test data, we call the predict method. For each data point in the test set, this computes its nearest neighbors in the training set and finds the most common class among these: In: print("Test set predictions: {}".format(clf.predict(X_test))) Out: Test set predictions: [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: In: print("Test set accuracy: {:.2f}".format(clf.score(X_test, y_test))) Out: Test set accuracy: 0.86 We see that our model is about 86% accurate, meaning the model predicted the class correctly for 86% of the samples in the test dataset. Analyzing KNeighborsClassifier For two-dimensional datasets, we can also illustrate the prediction for all possible test points in the xy-plane. We color the plane according to the class that would be assigned to a point in this region. This lets us view the decision boundary, which is the divide between where the algorithm assigns class 0 versus where it assigns class 1. Supervised Machine Learning Algorithms | 37 The following code produces the visualizations of the decision boundaries for one, three, and nine neighbors shown in Figure 2-6: In: fig, axes = plt.subplots(1, 3, figsize=(10, 3)) for n_neighbors, ax in zip([1, 3, 9], axes): # the fit method returns the object self, so we can instantiate # and fit in one line 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) mglearn.discrete_scatter(X[:, 0], X[:, 1], y, ax=ax) ax.set_title("{} neighbor(s)".format(n_neighbors)) ax.set_xlabel("feature 0") ax.set_ylabel("feature 1") axes.legend(loc=3) Figure 2-6. Decision boundaries created by the nearest neighbors model for different val‐ ues of n_neighbors As you can see on the left in the figure, using a single neighbor results in a decision boundary that follows the training data closely. Considering more and more neigh‐ bors leads to a smoother decision boundary. A smoother boundary corresponds to a simpler model. In other words, using few neighbors corresponds to high model com‐ plexity (as shown on the right side of Figure 2-1), and using many neighbors corre‐ sponds to low model complexity (as shown on the left side of Figure 2-1). If you consider the extreme case where the number of neighbors is the number of all data points in the training set, each test point would have exactly the same neighbors (all training points) and all predictions would be the same: the class that is most frequent in the training set. Let’s investigate whether we can confirm the connection between model complexity and generalization that we discussed earlier. 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 38 | Chapter 2: Supervised Learning we evaluate training and test set performance with different numbers of neighbors. The results are shown in Figure 2-7: In: 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.ylabel("Accuracy") plt.xlabel("n_neighbors") plt.legend() The plot shows the training and test set accuracy on the y-axis against the setting of n_neighbors on the x-axis. While real-world plots are rarely very smooth, we can still recognize some of the characteristics of overfitting and underfitting (note that because considering fewer neighbors corresponds to a more complex model, the plot is horizontally flipped relative to the illustration in Figure 2-1). Considering a single nearest neighbor, the prediction on the training set is perfect. But when more neigh‐ bors are considered, the model becomes simpler and the training accuracy drops. The test set accuracy for using a single neighbor is lower than when using more neigh‐ bors, indicating that using the single nearest neighbor leads to a model that is too complex. On the other hand, when considering 10 neighbors, the model is too simple and performance is even worse. The best performance is somewhere in the middle, using around 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. Supervised Machine Learning Algorithms | 39 Figure 2-7. Comparison of training and test accuracy as a function of n_neighbors k-neighbors regression There is also a regression variant of the k-nearest neighbors algorithm. Again, let’s start by using the single nearest neighbor, this time using the wave dataset. We’ve added three test data points as green stars on the x-axis. The prediction using a single neighbor is just the target value of the nearest neighbor. These are shown as blue stars in Figure 2-8: In: mglearn.plots.plot_knn_regression(n_neighbors=1) 40 | Chapter 2: Supervised Learning Figure 2-8. Predictions made by one-nearest-neighbor regression on the wave dataset Again, we can use more than the single closest neighbor for regression. When using multiple nearest neighbors, the prediction is the average, or mean, of the relevant neighbors (Figure 2-9): In: mglearn.plots.plot_knn_regression(n_neighbors=3) Supervised Machine Learning Algorithms | 41 Figure 2-9. Predictions made by three-nearest-neighbors regression on the wave dataset The k-nearest neighbors algorithm for regression