Morten Hjorth-Jensen
1081 lines
37 KiB
Plaintext
1081 lines
37 KiB
Plaintext
======= Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods =======
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As stated previously and seen in many of the examples discussed in the previous chapter about
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a single decision tree, we often end up overfitting our training
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data. This normally means that we have a high variance. Can we reduce
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the variance of a statistical learning method?
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This leads us to a set of different methods that can combine different
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machine learning algorithms or just use one of them to construct
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forests and jungles of trees, homogeneous ones or heterogenous
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ones. These methods are recognized by different names which we will
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try to explain here. These are
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o Voting classifiers
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o Bagging and Pasting
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o Random forests
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o Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)
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We discuss these methods here.
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=== An Overview of Ensemble Methods ===
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#FIGURE: [DataFiles/ensembleoverview.png, width=600 frac=0.8]
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===== Bagging =====
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The _plain_ decision trees suffer from high
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variance. This means that if we split the training data into two parts
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at random, and fit a decision tree to both halves, the results that we
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get could be quite different. In contrast, a procedure with low
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variance will yield similar results if applied repeatedly to distinct
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data sets; linear regression tends to have low variance, if the ratio
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of $n$ to $p$ is moderately large.
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_Bootstrap aggregation_, or just _bagging_, is a
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general-purpose procedure for reducing the variance of a statistical
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learning method.
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Bagging typically results in improved accuracy
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over prediction using a single tree. Unfortunately, however, it can be
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difficult to interpret the resulting model. Recall that one of the
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advantages of decision trees is the attractive and easily interpreted
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diagram that results.
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However, when we bag a large number of trees, it is no longer
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possible to represent the resulting statistical learning procedure
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using a single tree, and it is no longer clear which variables are
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most important to the procedure. Thus, bagging improves prediction
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accuracy at the expense of interpretability. Although the collection
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of bagged trees is much more difficult to interpret than a single
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tree, one can obtain an overall summary of the importance of each
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predictor using the MSE (for bagging regression trees) or the Gini
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index (for bagging classification trees). In the case of bagging
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regression trees, we can record the total amount that the MSE is
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decreased due to splits over a given predictor, averaged over all $B$ possible
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trees. A large value indicates an important predictor. Similarly, in
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the context of bagging classification trees, we can add up the total
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amount that the Gini index is decreased by splits over a given
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predictor, averaged over all $B$ trees.
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!bc pycod
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heads_proba = 0.51
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coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
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cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
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plt.figure(figsize=(8,3.5))
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plt.plot(cumulative_heads_ratio)
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plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
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plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
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plt.xlabel("Number of coin tosses")
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plt.ylabel("Heads ratio")
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plt.legend(loc="lower right")
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plt.axis([0, 10000, 0.42, 0.58])
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save_fig("votingsimple")
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plt.show()
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!ec
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!bc pycod
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import make_moons
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X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
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X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
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from sklearn.ensemble import RandomForestClassifier
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from sklearn.ensemble import VotingClassifier
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from sklearn.linear_model import LogisticRegression
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from sklearn.svm import SVC
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log_clf = LogisticRegression(solver="liblinear", random_state=42)
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rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
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svm_clf = SVC(gamma="auto", random_state=42)
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voting_clf = VotingClassifier(
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estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
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voting='hard')
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voting_clf.fit(X_train, y_train)
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from sklearn.metrics import accuracy_score
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for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
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clf.fit(X_train, y_train)
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y_pred = clf.predict(X_test)
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print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
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log_clf = LogisticRegression(solver="liblinear", random_state=42)
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rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
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svm_clf = SVC(gamma="auto", probability=True, random_state=42)
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voting_clf = VotingClassifier(
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estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
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voting='soft')
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voting_clf.fit(X_train, y_train)
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from sklearn.metrics import accuracy_score
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for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
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clf.fit(X_train, y_train)
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y_pred = clf.predict(X_test)
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print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
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!ec
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!bc pycod
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import make_moons
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X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
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X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
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from sklearn.ensemble import RandomForestClassifier
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from sklearn.ensemble import VotingClassifier
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from sklearn.linear_model import LogisticRegression
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from sklearn.svm import SVC
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log_clf = LogisticRegression(random_state=42)
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rnd_clf = RandomForestClassifier(random_state=42)
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svm_clf = SVC(random_state=42)
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voting_clf = VotingClassifier(
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estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
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voting='hard')
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voting_clf.fit(X_train, y_train)
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!ec
