參考：作者的Jupyter Notebook
Chapter 6 – Decision Trees
- 保存圖片
from __future__ import division, print_function, unicode_literals import numpy as np import matplotlib as mpl import matplotlib.pyplot as plt import os np.random.seed(42) mpl.rc('axes', labelsize=14) mpl.rc('xtick', labelsize=12) mpl.rc('ytick', labelsize=12) # Where to save the figures PROJECT_ROOT_DIR = "images" CHAPTER_ID = "decision_trees" def save_fig(fig_id, tight_layout=True): path = os.path.join(PROJECT_ROOT_DIR, CHAPTER_ID, fig_id + ".png") print("Saving figure", fig_id) if tight_layout: plt.tight_layout() plt.savefig(path, format='png', dpi=600)
決策樹訓練和可視化
-
要了解決策樹，讓我們先構建一個決策樹，看看它是如何做出預測的，下面的代碼在鳶尾花資料集（見第4章）上訓練了一個DecisionTreeClassifier：
from sklearn.datasets import load_iris from sklearn.tree import DecisionTreeClassifier iris = load_iris() X = iris.data[:, 2:] # petal length and width y = iris.target tree_clf = DecisionTreeClassifier(max_depth=2, random_state=42) tree_clf.fit(X, y) #print(tree_clf.fit(X, y)) -
要將決策樹可視化，首先，使用export_graphviz（）方法輸出一個圖形定義檔案，命名為iris_tree.dot：
from sklearn.tree import export_graphviz export_graphviz( tree_clf, out_file=image_path("iris_tree.dot"), feature_names=iris.feature_names[2:], class_names=iris.target_names, rounded=True, filled=True ) #下面這行命令將.dot檔案轉換為.png影像檔案： #$ dot -Tpng iris_tree.dot -o iris_tree.png
做出預測
```
from matplotlib.colors import ListedColormap
def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=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 not iris:
custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
if plot_training:
plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica")
plt.axis(axes)
if iris:
plt.xlabel("Petal length", fontsize=14)
plt.ylabel("Petal width", fontsize=14)
else:
plt.xlabel(r"$x_1$", fontsize=18)
plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
if legend:
plt.legend(loc="lower right", fontsize=14)
plt.figure(figsize=(8, 4))
plot_decision_boundary(tree_clf, X, y)
plt.plot([2.45, 2.45], [0, 3], "k-", linewidth=2)
plt.plot([2.45, 7.5], [1.75, 1.75], "k--", linewidth=2)
plt.plot([4.95, 4.95], [0, 1.75], "k:", linewidth=2)
plt.plot([4.85, 4.85], [1.75, 3], "k:", linewidth=2)
plt.text(1.40, 1.0, "Depth=0", fontsize=15)
plt.text(3.2, 1.80, "Depth=1", fontsize=13)
plt.text(4.05, 0.5, "(Depth=2)", fontsize=11)
save_fig("decision_tree_decision_boundaries_plot")
plt.show()
```
估算類別概率
- 決策樹同樣可以估算某個實體屬于特定類別k的概率
#print(tree_clf.predict_proba([[5, 1.5]])) #print(tree_clf.predict([[5, 1.5]]))0
CART訓練演算法
Scikit-Learn使用的是分類與回歸樹（Classification And Regression Tree，簡稱CART）演算法來訓練決策樹（也叫作“生長”樹），
計算復雜度
基尼不純度還是資訊熵
正則化超引數
```
from sklearn.datasets import make_moons
Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)
deep_tree_clf1 = DecisionTreeClassifier(random_state=42)
deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)
deep_tree_clf1.fit(Xm, ym)
deep_tree_clf2.fit(Xm, ym)
plt.figure(figsize=(11, 4))
plt.subplot(121)
plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
plt.title("No restrictions", fontsize=16)
plt.subplot(122)
plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14)
save_fig("min_samples_leaf_plot正則化超引數")
plt.show()
```
左圖使用默認引數（即無約束）來訓練決策樹，右圖的決策樹應用min_samples_leaf=4進行訓練，很明顯，左圖模型過度擬合，右圖的泛化效果更佳，
