Python Machine Learning 3rd Edition by Sebastian Raschka, Packt Publishing Ltd. 2019

Code Repository: https://github.com/rasbt/python-machine-learning-book-3rd-edition

Python Machine Learning - Code Examples¶

Chapter 3 - A Tour of Machine Learning Classifiers Using Scikit-Learn¶

Note that the optional watermark extension is a small IPython notebook plugin that I developed to make the code reproducible. You can just skip the following line(s).

In [1]:

%load_ext watermark
%watermark -a "Sebastian Raschka" -u -d -p numpy,pandas,matplotlib,sklearn

Sebastian Raschka 
last updated: 2019-07-08 

numpy 1.16.2
pandas 0.24.2
matplotlib 3.0.3
sklearn 0.21.1

The use of watermark is optional. You can install this Jupyter extension via

conda install watermark -c conda-forge

or

pip install watermark

For more information, please see: https://github.com/rasbt/watermark.

Overview¶

Choosing a classification algorithm
First steps with scikit-learn
- Training a perceptron via scikit-learn
Modeling class probabilities via logistic regression
Maximum margin classification with support vector machines
Solving nonlinear problems using a kernel SVM
- Using the kernel trick to find separating hyperplanes in higher dimensional space
Decision tree learning
K-nearest neighbors – a lazy learning algorithm
Summary

In [2]:

from IPython.display import Image
%matplotlib inline

Choosing a classification algorithm¶

...

First steps with scikit-learn¶

Loading the Iris dataset from scikit-learn. Here, the third column represents the petal length, and the fourth column the petal width of the flower examples. The classes are already converted to integer labels where 0=Iris-Setosa, 1=Iris-Versicolor, 2=Iris-Virginica.

In [3]:

from sklearn import datasets
import numpy as np

iris = datasets.load_iris()
X = iris.data[:, [2, 3]]
y = iris.target

print('Class labels:', np.unique(y))

Class labels: [0 1 2]

Splitting data into 70% training and 30% test data:

In [4]:

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.3, random_state=1, stratify=y)

In [5]:

print('Labels counts in y:', np.bincount(y))
print('Labels counts in y_train:', np.bincount(y_train))
print('Labels counts in y_test:', np.bincount(y_test))

Labels counts in y: [50 50 50]
Labels counts in y_train: [35 35 35]
Labels counts in y_test: [15 15 15]

Standardizing the features:

In [6]:

from sklearn.preprocessing import StandardScaler

sc = StandardScaler()
sc.fit(X_train)
X_train_std = sc.transform(X_train)
X_test_std = sc.transform(X_test)

Training a perceptron via scikit-learn¶

In [7]:

from sklearn.linear_model import Perceptron

ppn = Perceptron(eta0=0.1, random_state=1)
ppn.fit(X_train_std, y_train)

Out[7]:

Perceptron(alpha=0.0001, class_weight=None, early_stopping=False, eta0=0.1,
           fit_intercept=True, max_iter=1000, n_iter_no_change=5, n_jobs=None,
           penalty=None, random_state=1, shuffle=True, tol=0.001,
           validation_fraction=0.1, verbose=0, warm_start=False)

Note

You can replace Perceptron(n_iter, ...) by Perceptron(max_iter, ...) in scikit-learn >= 0.19. The n_iter parameter is used here deriberately, because some people still use scikit-learn 0.18.