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!bc pycod
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from sklearn.metrics import accuracy_score
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for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
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clf.fit(X_train, y_train)
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y_pred = clf.predict(X_test)
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print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
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!ec
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!bc pycod
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log_clf = LogisticRegression(random_state=42)
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rnd_clf = RandomForestClassifier(random_state=42)
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svm_clf = SVC(probability=True, random_state=42)
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voting_clf = VotingClassifier(
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estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
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voting='soft')
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voting_clf.fit(X_train, y_train)
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!ec
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!bc pycod
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from sklearn.metrics import accuracy_score
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for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
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clf.fit(X_train, y_train)
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y_pred = clf.predict(X_test)
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print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
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!ec
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===== Bagging Examples =====
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!bc pycod
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from sklearn.ensemble import BaggingClassifier
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from sklearn.tree import DecisionTreeClassifier
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bag_clf = BaggingClassifier(
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DecisionTreeClassifier(random_state=42), n_estimators=500,
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max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)
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bag_clf.fit(X_train, y_train)
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y_pred = bag_clf.predict(X_test)
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!ec
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!bc pycod
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from sklearn.metrics import accuracy_score
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print(accuracy_score(y_test, y_pred))
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!ec
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!bc pycod
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tree_clf = DecisionTreeClassifier(random_state=42)
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tree_clf.fit(X_train, y_train)
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y_pred_tree = tree_clf.predict(X_test)
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print(accuracy_score(y_test, y_pred_tree))
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!ec
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!bc pycod
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from matplotlib.colors import ListedColormap
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def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
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x1s = np.linspace(axes[0], axes[1], 100)
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x2s = np.linspace(axes[2], axes[3], 100)
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x1, x2 = np.meshgrid(x1s, x2s)
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X_new = np.c_[x1.ravel(), x2.ravel()]
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y_pred = clf.predict(X_new).reshape(x1.shape)
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custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
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plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
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if contour:
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custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
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plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
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plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
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plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
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plt.axis(axes)
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plt.xlabel(r"$x_1$", fontsize=18)
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plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
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plt.figure(figsize=(11,4))
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plt.subplot(121)
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plot_decision_boundary(tree_clf, X, y)
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plt.title("Decision Tree", fontsize=14)
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plt.subplot(122)
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plot_decision_boundary(bag_clf, X, y)
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plt.title("Decision Trees with Bagging", fontsize=14)
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save_fig("baggingtree")
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plt.show()
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!ec
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=== Making your own Bootstrap: Changing the Level of the Decision Tree ===
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Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
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a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points $n$).
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!bc pycod
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.pipeline import make_pipeline
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from sklearn.utils import resample
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from sklearn.tree import DecisionTreeRegressor
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n = 100
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n_boostraps = 100
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maxdepth = 8
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# Make data set.
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x = np.linspace(-3, 3, n).reshape(-1, 1)
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y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
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error = np.zeros(maxdepth)
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bias = np.zeros(maxdepth)
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variance = np.zeros(maxdepth)
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polydegree = np.zeros(maxdepth)
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X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
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from sklearn.preprocessing import StandardScaler
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scaler = StandardScaler()
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scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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# we produce a simple tree first as benchmark
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simpletree = DecisionTreeRegressor(max_depth=3)
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simpletree.fit(X_train_scaled, y_train)
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simpleprediction = simpletree.predict(X_test_scaled)
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for degree in range(1,maxdepth):
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model = DecisionTreeRegressor(max_depth=degree)
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y_pred = np.empty((y_test.shape[0], n_boostraps))
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for i in range(n_boostraps):
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x_, y_ = resample(X_train_scaled, y_train)
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model.fit(x_, y_)
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y_pred[:, i] = model.predict(X_test_scaled)#.ravel()
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polydegree[degree] = degree
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error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
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bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
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variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
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print('Polynomial degree:', degree)
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print('Error:', error[degree])
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print('Bias^2:', bias[degree])
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print('Var:', variance[degree])
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print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
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mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2)
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print(mse_simpletree)
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plt.xlim(1,maxdepth)
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plt.plot(polydegree, error, label='MSE')
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plt.plot(polydegree, bias, label='bias')
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plt.plot(polydegree, variance, label='Variance')
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plt.legend()
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save_fig("baggingboot")
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plt.show()
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!ec
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===== Random forests =====
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Random forests provide an improvement over bagged trees by way of a
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small tweak that decorrelates the trees.
