Chapter 7 – Ensemble Learning and Random Forests

This notebook contains all the sample code and solutions to the exercises in chapter 7.

# Setup¶

First, let's import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures. We also check that Python 3.5 or later is installed (although Python 2.x may work, it is deprecated so we strongly recommend you use Python 3 instead), as well as Scikit-Learn ≥0.20.

In [1]:
# Python ≥3.5 is required
import sys
assert sys.version_info >= (3, 5)

# Scikit-Learn ≥0.20 is required
import sklearn
assert sklearn.__version__ >= "0.20"

# Common imports
import numpy as np
import os

# to make this notebook's output stable across runs
np.random.seed(42)

# To plot pretty figures
%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rc('axes', labelsize=14)
mpl.rc('xtick', labelsize=12)
mpl.rc('ytick', labelsize=12)

# Where to save the figures
PROJECT_ROOT_DIR = "."
CHAPTER_ID = "ensembles"
IMAGES_PATH = os.path.join(PROJECT_ROOT_DIR, "images", CHAPTER_ID)
os.makedirs(IMAGES_PATH, exist_ok=True)

def save_fig(fig_id, tight_layout=True, fig_extension="png", resolution=300):
path = os.path.join(IMAGES_PATH, fig_id + "." + fig_extension)
print("Saving figure", fig_id)
if tight_layout:
plt.tight_layout()
plt.savefig(path, format=fig_extension, dpi=resolution)


# Voting classifiers¶

In [2]:
heads_proba = 0.51
coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)

In [3]:
plt.figure(figsize=(8,3.5))
plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
plt.xlabel("Number of coin tosses")
plt.legend(loc="lower right")
plt.axis([0, 10000, 0.42, 0.58])
save_fig("law_of_large_numbers_plot")
plt.show()

Saving figure law_of_large_numbers_plot

In [4]:
from sklearn.model_selection import train_test_split
from sklearn.datasets import make_moons

X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)


Note: to be future-proof, we set solver="lbfgs", n_estimators=100, and gamma="scale" since these will be the default values in upcoming Scikit-Learn versions.

In [5]:
from sklearn.ensemble import RandomForestClassifier
from sklearn.ensemble import VotingClassifier
from sklearn.linear_model import LogisticRegression
from sklearn.svm import SVC

log_clf = LogisticRegression(solver="lbfgs", random_state=42)
rnd_clf = RandomForestClassifier(n_estimators=100, random_state=42)
svm_clf = SVC(gamma="scale", random_state=42)

voting_clf = VotingClassifier(
estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
voting='hard')

In [6]:
voting_clf.fit(X_train, y_train)

Out[6]:
VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
('rf', RandomForestClassifier(random_state=42)),
('svc', SVC(random_state=42))])
In [7]:
from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

LogisticRegression 0.864
RandomForestClassifier 0.896
SVC 0.896
VotingClassifier 0.912


Note: the results in this notebook may differ slightly from the book, as Scikit-Learn algorithms sometimes get tweaked.

Soft voting:

In [8]:
log_clf = LogisticRegression(solver="lbfgs", random_state=42)
rnd_clf = RandomForestClassifier(n_estimators=100, random_state=42)
svm_clf = SVC(gamma="scale", probability=True, random_state=42)

voting_clf = VotingClassifier(
estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
voting='soft')
voting_clf.fit(X_train, y_train)

Out[8]:
VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
('rf', RandomForestClassifier(random_state=42)),
('svc', SVC(probability=True, random_state=42))],
voting='soft')
In [9]:
from sklearn.metrics import accuracy_score

for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
print(clf.__class__.__name__, accuracy_score(y_test, y_pred))

LogisticRegression 0.864
RandomForestClassifier 0.896
SVC 0.896
VotingClassifier 0.92


# Bagging ensembles¶

In [10]:
from sklearn.ensemble import BaggingClassifier
from sklearn.tree import DecisionTreeClassifier

bag_clf = BaggingClassifier(
DecisionTreeClassifier(), n_estimators=500,
max_samples=100, bootstrap=True, random_state=42)
bag_clf.fit(X_train, y_train)
y_pred = bag_clf.predict(X_test)

In [11]:
from sklearn.metrics import accuracy_score
print(accuracy_score(y_test, y_pred))

0.904

In [12]:
tree_clf = DecisionTreeClassifier(random_state=42)
tree_clf.fit(X_train, y_train)
y_pred_tree = tree_clf.predict(X_test)
print(accuracy_score(y_test, y_pred_tree))

