perform both classification and regression tasks and even mulioutput tasks
how to train, visualize, and make predictions with Decision Tree
CART training algorithm
limitation of Decision Tree

Training and Visualizing a Decision Tree¶

from sklearn.tree import DecisionTreeClassifier
from sklearn.datasets import load_iris

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

tree_clf = DecisionTreeClassifier(max_depth=2)
tree_clf.fit(X, y)

DecisionTreeClassifier(class_weight=None, criterion='gini', max_depth=2,
            max_features=None, max_leaf_nodes=None,
            min_impurity_split=1e-07, min_samples_leaf=1,
            min_samples_split=2, min_weight_fraction_leaf=0.0,
            presort=False, random_state=None, splitter='best')

from sklearn.tree import export_graphviz
import os

PROJECT_ROOT_DIR = "."
CHAPTER_ID = "decision_trees"
def image_path(fig_id):
    return os.path.join(PROJECT_ROOT_DIR, "images", CHAPTER_ID, fig_id)

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
)

Making Predictions¶

$$ Gini\ impurity:\ G_i=1-\sum_{k=1}^{n}p_{i,k}^{\ \ \ \ \ 2} $$
$p_{i,k}$是第i個實例分到第k類的比例
white box

Estimating Class Probabilities¶

輸出的時所屬的非葉子節點的實例的比例javascript

tree_clf.predict_proba([[5, 1.5]])

array([[ 0.        ,  0.90740741,  0.09259259]])

tree_clf.predict([[5,1.5]])

array([1])

The CART Training Algorithm¶

sklearn use Classification And Regression Tree(CART) to train Decision Tree
splite the train set in two purest subsets:$$ J(k,t_k) = \frac{m_{left}}{m} G_{left} + \frac{m_{right}}{m} G_{right} \\ where \left\{\begin{matrix} G_{left/right}\ measures\ the\ impurity\ of\ the\ left/right\ subset\\ m_{{left}/{right}}\ is\ the\ number\ of\ instances\ in\ the\ left/right\ subsets. \end{matrix}\right. $$
reasonably good solution

Computational Complexity¶

$O(log_2(m))$ for prediction and $O(n \times log_2{m})$ for train

Gini Impurity or Entropy¶

entropy origined thermodynamics to measure molecular disorder
in ml, it is frequently used as an impurity measure.
$$ H_i=-\sum_{k=1,p_{i,k}\neq 0}^{n}p_{i,k}log(p_{i,k}) $$
Gini is fast, Gini and Entropy lead to similar tree
when they differ, Gini impurity tend to isolate the most frequent class in its own bench of tree, while entropy trens to produce slightly more balanced trees

Regularization Hyperparameters¶

nonhyperparametric model:the number of parameters is not determined prior to training
most likely to overfitting, so need to restrict the freedom called regularization
generally this is controlled by max_depth
DecisionTreeClassifier class has min_samples_split, min_samples_leaf, min_weight_fraction_leaf, max_leaf_nodes, max_features
也可先不加限制生成決策樹，再剪枝。若是統計學不顯著，就剪掉

from sklearn.datasets import make_moons
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
import numpy as np

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, linewidth=10)
    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)
        
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)

plt.show()

Regression¶

from sklearn.tree import DecisionTreeRegressor

tree_reg = DecisionTreeRegressor(max_depth=2)
tree_reg.fit(X, y)

DecisionTreeRegressor(criterion='mse', max_depth=2, max_features=None,
           max_leaf_nodes=None, min_impurity_split=1e-07,
           min_samples_leaf=1, min_samples_split=2,
           min_weight_fraction_leaf=0.0, presort=False, random_state=None,
           splitter='best')

export_graphviz(
    tree_reg,
    out_file=image_path('moons_tree1.dot'),
    rounded=True,
    filled=True
)

$$ cost\ function=J(k,t_k)=\frac{m_{left}}{m}MSE_{left}+\frac{m_{right}}{m}MSE_{right}\\ where\left\{\begin{matrix} MSE_{node}=\sum_{i \in {node}}(\widehat{y}_{node}-y^{(i)})^2 \\ \widehat{y}_{node}=\frac{1}{m_{node}}\sum{i \in {node}}y^{(i)} \end{matrix}\right. $$

Instability¶

Decision Tree love orthogonal decision boundaries which make them sensitive to training set rotation; One way to limit this problem is to use PCA
they are very sensitive to small variations in training set; Random Forests can limit this instability by averaging predictions over many trees

Notes ： Chapter 6