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As in bagging, we build a
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number of decision trees on bootstrapped training samples. But when
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building these decision trees, each time a split in a tree is
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considered, a random sample of $m$ predictors is chosen as split
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candidates from the full set of $p$ predictors. The split is allowed to
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use only one of those $m$ predictors.
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A fresh sample of $m$ predictors is
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taken at each split, and typically we choose
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!bt
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\[
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m\approx \sqrt{p}.
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\]
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!et
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In building a random forest, at
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each split in the tree, the algorithm is not even allowed to consider
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a majority of the available predictors.
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The reason for this is rather clever. Suppose that there is one very
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strong predictor in the data set, along with a number of other
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moderately strong predictors. Then in the collection of bagged
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variable importance random forest trees, most or all of the trees will
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use this strong predictor in the top split. Consequently, all of the
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bagged trees will look quite similar to each other. Hence the
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predictions from the bagged trees will be highly correlated.
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Unfortunately, averaging many highly correlated quantities does not
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lead to as large of a reduction in variance as averaging many
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uncorrelated quantities. In particular, this means that bagging will
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not lead to a substantial reduction in variance over a single tree in
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this setting.
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The algorithm described here can be applied to both classification and regression problems.
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We will grow of forest of say $B$ trees.
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o For $b=1:B$
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* Draw a bootstrap sample from the training data organized in our $\bm{X}$ matrix.
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* We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached
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o we select $m \le p$ variables at random from the $p$ predictors/features
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o pick the best split point among the $m$ features using for example the CART algorithm and create a new node
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o split the node into daughter nodes
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o Output then the ensemble of trees $\{T_b\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem.
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!bc pycod
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import load_breast_cancer
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from sklearn.svm import SVC
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from sklearn.linear_model import LogisticRegression
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from sklearn.tree import DecisionTreeClassifier
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from sklearn.ensemble import BaggingClassifier
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# Load the data
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cancer = load_breast_cancer()
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X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
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print(X_train.shape)
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print(X_test.shape)
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# Logistic Regression
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logreg = LogisticRegression(solver='lbfgs')
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logreg.fit(X_train, y_train)
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print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
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# Support vector machine
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svm = SVC(gamma='auto', C=100)
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svm.fit(X_train, y_train)
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print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
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# Decision Trees
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deep_tree_clf = DecisionTreeClassifier(max_depth=None)
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deep_tree_clf.fit(X_train, y_train)
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print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
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#now scale the data
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from sklearn.preprocessing import StandardScaler
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scaler = StandardScaler()
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scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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# Logistic Regression
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logreg.fit(X_train_scaled, y_train)
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print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
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# Support Vector Machine
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svm.fit(X_train_scaled, y_train)
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print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
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# Decision Trees
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deep_tree_clf.fit(X_train_scaled, y_train)
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print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
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from sklearn.ensemble import RandomForestClassifier
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from sklearn.preprocessing import LabelEncoder
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from sklearn.model_selection import cross_validate
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# Data set not specificied
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#Instantiate the model with 500 trees and entropy as splitting criteria
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Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
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Random_Forest_model.fit(X_train_scaled, y_train)
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#Cross validation
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accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
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print(accuracy)
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print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
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import scikitplot as skplt
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y_pred = Random_Forest_model.predict(X_test_scaled)
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skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
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plt.show()
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y_probas = Random_Forest_model.predict_proba(X_test_scaled)
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skplt.metrics.plot_roc(y_test, y_probas)
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plt.show()
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skplt.metrics.plot_cumulative_gain(y_test, y_probas)
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plt.show()
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!ec
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Recall that the cumulative gains curve shows the percentage of the
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overall number of cases in a given category *gained* by targeting a
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percentage of the total number of cases.