0.856

In [13]:
from matplotlib.colors import ListedColormap

def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.45, -1, 1.5], alpha=0.5, contour=True):
x1s = np.linspace(axes[0], axes[1], 100)
x2s = np.linspace(axes[2], axes[3], 100)
x1, x2 = np.meshgrid(x1s, x2s)
X_new = np.c_[x1.ravel(), x2.ravel()]
y_pred = clf.predict(X_new).reshape(x1.shape)
custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
if contour:
custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
plt.axis(axes)
plt.xlabel(r"$x_1$", fontsize=18)
plt.ylabel(r"$x_2$", fontsize=18, rotation=0)

In [14]:
fix, axes = plt.subplots(ncols=2, figsize=(10,4), sharey=True)
plt.sca(axes[0])
plot_decision_boundary(tree_clf, X, y)
plt.title("Decision Tree", fontsize=14)
plt.sca(axes[1])
plot_decision_boundary(bag_clf, X, y)
plt.title("Decision Trees with Bagging", fontsize=14)
plt.ylabel("")
save_fig("decision_tree_without_and_with_bagging_plot")
plt.show()

Saving figure decision_tree_without_and_with_bagging_plot


# Random Forests¶

In [15]:
bag_clf = BaggingClassifier(
DecisionTreeClassifier(max_features="sqrt", max_leaf_nodes=16),
n_estimators=500, random_state=42)

In [16]:
bag_clf.fit(X_train, y_train)
y_pred = bag_clf.predict(X_test)

In [17]:
from sklearn.ensemble import RandomForestClassifier

rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, random_state=42)
rnd_clf.fit(X_train, y_train)

y_pred_rf = rnd_clf.predict(X_test)

In [18]:
np.sum(y_pred == y_pred_rf) / len(y_pred)  # very similar predictions

Out[18]:
1.0
In [19]:
from sklearn.datasets import load_iris
rnd_clf = RandomForestClassifier(n_estimators=500, random_state=42)
rnd_clf.fit(iris["data"], iris["target"])
for name, score in zip(iris["feature_names"], rnd_clf.feature_importances_):
print(name, score)

sepal length (cm) 0.11249225099876375
sepal width (cm) 0.02311928828251033
petal length (cm) 0.4410304643639577
petal width (cm) 0.4233579963547682

In [20]:
rnd_clf.feature_importances_

Out[20]:
array([0.11249225, 0.02311929, 0.44103046, 0.423358  ])
In [21]:
plt.figure(figsize=(6, 4))

for i in range(15):
tree_clf = DecisionTreeClassifier(max_leaf_nodes=16, random_state=42 + i)
indices_with_replacement = np.random.randint(0, len(X_train), len(X_train))
tree_clf.fit(X[indices_with_replacement], y[indices_with_replacement])
plot_decision_boundary(tree_clf, X, y, axes=[-1.5, 2.45, -1, 1.5], alpha=0.02, contour=False)

plt.show()


## Out-of-Bag evaluation¶

In [22]:
bag_clf = BaggingClassifier(
DecisionTreeClassifier(), n_estimators=500,
bootstrap=True, oob_score=True, random_state=40)
bag_clf.fit(X_train, y_train)
bag_clf.oob_score_