is implemented in the KNeighbors Regressor class in scikit-learn. It’s used similarly to KNeighborsClassifier: In: 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 and 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) Now we can make predictions on the test set: In: print("Test set predictions:\n{}".format(reg.predict(X_test))) 42 | Chapter 2: Supervised Learning Out: Test set predictions: [-0.054 0.357 1.137 -1.894 -1.139 -1.631 0.357 0.912 -0.447 -1.139] We can also evaluate the model using the score method, which for regressors returns the R2 score. The R2 score, also known as the coefficient of determination, is a meas‐ ure of goodness of a prediction for a regression model, and yields a score between 0 and 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: In: print("Test set R^2: {:.2f}".format(reg.score(X_test, y_test))) Out: Test set R^2: 0.83 Here, the score is 0.83, which indicates a relatively good model fit. Analyzing KNeighborsRegressor For our one-dimensional dataset, we can see what the predictions look like for all possible feature values (Figure 2-10). To do this, we create a test dataset consisting of many points on the line: In: fig, axes = plt.subplots(1, 3, figsize=(15, 4)) # create 1,000 data points, evenly spaced between -3 and 3 line = np.linspace(-3, 3, 1000).reshape(-1, 1) for n_neighbors, ax in zip([1, 3, 9], axes): # make predictions using 1, 3, or 9 neighbors reg = KNeighborsRegressor(n_neighbors=n_neighbors) reg.fit(X_train, y_train) ax.plot(line, reg.predict(line)) ax.plot(X_train, y_train, '^', c=mglearn.cm2(0), markersize=8) ax.plot(X_test, y_test, 'v', c=mglearn.cm2(1), markersize=8) ax.set_title( "{} neighbor(s)\n train score: {:.2f} test score: {:.2f}".format( n_neighbors, reg.score(X_train, y_train), reg.score(X_test, y_test))) ax.set_xlabel("Feature") ax.set_ylabel("Target") axes.legend(["Model predictions", "Training data/target", "Test data/target"], loc="best") Supervised Machine Learning Algorithms | 43 Figure 2-10. Comparing predictions made by nearest neighbors regression for different values of n_neighbors As we can see from the plot, 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. Strengths, weaknesses, and parameters In principle, 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 three or five often works well, but you should certainly 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 k-NN is that the model is very easy to understand, and often gives reasonable performance without a lot of adjustments. Using this algorithm is a good baseline method to try before considering more advanced techniques. Building the nearest neighbors model is usually very fast, but when your training set is very large (either in number of features or in number of samples) prediction can be slow. When using the k-NN algorithm, it’s important to preprocess your data (see Chap‐ ter 3). This approach often does not perform well on datasets with many features (hundreds or more), and it does particularly badly with datasets where most features are 0 most of the time (so-called sparse datasets). So, while the nearest k-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. 44 | Chapter 2: Supervised Learning 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 make a prediction using a linear function of the input features, which we will explain shortly. Linear models for regression For regression, the general prediction formula for a linear model looks as follows: ŷ = w * x + w * x +... + w[p] * x[p] + b Here, x to x[p] denotes the features (in this example, the number of features is p) of a single data point, w and b are parameters of the model that are learned, and ŷ is the prediction the model makes. For a dataset with a single feature, this is: ŷ = w * x + b which you might remember from high school mathematics as the equation for a line. Here, w 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 and b on our one-dimensional wave dataset might lead to the following line (see Figure 2-11): In: mglearn.plots.plot_linear_regression_wave() Out: w: 0.393906 b: -0.031804 Supervised Machine Learning Algorithms | 45 Figure 2-11. Predictions of a linear model on the wave dataset We added a coordinate cross into the plot to make it easier to understand the line. Looking at w we see that the slope should be around 0.4, which we can confirm visually in the plot. The intercept is where the prediction line should cross the y-axis: this is slightly below zero, 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 using more features). If you compare the predictions made by the straight line with those made by the KNeighborsRegressor in Figure 2-10, 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 46 | Chapter 2: Supervised Learning 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 pow‐ erful. 