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Similarly, the receiver operating characteristic curve, or ROC curve,
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displays the diagnostic ability of a binary classifier system as its
|
|
discrimination threshold is varied. It plots the true positive rate against the false positive rate.
|
|
|
|
|
|
=== Compare Bagging on Trees with Random Forests ===
|
|
!bc pycod
|
|
bag_clf = BaggingClassifier(
|
|
DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
|
|
n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
|
|
!ec
|
|
|
|
|
|
|
|
!bc pycod
|
|
bag_clf.fit(X_train, y_train)
|
|
y_pred = bag_clf.predict(X_test)
|
|
from sklearn.ensemble import RandomForestClassifier
|
|
rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
|
|
rnd_clf.fit(X_train, y_train)
|
|
y_pred_rf = rnd_clf.predict(X_test)
|
|
np.sum(y_pred == y_pred_rf) / len(y_pred)
|
|
!ec
|
|
|
|
|
|
|
|
|
|
===== Boosting, a Bird's Eye View =====
|
|
|
|
The basic idea is to combine weak classifiers in order to create a good
|
|
classifier. With a weak classifier we often intend a classifier which
|
|
produces results which are only slightly better than we would get by
|
|
random guesses.
|
|
|
|
This is done by applying in an iterative way a weak (or a standard
|
|
classifier like decision trees) to modify the data. In each iteration
|
|
we emphasize those observations which are misclassified by weighting
|
|
them with a factor.
|
|
|
|
|
|
|
|
Boosting is a way of fitting an additive expansion in a set of
|
|
elementary basis functions like for example some simple polynomials.
|
|
Assume for example that we have a function
|
|
!bt
|
|
\[
|
|
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
|
\]
|
|
!et
|
|
|
|
where $\beta_m$ are the expansion parameters to be determined in a
|
|
minimization process and $b(x;\gamma_m)$ are some simple functions of
|
|
the multivariable parameter $x$ which is characterized by the
|
|
parameters $\gamma_m$.
|
|
|
|
As an example, consider the Sigmoid function we used in logistic
|
|
regression. In that case, we can translate the function
|
|
$b(x;\gamma_m)$ into the Sigmoid function
|
|
|
|
|
|
!bt
|
|
\[
|
|
\sigma(t) = \frac{1}{1+\exp{(-t)}},
|
|
\]
|
|
!et
|
|
|
|
where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and
|
|
$\gamma_1$ were determined by the Logistic Regression fitting
|
|
algorithm.
|
|
|
|
As another example, consider the cost function we defined for linear regression
|
|
!bt
|
|
\[
|
|
C(\bm{y},\bm{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
\]
|
|
!et
|
|
|
|
In this case the function $f(x)$ was replaced by the design matrix
|
|
$\bm{X}$ and the unknown linear regression parameters $\bm{\beta}$,
|
|
that is $\bm{f}=\bm{X}\bm{\beta}$. In linear regression we can
|
|
simply invert a matrix and obtain the parameters $\beta$ by
|
|
|
|
!bt
|
|
\[
|
|
\bm{\beta}=\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}.
|
|
\]
|
|
!et
|
|
|
|
In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$.
|
|
|
|
|
|
=== Iterative Fitting, Regression and Squared-error Cost Function ===
|
|
|
|
The way we proceed is as follows (here we specialize to the squared-error cost function)
|
|
|
|
o Establish a cost function, here $\cal{C}(\bm{y},\bm{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$.
|
|
o Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.
|
|
o For $m=1:M$
|
|
o minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$
|
|
o This gives the optimal values $\beta_m$ and $\gamma_m$
|
|
o Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$
|
|
|
|
We could use any of the algorithms we have discussed till now. If we
|
|
use trees, $\gamma$ parameterizes the split variables and split points
|
|
at the internal nodes, and the predictions at the terminal nodes.
|
|
|
|
|
|
|
|
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
|
|
|
|
For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$.
|
|
|
|
This means that for every iteration $m$, we need to optimize
|
|
|
|
!bt
|
|
\[
|
|
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
|
|
\]
|
|
!et
|
|
|
|
We start our iteration by simply setting $f_0(x)=0$.