Out[22]:
0.8986666666666666
In [23]:
bag_clf.oob_decision_function_

Out[23]:
array([[0.32275132, 0.67724868],
[0.34117647, 0.65882353],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.09497207, 0.90502793],
[0.31147541, 0.68852459],
[0.01754386, 0.98245614],
[0.97109827, 0.02890173],
[0.97765363, 0.02234637],
[0.74404762, 0.25595238],
[0.        , 1.        ],
[0.7173913 , 0.2826087 ],
[0.85026738, 0.14973262],
[0.97222222, 0.02777778],
[0.0625    , 0.9375    ],
[0.        , 1.        ],
[0.97837838, 0.02162162],
[0.94642857, 0.05357143],
[1.        , 0.        ],
[0.01704545, 0.98295455],
[0.39473684, 0.60526316],
[0.88700565, 0.11299435],
[1.        , 0.        ],
[0.97790055, 0.02209945],
[0.        , 1.        ],
[0.99428571, 0.00571429],
[1.        , 0.        ],
[0.        , 1.        ],
[0.62569832, 0.37430168],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.13402062, 0.86597938],
[1.        , 0.        ],
[0.        , 1.        ],
[0.38251366, 0.61748634],
[0.        , 1.        ],
[1.        , 0.        ],
[0.27093596, 0.72906404],
[0.34146341, 0.65853659],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.00531915, 0.99468085],
[0.98843931, 0.01156069],
[0.91428571, 0.08571429],
[0.97282609, 0.02717391],
[0.98019802, 0.01980198],
[0.        , 1.        ],
[0.07361963, 0.92638037],
[0.98019802, 0.01980198],
[0.0052356 , 0.9947644 ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.97790055, 0.02209945],
[0.8       , 0.2       ],
[0.42424242, 0.57575758],
[1.        , 0.        ],
[0.        , 1.        ],
[0.66477273, 0.33522727],
[1.        , 0.        ],
[1.        , 0.        ],
[0.86781609, 0.13218391],
[1.        , 0.        ],
[0.56725146, 0.43274854],
[0.1576087 , 0.8423913 ],
[0.66492147, 0.33507853],
[0.91709845, 0.08290155],
[0.        , 1.        ],
[0.16759777, 0.83240223],
[0.87434555, 0.12565445],
[1.        , 0.        ],
[0.        , 1.        ],
[0.995     , 0.005     ],
[0.        , 1.        ],
[0.07878788, 0.92121212],
[0.05418719, 0.94581281],
[0.29015544, 0.70984456],
[1.        , 0.        ],
[0.        , 1.        ],
[0.83040936, 0.16959064],
[0.01092896, 0.98907104],
[0.        , 1.        ],
[0.        , 1.        ],
[0.21465969, 0.78534031],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.94660194, 0.05339806],
[0.77094972, 0.22905028],
[0.        , 1.        ],
[1.        , 0.        ],
[0.16574586, 0.83425414],
[0.65306122, 0.34693878],
[0.        , 1.        ],
[0.02564103, 0.97435897],
[0.50555556, 0.49444444],
[1.        , 0.        ],
[0.03208556, 0.96791444],
[0.99435028, 0.00564972],
[0.23699422, 0.76300578],
[0.49509804, 0.50490196],
[0.9947644 , 0.0052356 ],
[0.00555556, 0.99444444],
[0.98963731, 0.01036269],
[0.26153846, 0.73846154],
[0.92972973, 0.07027027],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.80113636, 0.19886364],
[1.        , 0.        ],
[0.0106383 , 0.9893617 ],
[1.        , 0.        ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.98181818, 0.01818182],
[1.        , 0.        ],
[0.01036269, 0.98963731],
[0.97752809, 0.02247191],
[0.99453552, 0.00546448],
[0.01960784, 0.98039216],
[0.17857143, 0.82142857],
[0.98387097, 0.01612903],
[0.29533679, 0.70466321],
[0.98295455, 0.01704545],
[0.        , 1.        ],
[0.00561798, 0.99438202],
[0.75690608, 0.24309392],
[0.38624339, 0.61375661],
[0.40625   , 0.59375   ],
[0.87368421, 0.12631579],
[0.92462312, 0.07537688],
[0.05181347, 0.94818653],
[0.82802548, 0.17197452],
[0.01546392, 0.98453608],
[0.        , 1.        ],
[0.02298851, 0.97701149],
[0.9726776 , 0.0273224 ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.01041667, 0.98958333],
[0.        , 1.        ],
[0.03804348, 0.96195652],
[0.02040816, 0.97959184],
[1.        , 0.        ],
[1.        , 0.        ],
[0.94915254, 0.05084746],
[1.        , 0.        ],
[1.        , 0.        ],
[0.99462366, 0.00537634],
[0.        , 1.        ],
[0.39378238, 0.60621762],
[0.33152174, 0.66847826],
[0.00609756, 0.99390244],
[0.        , 1.        ],
[0.3172043 , 0.6827957 ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.00588235, 0.99411765],
[0.        , 1.        ],
[0.98924731, 0.01075269],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.62893082, 0.37106918],
[0.92344498, 0.07655502],
[0.        , 1.        ],
[0.99526066, 0.00473934],
[1.        , 0.        ],
[0.98888889, 0.01111111],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.06989247, 0.93010753],
[1.        , 0.        ],
[0.03608247, 0.96391753],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.02185792, 0.97814208],