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.6 There are many different linear models for regression. The difference between these models lies in how the model parameters w and b are learned from the training data, and how model complexity can be controlled. We will now take a look at the most popular linear models for regression. Linear regression (aka ordinary least squares) Linear regression, or ordinary least squares (OLS), is the simplest and most classic lin‐ ear method for regression. Linear regression finds the parameters w and b that mini‐ mize 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 2-11: In: 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 coefficients, are stored in the coef_ attribute, while the offset or intercept (b) is stored in the intercept_ attribute: In: print("lr.coef_: {}".format(lr.coef_)) print("lr.intercept_: {}".format(lr.intercept_)) Out: lr.coef_: [ 0.394] lr.intercept_: -0.031804343026759746 6 This is easy to see if you know some linear algebra. Supervised Machine Learning Algorithms | 47  You might notice the strange-looking trailing underscore at the end of coef_ and intercept_. scikit-learn always 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. 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. Let’s look at the training set and test set performance: In: print("Training set score: {:.2f}".format(lr.score(X_train, y_train))) print("Test set score: {:.2f}".format(lr.score(X_test, y_test))) Out: Training set score: 0.67 Test set score: 0.66 An R2 of around 0.66 is not very good, but we can see that the scores on the training and test sets are very close together. This means we are likely underfitting, not over‐ fitting. 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 datasets with a large number of features), linear models become more pow‐ erful, and there is a higher chance of overfitting. Let’s take a look at how LinearRe gression performs on a more complex dataset, like the Boston Housing dataset. Remember that this dataset has 506 samples and 105 derived features. First, we load the dataset and split it into a training and a test set. Then we build the linear regres‐ sion model as before: In: 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 scores, we find that we predict very accu‐ rately on the training set, but the R2 on the test set is much worse: In: print("Training set score: {:.2f}".format(lr.score(X_train, y_train))) print("Test set score: {:.2f}".format(lr.score(X_test, y_test))) 48 | Chapter 2: Supervised Learning Out: Training set score: 0.95 Test set score: 0.61 This discrepancy between performance on the training set and the test set is a clear sign of overfitting, and therefore we should try to find a model that allows us to con‐ trol 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 the same one used for ordinary least squares. In ridge regression, though, the coefficients (w) are chosen not only so that they predict well on the train‐ ing data, but also to fit an additional constraint. We also want the magnitude of coef‐ ficients to be as small as possible; in other words, all entries of w should be close to zero. Intuitively, this means each feature should have as little effect on the outcome as possible (which translates to having a small slope), while still predicting well. This constraint is an example of what is called regularization. Regularization means explic‐ itly restricting a model to avoid overfitting. The particular kind used by ridge regres‐ sion is known as L2 regularization.7 Ridge regression is implemented in linear_model.Ridge. Let’s see how well it does on the extended Boston Housing dataset: In: from sklearn.linear_model import Ridge ridge = Ridge().fit(X_train, y_train) print("Training set score: {:.2f}".format(ridge.score(X_train, y_train))) print("Test set score: {:.2f}".format(ridge.score(X_test, y_test))) Out: Training set score: 0.89 Test set score: 0.75 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 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 perfor‐ mance, we should choose the Ridge model over the LinearRegression model. 7 Mathematically, Ridge penalizes the L2 norm of the coefficients, or the Euclidean length of w. Supervised Machine Learning Algorithms | 49 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. In the previous example, we used the default param‐ eter alpha=1.0. There is no reason why this will give us the best trade-off, though. The optimum setting of alpha depends on the particular dataset we are using. Increasing alpha forces coefficients to move more toward zero, which decreases training set performance but might help generalization. For example: In: ridge10 = Ridge(alpha=10).fit(X_train, y_train) print("Training set score: {:.2f}".format(ridge10.score(X_train, y_train))) print("Test set score: {:.2f}".format(ridge10.score(X_test, y_test))) Out: Training set score: 0.79 Test set score: 0.64 Decreasing alpha allows the coefficients to be less restricted, meaning we move right in Figure 2-1. For very small values of alpha, coefficients are barely restricted at all, and we end up with a model that resembles LinearRegression: In: ridge01 = Ridge(alpha=0.1).fit(X_train, y_train) print("Training set score: {:.2f}".format(ridge01.score(X_train, y_train))) print("Test set score: {:.2f}".format(ridge01.score(X_test, y_test))) Out: Training set score: 0.93 Test set score: 0.77 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 2-1. We will discuss methods to properly select parameters in Chapter 5. 