|
|
Taking the derivatives with respect to $\beta$ and $\gamma$ we obtain
|
|
!bt
|
|
\[
|
|
\frac{\partial \cal{C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
|
|
\]
|
|
!et
|
|
and
|
|
!bt
|
|
\[
|
|
\frac{\partial \cal{C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
|
|
\]
|
|
!et
|
|
We can then rewrite these equations as (defining $\bm{w}=\bm{e}+\gamma \bm{x})$ with $\bm{e}$ being the unit vector)
|
|
!bt
|
|
\[
|
|
\gamma \bm{w}^T(\bm{y}-\beta\gamma \bm{w})=0,
|
|
\]
|
|
!et
|
|
which gives us $\beta = \bm{w}^T\bm{y}/(\bm{w}^T\bm{w})$. Similarly we have
|
|
!bt
|
|
\[
|
|
\beta\gamma \bm{x}^T(\bm{y}-\beta(1+\gamma \bm{x}))=0,
|
|
\]
|
|
!et
|
|
|
|
which leads to $\gamma =(\bm{x}^T\bm{y}-\beta\bm{x}^T\bm{e})/(\beta\bm{x}^T\bm{x})$. Inserting
|
|
for $\beta$ gives us an equation for $\gamma$. This is a non-linear equation in the unknown $\gamma$ and has to be solved numerically.
|
|
|
|
The solution to these two equations gives us in turn $\beta_1$ and $\gamma_1$ leading to the new expression for $f_1(x)$ as
|
|
$f_1(x) = \beta_1(1+\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$.
|
|
|
|
|
|
|
|
=== Iterative Fitting, Classification and AdaBoost ===
|
|
|
|
Let us consider a binary classification problem with two outcomes $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of
|
|
observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values
|
|
$\{-1,1\}$.
|
|
|
|
The error rate of the training sample is then
|
|
|
|
!bt
|
|
\[
|
|
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
|
|
\]
|
|
!et
|
|
|
|
The iterative procedure starts with defining a weak classifier whose
|
|
error rate is barely better than random guessing. The iterative
|
|
procedure in boosting is to sequentially apply a weak
|
|
classification algorithm to repeatedly modified versions of the data
|
|
producing a sequence of weak classifiers $G_m(x)$.
|
|
|
|
Here we will express our function $f(x)$ in terms of $G(x)$. That is
|
|
!bt
|
|
\[
|
|
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
|
\]
|
|
!et
|
|
will be a function of
|
|
!bt
|
|
\[
|
|
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
|
|
\]
|
|
!et
|
|
|
|
|
|
|
|
In our iterative procedure we define thus
|
|
!bt
|
|
\[
|
|
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
|
|
\]
|
|
!et
|
|
|
|
The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
|
|
exponential cost/loss function defined as
|
|
!bt
|
|
\[
|
|
C(\bm{y},\bm{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
|
|
\]
|
|
!et
|
|
|
|
We optimize $\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.
|
|
This is normally done in two steps. Let us however first rewrite the cost function as
|
|
|
|
!bt
|
|
\[
|
|
C(\bm{y},\bm{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
|
|
\]
|
|
!et
|
|
where we have defined $w_i^m= \exp{(-y_if_{m-1}(x_i))}$.
|
|
|
|
|
|
|
|
First, for any $\beta > 0$, we optimize $G$ by setting
|
|
!bt
|
|
\[
|
|
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
|
|
\]
|
|
!et
|
|
which is the classifier that minimizes the weighted error rate in predicting $y$.
|
|
|
|
We can do this by rewriting
|
|
!bt
|
|
\[
|
|
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
|
|
\]
|
|
!et
|
|
which can be rewritten as
|
|
!bt
|
|
\[
|
|
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
|
|
\]
|
|
!et
|
|
which leads to
|
|
!bt
|
|
\[
|
|
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
|
|
\]
|
|
!et
|
|
where we have redefined the error as
|
|
!bt
|
|
\[
|
|
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
|
|
\]
|
|
!et
|
|
which leads to an update of
|
|
!bt
|
|
\[
|
|
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
|
|
\]
|
|
!et
|
|
This leads to the new weights
|
|
!bt
|
|
\[
|
|
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
|
|
\]
|
|
!et
|
|
|
|
|
|
=== Adaptive boosting: AdaBoost, Basic Algorithm ===
|
|
|
|
The algorithm here is rather straightforward. Assume that our weak
|
|
classifier is a decision tree and we consider a binary set of outputs
|
|
with $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of
|
|
observations. Our design matrix is given in terms of the
|
|
feature/predictor vectors
|
|
$\bm{X}=[\bm{x}_0\bm{x}_1\dots\bm{x}_{p-1}]$. Finally, we define also a
|
|
classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\bm{y}$.