[1.        , 0.        ],
[0.95808383, 0.04191617],
[0.78362573, 0.21637427],
[0.56650246, 0.43349754],
[0.        , 1.        ],
[0.18023256, 0.81976744],
[1.        , 0.        ],
[0.93121693, 0.06878307],
[0.97175141, 0.02824859],
[1.        , 0.        ],
[0.00531915, 0.99468085],
[0.        , 1.        ],
[0.43010753, 0.56989247],
[0.85858586, 0.14141414],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.00558659, 0.99441341],
[0.        , 1.        ],
[0.96923077, 0.03076923],
[0.        , 1.        ],
[0.21649485, 0.78350515],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.98477157, 0.01522843],
[0.8       , 0.2       ],
[0.99441341, 0.00558659],
[0.        , 1.        ],
[0.09497207, 0.90502793],
[0.99492386, 0.00507614],
[0.01714286, 0.98285714],
[0.        , 1.        ],
[0.02747253, 0.97252747],
[1.        , 0.        ],
[0.77005348, 0.22994652],
[0.        , 1.        ],
[0.90229885, 0.09770115],
[0.98387097, 0.01612903],
[0.22222222, 0.77777778],
[0.20348837, 0.79651163],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.20338983, 0.79661017],
[0.98181818, 0.01818182],
[0.        , 1.        ],
[1.        , 0.        ],
[0.98969072, 0.01030928],
[0.        , 1.        ],
[0.48663102, 0.51336898],
[1.        , 0.        ],
[0.00529101, 0.99470899],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.08379888, 0.91620112],
[0.12352941, 0.87647059],
[0.99415205, 0.00584795],
[0.03517588, 0.96482412],
[1.        , 0.        ],
[0.39790576, 0.60209424],
[0.05434783, 0.94565217],
[0.53191489, 0.46808511],
[0.51898734, 0.48101266],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.        , 1.        ],
[0.60869565, 0.39130435],
[0.        , 1.        ],
[1.        , 0.        ],
[0.24157303, 0.75842697],
[0.81578947, 0.18421053],
[0.08717949, 0.91282051],
[0.99453552, 0.00546448],
[0.82142857, 0.17857143],
[0.        , 1.        ],
[0.        , 1.        ],
[0.11904762, 0.88095238],
[0.04188482, 0.95811518],
[0.        , 1.        ],
[1.        , 0.        ],
[0.89150943, 0.10849057],
[0.19230769, 0.80769231],
[0.95238095, 0.04761905],
[0.00515464, 0.99484536],
[0.59375   , 0.40625   ],
[0.07692308, 0.92307692],
[0.99484536, 0.00515464],
[0.83684211, 0.16315789],
[0.        , 1.        ],
[0.99484536, 0.00515464],
[0.95360825, 0.04639175],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.26395939, 0.73604061],
[0.98461538, 0.01538462],
[1.        , 0.        ],
[0.        , 1.        ],
[0.00574713, 0.99425287],
[0.85142857, 0.14857143],
[0.        , 1.        ],
[1.        , 0.        ],
[0.75301205, 0.24698795],
[0.8969697 , 0.1030303 ],
[1.        , 0.        ],
[0.75555556, 0.24444444],
[0.48863636, 0.51136364],
[0.        , 1.        ],
[0.92473118, 0.07526882],
[0.        , 1.        ],
[1.        , 0.        ],
[0.87709497, 0.12290503],
[1.        , 0.        ],
[1.        , 0.        ],
[0.74752475, 0.25247525],
[0.09146341, 0.90853659],
[0.42268041, 0.57731959],
[0.22395833, 0.77604167],
[0.        , 1.        ],
[0.87046632, 0.12953368],
[0.78212291, 0.21787709],
[0.00507614, 0.99492386],
[1.        , 0.        ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.02884615, 0.97115385],
[0.96      , 0.04      ],
[0.93478261, 0.06521739],
[1.        , 0.        ],
[0.50731707, 0.49268293],
[1.        , 0.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.01604278, 0.98395722],
[1.        , 0.        ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.96987952, 0.03012048],
[0.        , 1.        ],
[0.05172414, 0.94827586],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[1.        , 0.        ],
[0.        , 1.        ],
[0.99494949, 0.00505051],
[0.01675978, 0.98324022],
[1.        , 0.        ],
[0.14583333, 0.85416667],
[0.        , 1.        ],
[0.00546448, 0.99453552],
[0.        , 1.        ],
[0.41836735, 0.58163265],
[0.13095238, 0.86904762],
[0.22110553, 0.77889447],
[1.        , 0.        ],
[0.97647059, 0.02352941],
[0.21195652, 0.78804348],
[0.98882682, 0.01117318],
[0.        , 1.        ],
[0.        , 1.        ],
[1.        , 0.        ],
[0.96428571, 0.03571429],
[0.34554974, 0.65445026],
[0.98235294, 0.01764706],
[1.        , 0.        ],
[0.        , 1.        ],
[0.99465241, 0.00534759],
[0.        , 1.        ],
[0.06043956, 0.93956044],
[0.98214286, 0.01785714],
[1.        , 0.        ],
[0.03108808, 0.96891192],
[0.58854167, 0.41145833]])
In [24]:
from sklearn.metrics import accuracy_score
y_pred = bag_clf.predict(X_test)
accuracy_score(y_test, y_pred)