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 the entries of coef_ to have smaller magnitude for a high value of alpha than for a low value of alpha. This is confirmed in the plot in Figure 2-12: 50 | Chapter 2: Supervised Learning In: plt.plot(ridge.coef_, 's', label="Ridge alpha=1") plt.plot(ridge10.coef_, '^', label="Ridge alpha=10") plt.plot(ridge01.coef_, 'v', label="Ridge alpha=0.1") plt.plot(lr.coef_, 'o', label="LinearRegression") plt.xlabel("Coefficient index") plt.ylabel("Coefficient magnitude") plt.hlines(0, 0, len(lr.coef_)) plt.ylim(-25, 25) plt.legend() Figure 2-12. Comparing coefficient magnitudes for ridge regression with different values of alpha and linear regression 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 values of the corresponding values of the coefficients. The main takeaway here is that for alpha=10, the coefficients are mostly between around –3 and 3. The coefficients for the Ridge model with alpha=1 are somewhat larger. The dots corresponding to alpha=0.1 have larger magnitude still, and many of the dots corresponding to linear regression without any regularization (which would be alpha=0) are so large they are outside of the chart. Supervised Machine Learning Algorithms | 51 Another way to understand the influence of regularization is to fix a value of alpha but vary the amount of training data available. For Figure 2-13, we subsampled the Boston Housing dataset and evaluated LinearRegression and Ridge(alpha=1) on subsets of increasing size (plots that show model performance as a function of dataset size are called learning curves): In: mglearn.plots.plot_ridge_n_samples() Figure 2-13. Learning curves for ridge regression and linear regression on the Boston Housing dataset As one would expect, the training score is higher than the test score for all dataset sizes, for both ridge and linear regression. Because ridge is regularized, the training score of ridge is lower than the training score for linear regression across the board. However, the test score for ridge is better, particularly for small subsets of the data. For less than 400 data points, linear regression is not able to learn anything. As more and more data becomes available to the model, both models improve, and linear regression catches up with ridge in the end. The lesson here is that with enough train‐ ing data, regularization becomes less important, and given enough data, ridge and 52 | Chapter 2: Supervised Learning linear regression will have the same performance (the fact that this happens here when using the full dataset is just by chance). Another interesting aspect of Figure 2-13 is the decrease in training performance for linear regression. If more data is added, it becomes harder for a model to overfit, or memorize the data. Lasso An alternative to Ridge for regularizing linear regression is Lasso. As with ridge regression, using the lasso also restricts coefficients to be close to zero, but in a slightly different way, called L1 regularization.8 The consequence of L1 regularization is that when using the lasso, some coefficients are exactly zero. This means some fea‐ tures are entirely ignored by the model. This can be seen as a form of automatic fea‐ ture selection. Having some coefficients be exactly zero often makes a model easier to interpret, and can reveal the most important features of your model. Let’s apply the lasso to the extended Boston Housing dataset: In: from sklearn.linear_model import Lasso lasso = Lasso().fit(X_train, y_train) print("Training set score: {:.2f}".format(lasso.score(X_train, y_train))) print("Test set score: {:.2f}".format(lasso.score(X_test, y_test))) print("Number of features used: {}".format(np.sum(lasso.coef_ != 0))) Out: Training set score: 0.29 Test set score: 0.21 Number of features used: 4 As you can see, Lasso does quite badly, both on the training and the test set. This indicates that we are underfitting, and we find that it used only 4 of the 105 features. Similarly to Ridge, the Lasso also has a regularization parameter, alpha, that controls how strongly coefficients are pushed toward zero. In the previous example, we used the default of alpha=1.0. To reduce underfitting, let’s try decreasing alpha. When we do this, we also need to increase the default setting of max_iter (the maximum num‐ ber of iterations to run): 8 The lasso penalizes the L1 norm of the coefficient vector—or in other words, the sum of the absolute values of the coefficients. Supervised Machine Learning Algori