回歸
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決策樹也可以執行回歸任務，我們用Scikit_Learn的DecisionTreeRegressor來構建一個回歸樹，在一個帶噪聲的二次資料集上進行訓練，其中max_depth=2：
np.random.seed(42) m = 200 X = np.random.rand(m, 1) y = 4 * (X - 0.5) ** 2 y = y + np.random.randn(m, 1) / 10 from sklearn.tree import DecisionTreeRegressor tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42) tree_reg.fit(X, y) #print(tree_reg.fit(X, y)) -
兩個決策樹回歸模型的預測對比
from sklearn.tree import DecisionTreeRegressor tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2) tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3) tree_reg1.fit(X, y) tree_reg2.fit(X, y) def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"): x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1) y_pred = tree_reg.predict(x1) plt.axis(axes) plt.xlabel("$x_1$", fontsize=18) if ylabel: plt.ylabel(ylabel, fontsize=18, rotation=0) plt.plot(X, y, "b.") plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$") plt.figure(figsize=(11, 4)) plt.subplot(121) plot_regression_predictions(tree_reg1, X, y) for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")): plt.plot([split, split], [-0.2, 1], style, linewidth=2) plt.text(0.21, 0.65, "Depth=0", fontsize=15) plt.text(0.01, 0.2, "Depth=1", fontsize=13) plt.text(0.65, 0.8, "Depth=1", fontsize=13) plt.legend(loc="upper center", fontsize=18) plt.title("max_depth=2", fontsize=14) plt.subplot(122) plot_regression_predictions(tree_reg2, X, y, ylabel=None) for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")): plt.plot([split, split], [-0.2, 1], style, linewidth=2) for split in (0.0458, 0.1298, 0.2873, 0.9040): plt.plot([split, split], [-0.2, 1], "k:", linewidth=1) plt.text(0.3, 0.5, "Depth=2", fontsize=13) plt.title("max_depth=3", fontsize=14) save_fig("tree_regression_plot兩個決策樹回歸模型的預測對比") plt.show()
不穩定性
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對資料旋轉敏感
np.random.seed(6) Xs = np.random.rand(100, 2) - 0.5 ys = (Xs[:, 0] > 0).astype(np.float32) * 2 angle = np.pi / 4 rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]]) Xsr = Xs.dot(rotation_matrix) tree_clf_s = DecisionTreeClassifier(random_state=42) tree_clf_s.fit(Xs, ys) tree_clf_sr = DecisionTreeClassifier(random_state=42) tree_clf_sr.fit(Xsr, ys) plt.figure(figsize=(11, 4)) plt.subplot(121) plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False) plt.subplot(122) plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False) save_fig("sensitivity_to_rotation_plot對資料旋轉敏感") plt.show() -
對訓練集細節敏感
X[(X[:, 1]==X[:, 1][y==1].max()) & (y==1)] # widest Iris-Versicolor flower not_widest_versicolor = (X[:, 1]!=1.8) | (y==2) X_tweaked = X[not_widest_versicolor] y_tweaked = y[not_widest_versicolor] tree_clf_tweaked = DecisionTreeClassifier(max_depth=2, random_state=40) tree_clf_tweaked.fit(X_tweaked, y_tweaked) plt.figure(figsize=(8, 4)) plot_decision_boundary(tree_clf_tweaked, X_tweaked, y_tweaked, legend=False) plt.plot([0, 7.5], [0.8, 0.8], "k-", linewidth=2) plt.plot([0, 7.5], [1.75, 1.75], "k--", linewidth=2) plt.text(1.0, 0.9, "Depth=0", fontsize=15) plt.text(1.0, 1.80, "Depth=1", fontsize=13) save_fig("decision_tree_inst ability_plot對訓練集細節敏感") plt.show()
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