Out[24]:
0.912

## Feature importance¶

Warning: since Scikit-Learn 0.24, fetch_openml() returns a Pandas DataFrame by default. To avoid this and keep the same code as in the book, we use as_frame=False.

In [25]:
from sklearn.datasets import fetch_openml

mnist = fetch_openml('mnist_784', version=1, as_frame=False)
mnist.target = mnist.target.astype(np.uint8)

In [26]:
rnd_clf = RandomForestClassifier(n_estimators=100, random_state=42)
rnd_clf.fit(mnist["data"], mnist["target"])

Out[26]:
RandomForestClassifier(random_state=42)
In [27]:
def plot_digit(data):
image = data.reshape(28, 28)
plt.imshow(image, cmap = mpl.cm.hot,
interpolation="nearest")
plt.axis("off")

In [28]:
plot_digit(rnd_clf.feature_importances_)

cbar = plt.colorbar(ticks=[rnd_clf.feature_importances_.min(), rnd_clf.feature_importances_.max()])
cbar.ax.set_yticklabels(['Not important', 'Very important'])

save_fig("mnist_feature_importance_plot")
plt.show()

Saving figure mnist_feature_importance_plot


In [29]:
from sklearn.ensemble import AdaBoostClassifier

DecisionTreeClassifier(max_depth=1), n_estimators=200,
algorithm="SAMME.R", learning_rate=0.5, random_state=42)

Out[29]:
AdaBoostClassifier(base_estimator=DecisionTreeClassifier(max_depth=1),
learning_rate=0.5, n_estimators=200, random_state=42)
In [30]:
plot_decision_boundary(ada_clf, X, y)

In [31]:
m = len(X_train)

fix, axes = plt.subplots(ncols=2, figsize=(10,4), sharey=True)
for subplot, learning_rate in ((0, 1), (1, 0.5)):
sample_weights = np.ones(m) / m
plt.sca(axes[subplot])
for i in range(5):
svm_clf = SVC(kernel="rbf", C=0.2, gamma=0.6, random_state=42)
svm_clf.fit(X_train, y_train, sample_weight=sample_weights * m)
y_pred = svm_clf.predict(X_train)

r = sample_weights[y_pred != y_train].sum() / sample_weights.sum() # equation 7-1
alpha = learning_rate * np.log((1 - r) / r) # equation 7-2
sample_weights[y_pred != y_train] *= np.exp(alpha) # equation 7-3
sample_weights /= sample_weights.sum() # normalization step

plot_decision_boundary(svm_clf, X, y, alpha=0.2)
plt.title("learning_rate = {}".format(learning_rate), fontsize=16)
if subplot == 0:
plt.text(-0.75, -0.95, "1", fontsize=14)
plt.text(-1.05, -0.95, "2", fontsize=14)
plt.text(1.0, -0.95, "3", fontsize=14)
plt.text(-1.45, -0.5, "4", fontsize=14)
plt.text(1.36,  -0.95, "5", fontsize=14)
else:
plt.ylabel("")

save_fig("boosting_plot")
plt.show()

Saving figure boosting_plot


In [32]:
np.random.seed(42)
X = np.random.rand(100, 1) - 0.5
y = 3*X[:, 0]**2 + 0.05 * np.random.randn(100)

In [33]:
from sklearn.tree import DecisionTreeRegressor

tree_reg1 = DecisionTreeRegressor(max_depth=2, random_state=42)
tree_reg1.fit(X, y)

Out[33]:
DecisionTreeRegressor(max_depth=2, random_state=42)
In [34]:
y2 = y - tree_reg1.predict(X)
tree_reg2 = DecisionTreeRegressor(max_depth=2, random_state=42)
tree_reg2.fit(X, y2)

Out[34]:
DecisionTreeRegressor(max_depth=2, random_state=42)
In [35]:
y3 = y2 - tree_reg2.predict(X)
tree_reg3 = DecisionTreeRegressor(max_depth=2, random_state=42)
tree_reg3.fit(X, y3)

Out[35]:
DecisionTreeRegressor(max_depth=2, random_state=42)
In [36]:
X_new = np.array([[0.8]])

In [37]:
y_pred = sum(tree.predict(X_new) for tree in (tree_reg1, tree_reg2, tree_reg3))

In [38]:
y_pred

Out[38]:
array([0.75026781])
In [39]:
def plot_predictions(regressors, X, y, axes, label=None, style="r-", data_style="b.", data_label=None):
x1 = np.linspace(axes[0], axes[1], 500)
y_pred = sum(regressor.predict(x1.reshape(-1, 1)) for regressor in regressors)
plt.plot(X[:, 0], y, data_style, label=data_label)
plt.plot(x1, y_pred, style, linewidth=2, label=label)
if label or data_label:
plt.legend(loc="upper center", fontsize=16)
plt.axis(axes)

In [40]:
plt.figure(figsize=(11,11))

plt.subplot(321)
plot_predictions([tree_reg1], X, y, axes=[-0.5, 0.5, -0.1, 0.8], label="$h_1(x_1)$", style="g-", data_label="Training set")
plt.ylabel("$y$", fontsize=16, rotation=0)
plt.title("Residuals and tree predictions", fontsize=16)

plt.subplot(322)
plot_predictions([tree_reg1], X, y, axes=[-0.5, 0.5, -0.1, 0.8], label="$h(x_1) = h_1(x_1)$", data_label="Training set")
plt.ylabel("$y$", fontsize=16, rotation=0)
plt.title("Ensemble predictions", fontsize=16)