|
|
|
|
We have already defined the misclassification error $\mathrm{err}$ as
|
|
!bt
|
|
\[
|
|
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
|
|
\]
|
|
!et
|
|
where the function $I()$ is one if we misclassify and zero if we classify correctly.
|
|
|
|
|
|
With the above definitions we are now ready to set up the algorithm for AdaBoost.
|
|
The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
|
|
o We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\dots n-1$. It is easy to see that we must have $\sum_{i=0}^{n-1}w_i = 1$.
|
|
o We rewrite the misclassification error as
|
|
!bt
|
|
\[
|
|
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
|
|
\]
|
|
!et
|
|
o Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.
|
|
o Fit then a given classifier to the training set using the weights $w_i$.
|
|
o Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.
|
|
o Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$
|
|
o Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$.
|
|
o Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$.
|
|
|
|
For the iterations with $m \le 2$ the weights are modified
|
|
individually at each steps. The observations which were misclassified
|
|
at iteration $m-1$ have a weight which is larger than those which were
|
|
classified properly. As this proceeds, the observations which were
|
|
difficult to classifiy correctly are given a larger influence. Each
|
|
new classification step $m$ is then forced to concentrate on those
|
|
observations that are missed in the previous iterations.
|
|
|
|
|
|
|
|
|
|
Using _Scikit-Learn_ it is easy to apply the adaptive boosting algorithm, as done here.
|
|
|
|
!bc pycod
|
|
from sklearn.ensemble import AdaBoostClassifier
|
|
|
|
ada_clf = AdaBoostClassifier(
|
|
DecisionTreeClassifier(max_depth=1), n_estimators=200,
|
|
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
|
|
ada_clf.fit(X_train, y_train)
|
|
|
|
from sklearn.ensemble import AdaBoostClassifier
|
|
|
|
ada_clf = AdaBoostClassifier(
|
|
DecisionTreeClassifier(max_depth=1), n_estimators=200,
|
|
algorithm="SAMME.R", learning_rate=0.5, random_state=42)
|
|
ada_clf.fit(X_train_scaled, y_train)
|
|
y_pred = ada_clf.predict(X_test_scaled)
|
|
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
|
|
plt.show()
|
|
y_probas = ada_clf.predict_proba(X_test_scaled)
|
|
skplt.metrics.plot_roc(y_test, y_probas)
|
|
plt.show()
|
|
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
|
|
plt.show()
|
|
!ec
|
|
|
|
|
|
|
|
===== Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent =====
|
|
|
|
Gradient boosting is again a similar technique to Adaptive boosting,
|
|
it combines so-called weak classifiers or regressors into a strong
|
|
method via a series of iterations.
|
|
|
|
In order to understand the method, let us illustrate its basics by
|
|
bringing back the essential steps in linear regression, where our cost
|
|
function was the least squares function.
|
|
|
|
|
|
We start again with our cost function $\cal{C}(\bm{y}m\bm{f})=\sum_{i=0}^{n-1}\cal{L}(y_i, f(x_i))$ where we want to minimize
|
|
This means that for every iteration, we need to optimize
|
|
|
|
!bt
|
|
\[
|
|
(\hat{\bm{f}}) = \mathrm{argmin}_{\bm{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
\]
|
|
!et
|
|
|
|
We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as
|
|
!bt
|
|
\[
|
|
f_M(x) = \sum_{m=0}^M h_m(x).
|
|
\]
|
|
!et
|
|
|
|
In the steepest decent approach we approximate $h_m(x) = -\rho_m g_m(x)$, where $\rho_m$ is a scalar and $g_m(x)$ the gradient defined as
|
|
!bt
|
|
\[
|
|
g_m(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
|
|
\]
|
|
!et
|
|
|
|
With the new gradient we can update $f_m(x) = f_{m-1}(x) -\rho_m g_m(x)$. Using the above squared-error function we see that
|
|
the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.
|
|
|
|
Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have
|
|
!bt
|
|
\[
|
|
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
|
|
\]
|
|
!et
|
|
|
|
|
|
Optimizing with respect to $\rho$ we obtain (taking the derivative) that $\rho_1 = -1/2$. We have then that
|
|
!bt
|
|
\[
|
|
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
|
|
\]
|
|
!et
|
|
We can then proceed and compute
|
|
!bt
|
|
\[
|
|
g_2(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
|
|
\]
|
|
!et
|
|
and find a new value for $\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called _gradient boosting_.
|
|
|
|
|
|
Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,
|
|
so we do not learn a function that can generalize. However, we can modify the algorithm by
|
|
fitting a weak learner to approximate the negative gradient signal.
|
|
|
|
Suppose we have a cost function $C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function
|
|
!bt
|
|
\[
|
|
C(\bm{y},\bm{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
\]
|
|
!et
|
|
|
|
The way we proceed in an iterative fashion is to
|
|
o Initialize our estimate $f_0(x)$.
|
|
o For $m=1:M$, we
|
|
o compute the negative gradient vector $\bm{u}_m = -\partial C(\bm{y},\bm{f})/\partial \bm{f}(x)$ at $f(x) = f_{m-1}(x)$;
|
|
o fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;
|
|
o update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;
|
|
o The final estimate is then $f_M(x) = \sum_{m=1}^M h_m(u_m,x)$.
|
|
|
|
|
|
|
|
===== Gradient Boosting, Examples of Regression =====
|
|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
from sklearn.model_selection import train_test_split
|
|
from sklearn.ensemble import GradientBoostingRegressor
|
|
from sklearn.preprocessing import StandardScaler
|
|
import scikitplot as skplt
|
|
from sklearn.metrics import mean_squared_error
|
|
|
|
n = 100
|
|
maxdegree = 6
|
|
|
|
# Make data set.
|
|
x = np.linspace(-3, 3, n).reshape(-1, 1)
|
|
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
|
|
|
|
error = np.zeros(maxdegree)
|
|
bias = np.zeros(maxdegree)
|
|
variance = np.zeros(maxdegree)
|
|
polydegree = np.zeros(maxdegree)
|
|
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
|
|
scaler = StandardScaler()
|
|
scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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for degree in range(1,maxdegree):
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model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)
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model.fit(X_train_scaled,y_train)
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y_pred = model.predict(X_test_scaled)
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polydegree[degree] = degree
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error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
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bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
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variance[degree] = np.mean( np.var(y_pred) )
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print('Max depth:', degree)
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print('Error:', error[degree])
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print('Bias^2:', bias[degree])
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print('Var:', variance[degree])
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print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
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plt.xlim(1,maxdegree-1)
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plt.plot(polydegree, error, label='Error')
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plt.plot(polydegree, bias, label='bias')
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plt.plot(polydegree, variance, label='Variance')
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plt.legend()
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save_fig("gdregression")
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plt.show()
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!ec
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===== Gradient Boosting, Classification Example =====
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!bc pycod
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.model_selection import train_test_split
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from sklearn.datasets import load_breast_cancer
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import scikitplot as skplt
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from sklearn.ensemble import GradientBoostingClassifier
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from sklearn.model_selection import cross_validate
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|
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# Load the data
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cancer = load_breast_cancer()
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X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
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print(X_train.shape)
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print(X_test.shape)
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#now scale the data
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from sklearn.preprocessing import StandardScaler
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scaler = StandardScaler()
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scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
|
|
|
|
gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)
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gd_clf.fit(X_train_scaled, y_train)
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|
#Cross validation
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accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
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print(accuracy)
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print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
|
|
|
|
import scikitplot as skplt
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|
y_pred = gd_clf.predict(X_test_scaled)
|
|
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
|
|
save_fig("gdclassiffierconfusion")
|
|
plt.show()
|
|
y_probas = gd_clf.predict_proba(X_test_scaled)
|
|
skplt.metrics.plot_roc(y_test, y_probas)
|
|
save_fig("gdclassiffierroc")
|
|
plt.show()
|
|
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
|
|
save_fig("gdclassiffiercgain")
|
|
plt.show()
|
|
!ec
|
|
|
|
|
|
|
|
===== XGBoost: Extreme Gradient Boosting =====
|
|
|
|
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|
"XGBoost":"https://github.com/dmlc/xgboost" or Extreme Gradient
|
|
Boosting, is an optimized distributed gradient boosting library
|
|
designed to be highly efficient, flexible and portable. It implements
|
|
machine learning algorithms under the Gradient Boosting
|
|
framework. XGBoost provides a parallel tree boosting that solve many
|
|
data science problems in a fast and accurate way. See the "article by Chen and Guestrin":"https://arxiv.org/abs/1603.02754".
|
|
|
|
The authors design and build a highly scalable end-to-end tree
|
|
boosting system. It has a theoretically justified weighted quantile
|
|
sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.
|
|
|
|
It is now the algorithm which wins essentially all ML competitions!!!
|
|
|
|
!split
|
|
===== Regression Case =====
|
|
|
|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
from sklearn.model_selection import train_test_split
|
|
import xgboost as xgb
|
|
from sklearn.preprocessing import StandardScaler
|
|
import scikitplot as skplt
|
|
from sklearn.metrics import mean_squared_error
|
|
|
|
n = 100
|
|
maxdegree = 6
|
|
|
|
# Make data set.
|
|
x = np.linspace(-3, 3, n).reshape(-1, 1)
|
|
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
|
|
|
|
error = np.zeros(maxdegree)
|
|
bias = np.zeros(maxdegree)
|
|
variance = np.zeros(maxdegree)
|
|
polydegree = np.zeros(maxdegree)
|
|
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
|
|
scaler = StandardScaler()
|
|
scaler.fit(X_train)
|
|
X_train_scaled = scaler.transform(X_train)
|
|
X_test_scaled = scaler.transform(X_test)
|
|
|
|
for degree in range(maxdegree):
|
|
model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)
|
|
|
|
model.fit(X_train_scaled,y_train)
|
|
y_pred = model.predict(X_test_scaled)
|
|
polydegree[degree] = degree
|
|
error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
|
|
bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
|
|
variance[degree] = np.mean( np.var(y_pred) )
|
|
print('Max depth:', degree)
|
|
print('Error:', error[degree])
|
|
print('Bias^2:', bias[degree])
|
|
print('Var:', variance[degree])
|
|
print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
|
|
|
|
plt.xlim(1,maxdegree-1)
|
|
plt.plot(polydegree, error, label='Error')
|
|
plt.plot(polydegree, bias, label='bias')
|
|
plt.plot(polydegree, variance, label='Variance')
|
|
plt.legend()
|
|
plt.show()
|
|
|
|
!ec
|
|
|
|
|
|
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
|
|
!bc pycod
|
|
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
from sklearn.model_selection import train_test_split
|
|
from sklearn.datasets import load_breast_cancer
|
|
from sklearn.preprocessing import LabelEncoder
|
|
from sklearn.model_selection import cross_validate
|
|
import scikitplot as skplt
|
|
import xgboost as xgb
|
|
# Load the data
|
|
cancer = load_breast_cancer()
|
|
|
|
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
|
print(X_train.shape)
|
|
print(X_test.shape)
|
|
#now scale the data
|
|
from sklearn.preprocessing import StandardScaler
|
|
scaler = StandardScaler()
|
|
scaler.fit(X_train)
|
|
X_train_scaled = scaler.transform(X_train)
|
|
X_test_scaled = scaler.transform(X_test)
|
|
|
|
xg_clf = xgb.XGBClassifier()
|
|
xg_clf.fit(X_train_scaled,y_train)
|
|
|
|
y_test = xg_clf.predict(X_test_scaled)
|
|
|
|
print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
|
|
|
|
import scikitplot as skplt
|
|
y_pred = xg_clf.predict(X_test_scaled)
|
|
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
|
|
save_fig("xdclassiffierconfusion")
|
|
plt.show()
|
|
y_probas = xg_clf.predict_proba(X_test_scaled)
|
|
skplt.metrics.plot_roc(y_test, y_probas)
|
|
save_fig("xdclassiffierroc")
|
|
plt.show()
|
|
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
|
|
save_fig("gdclassiffiercgain")
|
|
plt.show()
|
|
|
|
|
|
xgb.plot_tree(xg_clf,num_trees=0)
|
|
plt.rcParams['figure.figsize'] = [50, 10]
|
|
save_fig("xgtree")
|
|
plt.show()
|
|
|
|
xgb.plot_importance(xg_clf)
|
|
plt.rcParams['figure.figsize'] = [5, 5]
|
|
save_fig("xgparams")
|
|
plt.show()
|
|
|
|
!ec
|
|
|
|
|
|
|
|
|
|
|
|
|