In [8]:

y_pred = ppn.predict(X_test_std)
print('Misclassified examples: %d' % (y_test != y_pred).sum())

Misclassified examples: 1

In [9]:

from sklearn.metrics import accuracy_score

print('Accuracy: %.3f' % accuracy_score(y_test, y_pred))

Accuracy: 0.978

In [10]:

print('Accuracy: %.3f' % ppn.score(X_test_std, y_test))

Accuracy: 0.978

In [11]:

from matplotlib.colors import ListedColormap
import matplotlib.pyplot as plt


def plot_decision_regions(X, y, classifier, test_idx=None, resolution=0.02):

    # setup marker generator and color map
    markers = ('s', 'x', 'o', '^', 'v')
    colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan')
    cmap = ListedColormap(colors[:len(np.unique(y))])

    # plot the decision surface
    x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1
    x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1
    xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution),
                           np.arange(x2_min, x2_max, resolution))
    Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T)
    Z = Z.reshape(xx1.shape)
    plt.contourf(xx1, xx2, Z, alpha=0.3, cmap=cmap)
    plt.xlim(xx1.min(), xx1.max())
    plt.ylim(xx2.min(), xx2.max())

    for idx, cl in enumerate(np.unique(y)):
        plt.scatter(x=X[y == cl, 0], 
                    y=X[y == cl, 1],
                    alpha=0.8, 
                    c=colors[idx],
                    marker=markers[idx], 
                    label=cl, 
                    edgecolor='black')

    # highlight test examples
    if test_idx:
        # plot all examples
        X_test, y_test = X[test_idx, :], y[test_idx]

        plt.scatter(X_test[:, 0],
                    X_test[:, 1],
                    c='',
                    edgecolor='black',
                    alpha=1.0,
                    linewidth=1,
                    marker='o',
                    s=100, 
                    label='test set')

Training a perceptron model using the standardized training data:

In [12]:

X_combined_std = np.vstack((X_train_std, X_test_std))
y_combined = np.hstack((y_train, y_test))

plot_decision_regions(X=X_combined_std, y=y_combined,
                      classifier=ppn, test_idx=range(105, 150))
plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')

plt.tight_layout()
#plt.savefig('images/03_01.png', dpi=300)
plt.show()

Modeling class probabilities via logistic regression¶

...

Logistic regression intuition and conditional probabilities¶

In [13]:

import matplotlib.pyplot as plt
import numpy as np


def sigmoid(z):
    return 1.0 / (1.0 + np.exp(-z))

z = np.arange(-7, 7, 0.1)
phi_z = sigmoid(z)

plt.plot(z, phi_z)
plt.axvline(0.0, color='k')
plt.ylim(-0.1, 1.1)
plt.xlabel('z')
plt.ylabel('$\phi (z)$')

# y axis ticks and gridline
plt.yticks([0.0, 0.5, 1.0])
ax = plt.gca()
ax.yaxis.grid(True)

plt.tight_layout()
#plt.savefig('images/03_02.png', dpi=300)
plt.show()

In [14]:

Image(filename='images/03_03.png', width=500) 

Out[14]:

Learning the weights of the logistic cost function¶

In [15]:

def cost_1(z):
    return - np.log(sigmoid(z))


def cost_0(z):
    return - np.log(1 - sigmoid(z))

z = np.arange(-10, 10, 0.1)
phi_z = sigmoid(z)

c1 = [cost_1(x) for x in z]
plt.plot(phi_z, c1, label='J(w) if y=1')

c0 = [cost_0(x) for x in z]
plt.plot(phi_z, c0, linestyle='--', label='J(w) if y=0')

plt.ylim(0.0, 5.1)
plt.xlim([0, 1])
plt.xlabel('$\phi$(z)')
plt.ylabel('J(w)')
plt.legend(loc='best')
plt.tight_layout()
#plt.savefig('images/03_04.png', dpi=300)
plt.show()

In [16]:

class LogisticRegressionGD(object):
    """Logistic Regression Classifier using gradient descent.

    Parameters
    ------------
    eta : float
      Learning rate (between 0.0 and 1.0)
    n_iter : int
      Passes over the training dataset.
    random_state : int
      Random number generator seed for random weight
      initialization.


    Attributes
    -----------
    w_ : 1d-array
      Weights after fitting.
    cost_ : list
      Logistic cost function value in each epoch.