plt.subplot(323)
plot_predictions([tree_reg2], X, y2, axes=[-0.5, 0.5, -0.5, 0.5], label="$h_2(x_1)$", style="g-", data_style="k+", data_label="Residuals")
plt.ylabel("$y - h_1(x_1)$", fontsize=16)

plt.subplot(324)
plot_predictions([tree_reg1, tree_reg2], X, y, axes=[-0.5, 0.5, -0.1, 0.8], label="$h(x_1) = h_1(x_1) + h_2(x_1)$")
plt.ylabel("$y$", fontsize=16, rotation=0)

plt.subplot(325)
plot_predictions([tree_reg3], X, y3, axes=[-0.5, 0.5, -0.5, 0.5], label="$h_3(x_1)$", style="g-", data_style="k+")
plt.ylabel("$y - h_1(x_1) - h_2(x_1)$", fontsize=16)
plt.xlabel("$x_1$", fontsize=16)

plt.subplot(326)
plot_predictions([tree_reg1, tree_reg2, tree_reg3], X, y, axes=[-0.5, 0.5, -0.1, 0.8], label="$h(x_1) = h_1(x_1) + h_2(x_1) + h_3(x_1)$")
plt.xlabel("$x_1$", fontsize=16)
plt.ylabel("$y$", fontsize=16, rotation=0)

plt.show()

Saving figure gradient_boosting_plot

In [41]:
from sklearn.ensemble import GradientBoostingRegressor

gbrt = GradientBoostingRegressor(max_depth=2, n_estimators=3, learning_rate=1.0, random_state=42)
gbrt.fit(X, y)

Out[41]:
GradientBoostingRegressor(learning_rate=1.0, max_depth=2, n_estimators=3,
random_state=42)
In [42]:
gbrt_slow = GradientBoostingRegressor(max_depth=2, n_estimators=200, learning_rate=0.1, random_state=42)
gbrt_slow.fit(X, y)

Out[42]:
GradientBoostingRegressor(max_depth=2, n_estimators=200, random_state=42)
In [43]:
fix, axes = plt.subplots(ncols=2, figsize=(10,4), sharey=True)

plt.sca(axes[0])
plot_predictions([gbrt], X, y, axes=[-0.5, 0.5, -0.1, 0.8], label="Ensemble predictions")
plt.title("learning_rate={}, n_estimators={}".format(gbrt.learning_rate, gbrt.n_estimators), fontsize=14)
plt.xlabel("$x_1$", fontsize=16)
plt.ylabel("$y$", fontsize=16, rotation=0)

plt.sca(axes[1])
plot_predictions([gbrt_slow], X, y, axes=[-0.5, 0.5, -0.1, 0.8])
plt.title("learning_rate={}, n_estimators={}".format(gbrt_slow.learning_rate, gbrt_slow.n_estimators), fontsize=14)
plt.xlabel("$x_1$", fontsize=16)

save_fig("gbrt_learning_rate_plot")
plt.show()

Saving figure gbrt_learning_rate_plot


## Gradient Boosting with Early stopping¶

In [44]:
import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error

X_train, X_val, y_train, y_val = train_test_split(X, y, random_state=49)

gbrt.fit(X_train, y_train)

errors = [mean_squared_error(y_val, y_pred)
for y_pred in gbrt.staged_predict(X_val)]
bst_n_estimators = np.argmin(errors) + 1

gbrt_best.fit(X_train, y_train)

Out[44]:
GradientBoostingRegressor(max_depth=2, n_estimators=56, random_state=42)
In [45]:
min_error = np.min(errors)

In [46]:
plt.figure(figsize=(10, 4))

plt.subplot(121)
plt.plot(errors, "b.-")
plt.plot([bst_n_estimators, bst_n_estimators], [0, min_error], "k--")
plt.plot([0, 120], [min_error, min_error], "k--")
plt.plot(bst_n_estimators, min_error, "ko")
plt.text(bst_n_estimators, min_error*1.2, "Minimum", ha="center", fontsize=14)
plt.axis([0, 120, 0, 0.01])
plt.xlabel("Number of trees")
plt.ylabel("Error", fontsize=16)
plt.title("Validation error", fontsize=14)

plt.subplot(122)
plot_predictions([gbrt_best], X, y, axes=[-0.5, 0.5, -0.1, 0.8])
plt.title("Best model (%d trees)" % bst_n_estimators, fontsize=14)
plt.ylabel("$y$", fontsize=16, rotation=0)
plt.xlabel("$x_1$", fontsize=16)

save_fig("early_stopping_gbrt_plot")
plt.show()

