This post looks into the problem of classification, a situation in which a response is a categorical variable. We will build upon the techniques that we previously discussed in the context of regression and show how they can be transferred to classification problems. This post introduces a number of classification techniques, and it will try to convey their corresponding strengths and weaknesses by visually inspecting the decision boundaries for each model.

*This is part of a series of blog posts showing how to do common statistical learning techniques with Python. We provide only a small amount of background on the concepts and techniques we cover, so if you’d like a more thorough explanation check out Introduction to Statistical Learning or sign up for the free online course run by the book's authors here.*

In this post we will use scikit-learn, an easy-to-use, general-purpose toolbox for machine learning in Python. We will use it extensively in the coming posts in this series so it's worth spending some time to introduce it thoroughtly.

Scikit-learn is a library that provides a variety of both supervised and unsupervised machine learning techniques. Supervised machine learning refers to the problem of inferring a function from labeled training data, and it comprises both regression and classification. Unsupervised machine learning, on the other hand, refers to the problem of finding interesting patterns or structure in the data; it comprises techniques such as clustering and dimensionality reduction. In addition to statistical learning techniques, scikit-learn provides utilities for common tasks such as model selection, feature extraction, and feature selection.

Scikit-learn provides an object-oriented interface centered around the concept of an `Estimator`

. According to the scikit-learn tutorial "*An estimator is any object that learns from data; it may be a classification, regression or clustering algorithm or a transformer that extracts/filters useful features from raw data.*" The API of an estimator looks roughly as follows:

In [2]:

```
class Estimator(object):
def fit(self, X, y=None):
"""Fits estimator to data. """
# set state of ``self``
return self
def predict(self, X):
"""Predict response of ``X``. """
# compute predictions ``pred``
return pred
```

The `Estimator.fit`

method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array `X`

of shape `(n_samples, n_predictors)`

that holds the so-called feature matrix and a one-dimensional numpy array `y`

that holds the responses.
Some estimators allow the user to control the fitting behavior. For example, the `sklearn.linear_model.LinearRegression`

estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object:

In [3]:

```
from sklearn.linear_model import LinearRegression
est = LinearRegression(fit_intercept=False)
```

The docstring of the estimator shows you all available arguments - in IPython simply use `LinearRegression?`

to view the docstring.

During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (`'_'`

). For example, the coefficients of a `LinearRegression`

estimator are stored in the attribute `coef_`

:

In [4]:

```
import numpy as np
# random training data
X = np.random.rand(10, 2)
y = np.random.randint(2, size=10)
est.fit(X, y)
est.coef_ # access coefficients
```

Out[4]:

Estimators that can generate predictions provide a `Estimator.predict`

method.
In the case of regression, `Estimator.predict`

will return the predicted regression values; it will return the corresponding class labels in the case of classification. Classifiers that can predict the probability of class membership have a method `Estimator.predict_proba`

that returns a two-dimensional numpy array of shape `(n_samples, n_classes)`

where the classes are lexicographically ordered.

Finally, there is a special type of `Estimator`

called `Transformer`

which transforms the input data -- e.g. selects a subset of the features or extracts new features based on the original ones. In addition to a `fit`

method, a `Transformer`

object provides the following methods:

In [5]:

```
class Transformer(Estimator):
def transform(self, X):
"""Transforms the input data. """
# transform ``X`` to ``X_prime``
return X_prime
```

Usually, a `Transformer`

does not provide a `predict`

method, but in some cases it may. One transformer that we will use in this posting is `sklearn.preprocessing.StandardScaler`

. This transformer centers each predictor in `X`

to have zero mean and unit variance:

In [6]:

```
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler(copy=True) # always copy input data (don't modify in-place)
X_centered = scaler.fit(X).transform(X)
scaler.mean_ # mean that will be subtracted upon transform
```

Out[6]:

For more information on scikit-learn please consult the detailed user guide or walk through the excellent tutorial.

Although regression and classification appear to be very different they are in fact similar problems.

In regression our predictions for the response are real-valued numbers; on the other hand, in classification the response is a mutually exclusive class label such as "Is the email spam/ham?" or "Is the credit card transaction fraudulent?". If the number of classes is equal to two, then we call it a binary classification problem; if there are more than two classes, then we call it a multiclass classification problem. In the following we will assume binary classification because it's the more general case, and — we can always represent a multiclass problem as a sequence of binary classification problems.