    """
    def __init__(self, eta=0.05, n_iter=100, random_state=1):
        self.eta = eta
        self.n_iter = n_iter
        self.random_state = random_state

    def fit(self, X, y):
        """ Fit training data.

        Parameters
        ----------
        X : {array-like}, shape = [n_examples, n_features]
          Training vectors, where n_examples is the number of examples and
          n_features is the number of features.
        y : array-like, shape = [n_examples]
          Target values.

        Returns
        -------
        self : object

        """
        rgen = np.random.RandomState(self.random_state)
        self.w_ = rgen.normal(loc=0.0, scale=0.01, size=1 + X.shape[1])
        self.cost_ = []

        for i in range(self.n_iter):
            net_input = self.net_input(X)
            output = self.activation(net_input)
            errors = (y - output)
            self.w_[1:] += self.eta * X.T.dot(errors)
            self.w_[0] += self.eta * errors.sum()
            
            # note that we compute the logistic `cost` now
            # instead of the sum of squared errors cost
            cost = -y.dot(np.log(output)) - ((1 - y).dot(np.log(1 - output)))
            self.cost_.append(cost)
        return self
    
    def net_input(self, X):
        """Calculate net input"""
        return np.dot(X, self.w_[1:]) + self.w_[0]

    def activation(self, z):
        """Compute logistic sigmoid activation"""
        return 1. / (1. + np.exp(-np.clip(z, -250, 250)))

    def predict(self, X):
        """Return class label after unit step"""
        return np.where(self.net_input(X) >= 0.0, 1, 0)
        # equivalent to:
        # return np.where(self.activation(self.net_input(X)) >= 0.5, 1, 0)

In [17]:

X_train_01_subset = X_train[(y_train == 0) | (y_train == 1)]
y_train_01_subset = y_train[(y_train == 0) | (y_train == 1)]

lrgd = LogisticRegressionGD(eta=0.05, n_iter=1000, random_state=1)
lrgd.fit(X_train_01_subset,
         y_train_01_subset)

plot_decision_regions(X=X_train_01_subset, 
                      y=y_train_01_subset,
                      classifier=lrgd)

plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')

plt.tight_layout()
#plt.savefig('images/03_05.png', dpi=300)
plt.show()

Training a logistic regression model with scikit-learn¶

In [18]:

from sklearn.linear_model import LogisticRegression

lr = LogisticRegression(C=100.0, random_state=1, solver='lbfgs', multi_class='ovr')
lr.fit(X_train_std, y_train)

plot_decision_regions(X_combined_std, y_combined,
                      classifier=lr, test_idx=range(105, 150))
plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('images/03_06.png', dpi=300)
plt.show()

In [19]:

lr.predict_proba(X_test_std[:3, :])

Out[19]:

array([[3.81527885e-09, 1.44792866e-01, 8.55207131e-01],
       [8.34020679e-01, 1.65979321e-01, 3.25737138e-13],
       [8.48831425e-01, 1.51168575e-01, 2.62277619e-14]])

In [20]:

lr.predict_proba(X_test_std[:3, :]).sum(axis=1)

Out[20]:

array([1., 1., 1.])

In [21]:

lr.predict_proba(X_test_std[:3, :]).argmax(axis=1)

Out[21]:

array([2, 0, 0])

In [22]:

lr.predict(X_test_std[:3, :])

Out[22]:

array([2, 0, 0])

In [23]:

lr.predict(X_test_std[0, :].reshape(1, -1))

Out[23]:

array([2])

Tackling overfitting via regularization¶

In [24]:

Image(filename='images/03_07.png', width=700) 

Out[24]:

In [25]:

weights, params = [], []
for c in np.arange(-5, 5):
    lr = LogisticRegression(C=10.**c, random_state=1,
                            solver='lbfgs',
                            multi_class='ovr')
    lr.fit(X_train_std, y_train)
    weights.append(lr.coef_[1])
    params.append(10.**c)

weights = np.array(weights)
plt.plot(params, weights[:, 0],
         label='petal length')
plt.plot(params, weights[:, 1], linestyle='--',
         label='petal width')
plt.ylabel('weight coefficient')
plt.xlabel('C')
plt.legend(loc='upper left')
plt.xscale('log')
#plt.savefig('images/03_08.png', dpi=300)
plt.show()

Maximum margin classification with support vector machines¶

In [26]:

Image(filename='images/03_09.png', width=700) 

Out[26]:

Maximum margin intuition¶

...