Saving figure early_stopping_gbrt_plot

In [47]:
gbrt = GradientBoostingRegressor(max_depth=2, warm_start=True, random_state=42)

min_val_error = float("inf")
error_going_up = 0
for n_estimators in range(1, 120):
gbrt.n_estimators = n_estimators
gbrt.fit(X_train, y_train)
y_pred = gbrt.predict(X_val)
val_error = mean_squared_error(y_val, y_pred)
if val_error < min_val_error:
min_val_error = val_error
error_going_up = 0
else:
error_going_up += 1
if error_going_up == 5:
break  # early stopping

In [48]:
print(gbrt.n_estimators)

61

In [49]:
print("Minimum validation MSE:", min_val_error)

Minimum validation MSE: 0.002712853325235463


## Using XGBoost¶

In [50]:
try:
import xgboost
except ImportError as ex:
print("Error: the xgboost library is not installed.")
xgboost = None

In [51]:
if xgboost is not None:  # not shown in the book
xgb_reg = xgboost.XGBRegressor(random_state=42)
xgb_reg.fit(X_train, y_train)
y_pred = xgb_reg.predict(X_val)
val_error = mean_squared_error(y_val, y_pred) # Not shown
print("Validation MSE:", val_error)           # Not shown

Validation MSE: 0.00400040950714611

In [52]:
if xgboost is not None:  # not shown in the book
xgb_reg.fit(X_train, y_train,
eval_set=[(X_val, y_val)], early_stopping_rounds=2)
y_pred = xgb_reg.predict(X_val)
val_error = mean_squared_error(y_val, y_pred)  # Not shown
print("Validation MSE:", val_error)            # Not shown

[0]	validation_0-rmse:0.22834
Will train until validation_0-rmse hasn't improved in 2 rounds.
[1]	validation_0-rmse:0.16224
[2]	validation_0-rmse:0.11843
[3]	validation_0-rmse:0.08760
[4]	validation_0-rmse:0.06848
[5]	validation_0-rmse:0.05709
[6]	validation_0-rmse:0.05297
[7]	validation_0-rmse:0.05129
[8]	validation_0-rmse:0.05155
[9]	validation_0-rmse:0.05211
Stopping. Best iteration:
[7]	validation_0-rmse:0.05129

Validation MSE: 0.0026308690413069744

In [53]:
%timeit xgboost.XGBRegressor().fit(X_train, y_train) if xgboost is not None else None

76.1 ms ± 5.64 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

In [54]:
%timeit GradientBoostingRegressor().fit(X_train, y_train)

20.8 ms ± 351 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)


# Exercise solutions¶

See Appendix A.

## 8. Voting Classifier¶

Exercise: Load the MNIST data and split it into a training set, a validation set, and a test set (e.g., use 50,000 instances for training, 10,000 for validation, and 10,000 for testing).

The MNIST dataset was loaded earlier.

In [55]:
from sklearn.model_selection import train_test_split

In [56]:
X_train_val, X_test, y_train_val, y_test = train_test_split(
mnist.data, mnist.target, test_size=10000, random_state=42)
X_train, X_val, y_train, y_val = train_test_split(
X_train_val, y_train_val, test_size=10000, random_state=42)


Exercise: Then train various classifiers, such as a Random Forest classifier, an Extra-Trees classifier, and an SVM.

In [57]:
from sklearn.ensemble import RandomForestClassifier, ExtraTreesClassifier
from sklearn.svm import LinearSVC
from sklearn.neural_network import MLPClassifier

In [58]:
random_forest_clf = RandomForestClassifier(n_estimators=100, random_state=42)
extra_trees_clf = ExtraTreesClassifier(n_estimators=100, random_state=42)
svm_clf = LinearSVC(max_iter=100, tol=20, random_state=42)
mlp_clf = MLPClassifier(random_state=42)

In [59]:
estimators = [random_forest_clf, extra_trees_clf, svm_clf, mlp_clf]
for estimator in estimators:
print("Training the", estimator)
estimator.fit(X_train, y_train)

Training the RandomForestClassifier(random_state=42)
Training the ExtraTreesClassifier(random_state=42)
Training the LinearSVC(max_iter=100, random_state=42, tol=20)
Training the MLPClassifier(random_state=42)

In [60]:
[estimator.score(X_val, y_val) for estimator in estimators]

Out[60]:
[0.9692, 0.9715, 0.859, 0.9639]

The linear SVM is far outperformed by the other classifiers. However, let's keep it for now since it may improve the voting classifier's performance.

Exercise: Next, try to combine them into an ensemble that outperforms them all on the validation set, using a soft or hard voting classifier.