We can also think of classification as a function estimation problem where the function that we want to estimate separates the two classes. This is illustrated in the example below where our goal is to predict whether or not a credit card transaction is fraudulent -- the dataset is provided by James et al., Introduction to Statistical Learning.

In [7]:

```
import pandas as pd
df = pd.read_csv('https://d1pqsl2386xqi9.cloudfront.net/notebooks/Default.csv', index_col=0)
# downsample negative cases -- there are many more negatives than positives
indices = np.where(df.default == 'No')[0]
rng = np.random.RandomState(13)
rng.shuffle(indices)
n_pos = (df.default == 'Yes').sum()
df = df.drop(df.index[indices[n_pos:]])
df.head()
```

Out[7]:

On the left you can see a scatter plot where fraudulent cases are marked as pluses and non-fraudulent cases as circles. A good separation seems to be a vertical line at around a *balance* of 1400 as indicated by the boxplots below.

In [8]:

```
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
%pylab inline
# setup figure
plt.figure(figsize=(10, 8))
# scatter plot of balance (x) and income (y)
ax1 = plt.subplot(221)
cm_bright = ListedColormap(['#FF0000', '#0000FF'])
ax1.scatter(df.balance, df.income, c=(df.default == 'Yes'), cmap=cm_bright)
ax1.set_xlim((df.balance.min(), df.balance.max()))
ax1.set_ylim((df.income.min(), df.income.max()))
ax1.set_xlabel('balance')
ax1.set_ylabel('income')
ax1.legend(loc='upper right')
# box plots for income
ax2 = plt.subplot(222)
ax2.boxplot([df.income[df.default == 'Yes'], df.income[df.default == 'No']])
ax2.set_ylim((df.income.min(), df.income.max()))
ax2.set_xticklabels(('Yes', 'No'))
ax2.set_ylabel('income')
# box plots for balance
ax3 = plt.subplot(223)
ax3.boxplot([df.balance[df.default == 'Yes'], df.balance[df.default == 'No']], vert=0)
ax3.set_xlim((df.balance.min(), df.balance.max()))
ax3.set_yticklabels(('Yes', 'No'))
ax3.set_xlabel('balance')
plt.tight_layout()
```

A simple approach to binary classification is to simply encode *default* as a numeric variable with `'Yes' == 1`

and `'No' == -1`

; fit an Ordinary Least Squares regression model like we introduced in the last post; and use this model to predict the response as`'Yes'`

if the regressed value is higher than 0.0 and `'No'`

otherwise. The points for which the regression model predicts 0.0 lie on the so-called *decision surface* -- since we are using a linear regression model, the decision surface is linear as well.

The example below illustrates this. Note that we use the `sklearn.linear_model.LinearRegression`

class in scikit-learn instead of the `statsmodels.api.OLS`

class in statsmodels - they both implement the same procedure.

In [9]:

```
from sklearn.linear_model import LinearRegression
# get feature/predictor matrix as numpy array
X = df[['balance', 'income']].values
# encode class labels
classes, y = np.unique(df.default.values, return_inverse=True)
y = (y * 2) - 1 # map {0, 1} to {-1, 1}
# fit OLS regression
est = LinearRegression(fit_intercept=True, normalize=True)
est.fit(X, y)
# plot data and decision surface
ax = plt.gca()
ax.scatter(df.balance, df.income, c=(df.default == 'Yes'), cmap=cm_bright)
try:
plot_surface(est, X[:, 0], X[:, 1], ax=ax)
except NameError:
print('Please run cells in Appendix first')
```

Points that lie on the left side of the decision boundary will be classified as negative; points on the right side, positive. The implementation of `plot_surface`

can be found in the Appendix.
We can asses the performance of the model by looking at the confusion matrix — a cross tabulation of the actual and the predicted class labels. The correct classifications are shown in the diagonal of the confusion matrix. The off-diagonal terms show you the classification errors. A condensed summary of the model performance is given by the *misclassification rate* determined simply by dividing the number of errors by the total number of cases.

In [10]:

```
from sklearn.metrics import confusion_matrix as sk_confusion_matrix
# the larger operator will return a boolean array which we will cast as integers for fancy indexing
y_pred = (2 * (est.predict(X) > 0.0)) - 1
def confusion_matrix(y_test, y_pred):
cm = sk_confusion_matrix(y, y_pred)
cm = pd.DataFrame(data=cm, columns=[-1, 1], index=[-1, 1])
cm.columns.name = 'Predicted label'
cm.index.name = 'True label'
error_rate = (y_pred != y).mean()
print('error rate: %.2f' % error_rate)
return cm
confusion_matrix(y, y_pred)
```

Out[10]:

In the above example we are assessing the model performance on the same data that we used to fit the model. This might be a biased estimate of the models performance, for a classifier that simply memorizes the training data has zero training error but would be totally useless to make predictions. It is much better to assess the model performance on a separate dataset called the test data or held-out data. Scikit-learn provides a number of ways to compute such held-out estimates of the model performance. One way is to simply split the data into a training and testing set.

In [11]:

```
from sklearn.cross_validation import train_test_split
# create 80%-20% train-test split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
# fit on training data
est = LinearRegression().fit(X_train, y_train)
# test on data that was not used for fitting
y_pred = (2 * (est.predict(X) > 0.0)) - 1
confusion_matrix(y_test, y_pred)
```

Out[11]:

Different classification techniques can often be compared using the type of decision surface they can learn. The decision surfaces describe for what values of the predictors the model changes its predictions and it can take several different shapes: piece-wise constant, linear, quadratic, vornoi tessellation, …

This section will introduce three popular classification techniques: Logistic Regression, Discriminant Analysis, and Nearest Neighbor. We will investigate what their strengths and weaknesses are by looking at the decision boundaries they can model. In the following we will use three synthetic datasets that we adopted from this scikit-learn example.

In [12]:

```
# Adopted from: Gael Varoqueux
# Andreas Mueller
from collections import OrderedDict
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import make_moons, make_circles, make_classification
# generate 3 synthetic datasets
X, y = make_classification(n_features=2, n_redundant=0, n_informative=2,
random_state=1, n_clusters_per_class=1)
rng = np.random.RandomState(2)
X += 2 * rng.uniform(size=X.shape)
linearly_separable = (X, y)
datasets = OrderedDict()
for name, (X, y) in [('moon', make_moons(noise=0.3, random_state=0)),
('circles', make_circles(noise=0.2, factor=0.5, random_state=1)),
('linear', linearly_separable)]:
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=.4, random_state=1)
# standardize data
scaler = StandardScaler().fit(X_train)
datasets[name] = {'X_train': scaler.transform(X_train), 'y_train': y_train,
'X_test': scaler.transform(X_test), 'y_test': y_test}
# plots the datasets - see Appendix
plot_datasets()
```

The task in each of the above examples is to separate the red from the blue points. Testing data points are plotted in lighter color. The left example contains two intertwined moon sickles; the middle example is a circle of blues framed by a ring of reds; and the right example shows two linearly separable gaussian blobs.

Logistic regression can be viewed as an extension of linear regression to classification problems. One of the limitations of linear regression is that it cannot provide class probability estimates. This is often useful, for example, when we want to inspect manually the most fraudulent cases. Basically, we would like to constrain the predictions of the model to the range $[0, 1]$ so that we can interpret them as probability estimates. In Logistic Regression, we use the logit function to clamp predictions from the range $[-\infty, \infty]$ to $[0, 1]$:

In [13]:

```
x = np.linspace(-10, 10, 100)
y = 1.0 / (1.0 + np.exp(-x))
plt.plot(x, y, 'r-', label='logit')
plt.legend(loc='lower right')
```

Out[13]:

Logistic regression is available in scikit-learn via the class `sklearn.linear_model.LogisticRegression`

. It uses liblinear, so it can be used for problems involving millions of samples and hundred of thousands of predictors.
Lets see how Logistic Regression does on our three toy datasets:

In [14]:

```
from sklearn.linear_model import LogisticRegression
est = LogisticRegression()
plot_datasets(est)
```

As we can see, a linear decision boundary is not a poor approximation for the *moon* datasets, although we fail to separate the two tips of the sickles in the center. The cicles dataset, on the other hand, is not well suited for a linear decision boundary. The error rate of $0.68$ is in fact worse than random guessing. For the linear dataset we picked in fact the correct model class -- the error rate of 10% is due to the noise component in our data.
The gradient shows you the probability of class membership -- white shows you that the model is very uncertain about its prediction.

Linear discriminant Analysis (LDA) is another popular technique which shares some similarities with `LogisticRegression`

. LDA too finds linear boundary between the two classes where points on side are classified as one class and those on the other as classified as the other class.

In [15]:

```
from sklearn.lda import LDA
est = LDA()
plot_datasets(est)
```

The major difference between `LDA`

and `LogisticRegression`

is the way each picks the linear decision boundary: Linear Discriminant Analysis models the decision boundary by making distributional assumptions about the data generating process while Logistic Regression models the probability of a sample being member of a class given its feature values.

Nearest Neighbor uses the notion of similarity to assign class labels; it is based on the smoothness assumption that points which are nearby in input space should have similar outputs. It does this by specifying a similarity (or distance) metric, and at prediction time it simply searches for the `k`

most similar among the training examples to a given test example. The prediction is then either a majority vote of those `k`

training examples or a vote weighted by similarity. The parameter `k`

specifies the smoothness of the decision surface. The decision surface of a k-nearest neighbor classifier can be illustrated by the Voronoi tesselation of the training data, that show you the regions of constant respones.

Yet Nearest Neighbor differs fundamentally from the above models in that it is a so-called non-parametric technique: the number of parameters of the model can grow infinitely as the size of the training data grows. Furthermore, it can model non-linear decision boundaries, something that is important for the first two datasets: moons and circles.

In [16]:

```
from sklearn.neighbors import KNeighborsClassifier
est = KNeighborsClassifier(n_neighbors=1)
plot_datasets(est)
```

If we increase `k`

we enforce the smoothness assumption. This can be seen by comparing the decision boundaries in the plots below where `k=5`

to those above where `k=1`

.

In [17]:

```
est = KNeighborsClassifier(n_neighbors=5)
plot_datasets(est)
```

The cell below contains some utility functions that we used to plot the decision surface of a classifier.

In [18]:

```
from matplotlib.colors import ListedColormap
# utility function to plot the decision surface
def plot_surface(est, x_1, x_2, ax=None, threshold=0.0, contourf=False):
"""Plots the decision surface of ``est`` on features ``x1`` and ``x2``. """
xx1, xx2 = np.meshgrid(np.linspace(x_1.min(), x_1.max(), 100),
np.linspace(x_2.min(), x_2.max(), 100))
# plot the hyperplane by evaluating the parameters on the grid
X_pred = np.c_[xx1.ravel(), xx2.ravel()] # convert 2d grid into seq of points
if hasattr(est, 'predict_proba'): # check if ``est`` supports probabilities
# take probability of positive class
pred = est.predict_proba(X_pred)[:, 1]
else:
pred = est.predict(X_pred)
Z = pred.reshape((100, 100)) # reshape seq to grid
if ax is None:
ax = plt.gca()
# plot line via contour plot
if contourf:
ax.contourf(xx1, xx2, Z, levels=np.linspace(0, 1.0, 10), cmap=plt.cm.RdBu, alpha=0.6)
ax.contour(xx1, xx2, Z, levels=[threshold], colors='black')
ax.set_xlim((x_1.min(), x_1.max()))
ax.set_ylim((x_2.min(), x_2.max()))
def plot_datasets(est=None):
"""Plotsthe decision surface of ``est`` on each of the three datasets. """
fig, axes = plt.subplots(1, 3, figsize=(10, 4))
for (name, ds), ax in zip(datasets.iteritems(), axes):
X_train = ds['X_train']
y_train = ds['y_train']
X_test = ds['X_test']
y_test = ds['y_test']
# plot test lighter than training
cm_bright = ListedColormap(['#FF0000', '#0000FF'])
# Plot the training points
ax.scatter(X_train[:, 0], X_train[:, 1], c=y_train, cmap=cm_bright)
# and testing points
ax.scatter(X_test[:, 0], X_test[:, 1], c=y_test, cmap=cm_bright, alpha=0.6)
# plot limits
ax.set_xlim(X_train[:, 0].min(), X_train[:, 0].max())
ax.set_ylim(X_train[:, 1].min(), X_train[:, 1].max())
# no ticks
ax.set_xticks(())
ax.set_yticks(())
ax.set_ylabel('$x_1$')
ax.set_xlabel('$x_0$')
ax.set_title(name)
if est is not None:
est.fit(X_train, y_train)
plot_surface(est, X_train[:, 0], X_train[:, 1], ax=ax, threshold=0.5, contourf=True)
err = (y_test != est.predict(X_test)).mean()
ax.text(0.88, 0.02, '%.2f' % err, transform=ax.transAxes)
fig.subplots_adjust(left=.02, right=.98)
```