Dealing with the nonlinearly separable case using slack variables¶

In [27]:

Image(filename='images/03_10.png', width=600) 

Out[27]:

In [28]:

from sklearn.svm import SVC

svm = SVC(kernel='linear', C=1.0, random_state=1)
svm.fit(X_train_std, y_train)

plot_decision_regions(X_combined_std, 
                      y_combined,
                      classifier=svm, 
                      test_idx=range(105, 150))
plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_11.png', dpi=300)
plt.show()

Alternative implementations in scikit-learn¶

In [29]:

from sklearn.linear_model import SGDClassifier

ppn = SGDClassifier(loss='perceptron')
lr = SGDClassifier(loss='log')
svm = SGDClassifier(loss='hinge')

Solving non-linear problems using a kernel SVM¶

In [30]:

import matplotlib.pyplot as plt
import numpy as np

np.random.seed(1)
X_xor = np.random.randn(200, 2)
y_xor = np.logical_xor(X_xor[:, 0] > 0,
                       X_xor[:, 1] > 0)
y_xor = np.where(y_xor, 1, -1)

plt.scatter(X_xor[y_xor == 1, 0],
            X_xor[y_xor == 1, 1],
            c='b', marker='x',
            label='1')
plt.scatter(X_xor[y_xor == -1, 0],
            X_xor[y_xor == -1, 1],
            c='r',
            marker='s',
            label='-1')

plt.xlim([-3, 3])
plt.ylim([-3, 3])
plt.legend(loc='best')
plt.tight_layout()
#plt.savefig('images/03_12.png', dpi=300)
plt.show()

In [31]:

Image(filename='images/03_13.png', width=700) 

Out[31]:

Using the kernel trick to find separating hyperplanes in higher dimensional space¶

In [32]:

svm = SVC(kernel='rbf', random_state=1, gamma=0.10, C=10.0)
svm.fit(X_xor, y_xor)
plot_decision_regions(X_xor, y_xor,
                      classifier=svm)

plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_14.png', dpi=300)
plt.show()

In [33]:

from sklearn.svm import SVC

svm = SVC(kernel='rbf', random_state=1, gamma=0.2, C=1.0)
svm.fit(X_train_std, y_train)

plot_decision_regions(X_combined_std, y_combined,
                      classifier=svm, test_idx=range(105, 150))
plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_15.png', dpi=300)
plt.show()

In [34]:

svm = SVC(kernel='rbf', random_state=1, gamma=100.0, C=1.0)
svm.fit(X_train_std, y_train)

plot_decision_regions(X_combined_std, y_combined, 
                      classifier=svm, test_idx=range(105, 150))
plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_16.png', dpi=300)
plt.show()

Decision tree learning¶

In [35]:

Image(filename='images/03_17.png', width=500) 

Out[35]:

In [36]:

Image(filename='images/03_18.png', width=500) 

Out[36]:

Maximizing information gain - getting the most bang for the buck¶

In [37]:

import matplotlib.pyplot as plt
import numpy as np


def gini(p):
    return p * (1 - p) + (1 - p) * (1 - (1 - p))


def entropy(p):
    return - p * np.log2(p) - (1 - p) * np.log2((1 - p))


def error(p):
    return 1 - np.max([p, 1 - p])

x = np.arange(0.0, 1.0, 0.01)

ent = [entropy(p) if p != 0 else None for p in x]
sc_ent = [e * 0.5 if e else None for e in ent]
err = [error(i) for i in x]

fig = plt.figure()
ax = plt.subplot(111)
for i, lab, ls, c, in zip([ent, sc_ent, gini(x), err], 
                          ['Entropy', 'Entropy (scaled)', 
                           'Gini impurity', 'Misclassification error'],
                          ['-', '-', '--', '-.'],
                          ['black', 'lightgray', 'red', 'green', 'cyan']):
    line = ax.plot(x, i, label=lab, linestyle=ls, lw=2, color=c)

ax.legend(loc='upper center', bbox_to_anchor=(0.5, 1.15),
          ncol=5, fancybox=True, shadow=False)

ax.axhline(y=0.5, linewidth=1, color='k', linestyle='--')
ax.axhline(y=1.0, linewidth=1, color='k', linestyle='--')
plt.ylim([0, 1.1])
plt.xlabel('p(i=1)')
plt.ylabel('impurity index')
#plt.savefig('images/03_19.png', dpi=300, bbox_inches='tight')
plt.show()

Building a decision tree¶

In [38]:

from sklearn.tree import DecisionTreeClassifier

tree_model = DecisionTreeClassifier(criterion='gini', 
                                    max_depth=4, 
                                    random_state=1)
tree_model.fit(X_train, y_train)

X_combined = np.vstack((X_train, X_test))
y_combined = np.hstack((y_train, y_test))
plot_decision_regions(X_combined, y_combined, 
                      classifier=tree_model,
                      test_idx=range(105, 150))

plt.xlabel('petal length [cm]')
plt.ylabel('petal width [cm]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_20.png', dpi=300)
plt.show()

In [39]:

from sklearn import tree

tree.plot_tree(tree_model)
#plt.savefig('images/03_21_1.pdf')
plt.show()

In [40]:

from pydotplus import graph_from_dot_data
from sklearn.tree import export_graphviz

dot_data = export_graphviz(tree_model,
                           filled=True, 
                           rounded=True,
                           class_names=['Setosa', 
                                        'Versicolor',
                                        'Virginica'],
                           feature_names=['petal length', 
                                          'petal width'],
                           out_file=None) 
graph = graph_from_dot_data(dot_data) 
graph.write_png('tree.png') 

Out[40]:

True

In [41]:

Image(filename='images/03_21.png', width=600) 

Out[41]:

Combining weak to strong learners via random forests¶

In [42]:

from sklearn.ensemble import RandomForestClassifier

forest = RandomForestClassifier(criterion='gini',
                                n_estimators=25, 
                                random_state=1,
                                n_jobs=2)
forest.fit(X_train, y_train)

plot_decision_regions(X_combined, y_combined, 
                      classifier=forest, test_idx=range(105, 150))

plt.xlabel('petal length [cm]')
plt.ylabel('petal width [cm]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_22.png', dpi=300)
plt.show()

K-nearest neighbors - a lazy learning algorithm¶

In [43]:

Image(filename='images/03_23.png', width=400) 

Out[43]:

In [44]:

from sklearn.neighbors import KNeighborsClassifier

knn = KNeighborsClassifier(n_neighbors=5, 
                           p=2, 
                           metric='minkowski')
knn.fit(X_train_std, y_train)

plot_decision_regions(X_combined_std, y_combined, 
                      classifier=knn, test_idx=range(105, 150))

plt.xlabel('petal length [standardized]')
plt.ylabel('petal width [standardized]')
plt.legend(loc='upper left')
plt.tight_layout()
#plt.savefig('images/03_24.png', dpi=300)
plt.show()

Summary¶

...

Readers may ignore the next cell.

In [45]:

! python ../.convert_notebook_to_script.py --input ch03.ipynb --output ch03.py

[NbConvertApp] Converting notebook ch03.ipynb to script
[NbConvertApp] Writing 20066 bytes to ch03.py