In [61]:
from sklearn.ensemble import VotingClassifier

In [62]:
named_estimators = [
("random_forest_clf", random_forest_clf),
("extra_trees_clf", extra_trees_clf),
("svm_clf", svm_clf),
("mlp_clf", mlp_clf),
]

In [63]:
voting_clf = VotingClassifier(named_estimators)

In [64]:
voting_clf.fit(X_train, y_train)

Out[64]:
VotingClassifier(estimators=[('random_forest_clf',
RandomForestClassifier(random_state=42)),
('extra_trees_clf',
ExtraTreesClassifier(random_state=42)),
('svm_clf',
LinearSVC(max_iter=100, random_state=42, tol=20)),
('mlp_clf', MLPClassifier(random_state=42))])
In [65]:
voting_clf.score(X_val, y_val)

Out[65]:
0.9711
In [66]:
[estimator.score(X_val, y_val) for estimator in voting_clf.estimators_]

Out[66]:
[0.9692, 0.9715, 0.859, 0.9639]

Let's remove the SVM to see if performance improves. It is possible to remove an estimator by setting it to None using set_params() like this:

In [67]:
voting_clf.set_params(svm_clf=None)

Out[67]:
VotingClassifier(estimators=[('random_forest_clf',
RandomForestClassifier(random_state=42)),
('extra_trees_clf',
ExtraTreesClassifier(random_state=42)),
('svm_clf', None),
('mlp_clf', MLPClassifier(random_state=42))])

This updated the list of estimators:

In [68]:
voting_clf.estimators

Out[68]:
[('random_forest_clf', RandomForestClassifier(random_state=42)),
('extra_trees_clf', ExtraTreesClassifier(random_state=42)),
('svm_clf', None),
('mlp_clf', MLPClassifier(random_state=42))]

However, it did not update the list of trained estimators:

In [69]:
voting_clf.estimators_

Out[69]:
[RandomForestClassifier(random_state=42),
ExtraTreesClassifier(random_state=42),
LinearSVC(max_iter=100, random_state=42, tol=20),
MLPClassifier(random_state=42)]

So we can either fit the VotingClassifier again, or just remove the SVM from the list of trained estimators:

In [70]:
del voting_clf.estimators_[2]


Now let's evaluate the VotingClassifier again:

In [71]:
voting_clf.score(X_val, y_val)

Out[71]:
0.9735

A bit better! The SVM was hurting performance. Now let's try using a soft voting classifier. We do not actually need to retrain the classifier, we can just set voting to "soft":

In [72]:
voting_clf.voting = "soft"

In [73]:
voting_clf.score(X_val, y_val)

Out[73]:
0.9693

Nope, hard voting wins in this case.

Once you have found one, try it on the test set. How much better does it perform compared to the individual classifiers?

In [74]:
voting_clf.voting = "hard"
voting_clf.score(X_test, y_test)

Out[74]:
0.9706
In [75]:
[estimator.score(X_test, y_test) for estimator in voting_clf.estimators_]

Out[75]:
[0.9645, 0.9691, 0.9624]

The voting classifier only very slightly reduced the error rate of the best model in this case.

## 9. Stacking Ensemble¶

Exercise: Run the individual classifiers from the previous exercise to make predictions on the validation set, and create a new training set with the resulting predictions: each training instance is a vector containing the set of predictions from all your classifiers for an image, and the target is the image's class. Train a classifier on this new training set.

In [76]:
X_val_predictions = np.empty((len(X_val), len(estimators)), dtype=np.float32)

for index, estimator in enumerate(estimators):
X_val_predictions[:, index] = estimator.predict(X_val)

In [77]:
X_val_predictions

Out[77]:
array([[5., 5., 5., 5.],
[8., 8., 8., 8.],
[2., 2., 3., 2.],
...,
[7., 7., 7., 7.],
[6., 6., 6., 6.],
[7., 7., 7., 7.]], dtype=float32)
In [78]:
rnd_forest_blender = RandomForestClassifier(n_estimators=200, oob_score=True, random_state=42)
rnd_forest_blender.fit(X_val_predictions, y_val)

Out[78]:
RandomForestClassifier(n_estimators=200, oob_score=True, random_state=42)
In [79]:
rnd_forest_blender.oob_score_

Out[79]:
0.9689

You could fine-tune this blender or try other types of blenders (e.g., an MLPClassifier), then select the best one using cross-validation, as always.

Exercise: Congratulations, you have just trained a blender, and together with the classifiers they form a stacking ensemble! Now let's evaluate the ensemble on the test set. For each image in the test set, make predictions with all your classifiers, then feed the predictions to the blender to get the ensemble's predictions. How does it compare to the voting classifier you trained earlier?

In [80]:
X_test_predictions = np.empty((len(X_test), len(estimators)), dtype=np.float32)

for index, estimator in enumerate(estimators):
X_test_predictions[:, index] = estimator.predict(X_test)

In [81]:
y_pred = rnd_forest_blender.predict(X_test_predictions)

In [82]:
from sklearn.metrics import accuracy_score

In [83]:
accuracy_score(y_test, y_pred)

Out[83]:
0.9683

This stacking ensemble does not perform as well as the voting classifier we trained earlier, it's not quite as good as the best individual classifier.

In [ ]: