Intro

import os
import numpy.random as rnd

Training & Visualization

# load iris dataset & train a DT classifier

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)
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')

# graph it into a .dot file

from sklearn.tree import export_graphviz

def image_path(fig_id):
    #return os.path...
    return 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
)

# convert to PDF or PNG using command-line tool.
! dot -Tpng iris_tree.dot -o iris_tree.png

Predictions

• DTs require very little data prep. No feature scaling & centering.
• SciKit uses CART algorithm. (only two children per node.) Other algos, ex ID3, can build DTs with >2 children per node.
• gini attribute refers to a node's "impurity" (gini=0 if all applicable training instances belong to same class.)

# Plot DT decision boundaries
# Depth=0: root node (petal length=2.45cm)
# Depth=1: right node splits @ 1.75cm
# Stops at max_depth = 2.
# Vertical dotted line shows boundary if max_depth set = 3.

import numpy as np
import matplotlib.pyplot as plt
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, 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)

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()

Estimating Class Probabilities

# Probability of instance 5cm long, 1.5cm wide belonging to any one of three nodes above:
print(tree_clf.predict_proba([[5, 1.5]]))

# Return class of highest probability (in this case, class #1.)
print(tree_clf.predict([[5, 1.5]]))

[[ 0.          0.90740741  0.09259259]]
[1]

Training: CART algorithm

• Split training set in two using feature k and threshold t_k.
• Searches for pair (k, t_k) that returns purest subsets, weighted by size.
• Cost function to minimize shown below.

• "Greedy" algorithm; searches for optimum at each level w/o regard for lower levels. Not guaranteed to find optimum solution.

Computational Complexity

• Typical: O(log2(m)) = independent of #features. (So: very fast prediction times.)

Gini Impurity, or Entropy?

• Can use entropy measure by setting criterion parameter to "entropy".

• Dataset's entropy = 0 when it contains instances of only one class.

• Can use either; Gini impurity = slightly faster. Entropy tends to build slightly more balanced trees.

Regularization Hyperparameters

• max_depth controls max depth of the DT. Reducing max_depth regularizes the model, therefore reduces risk of overfit.
• Also: min_samples_split, min_samples_leaf, min_weight_fraction_leaf, max_leaf_nodes, max_features -- increasing min* or reducing max* params will regularize the model.

# Train two DTs on moons dataset.
# left: default params = no restrictions (case of overfitting)
# right: min_samples_leaf = 4. (better generalization)

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")
plt.show()

Regression

• Task: Predict a value (instead of a class) for each node.

from sklearn.tree import DecisionTreeRegressor

# Quadrat!ic training set + noise
rnd.seed(42)
m = 200
X = rnd.rand(m, 1)
y = 4 * (X - 0.5) ** 2
y = y + rnd.randn(m, 1) / 10

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)

plt.show()

# Predicted value for each region (red line) = avg target value of instances in that region.

• Instead of trying to minimize impurity (classification) DTs now try to minimize MSE:

tree_reg1 = DecisionTreeRegressor(random_state=42)
tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
tree_reg1.fit(X, y)
tree_reg2.fit(X, y)

x1 = np.linspace(0, 1, 500).reshape(-1, 1)
y_pred1 = tree_reg1.predict(x1)
y_pred2 = tree_reg2.predict(x1)

plt.figure(figsize=(11, 4))

plt.subplot(121)
plt.plot(X, y, "b.")
plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
plt.axis([0, 1, -0.2, 1.1])
plt.xlabel("$x_1$", fontsize=18)
plt.ylabel("$y$", fontsize=18, rotation=0)
plt.legend(loc="upper center", fontsize=18)
plt.title("No restrictions", fontsize=14)

plt.subplot(122)
plt.plot(X, y, "b.")
plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
plt.axis([0, 1, -0.2, 1.1])
plt.xlabel("$x_1$", fontsize=18)
plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)

#save_fig("tree_regression_regularization_plot")
plt.show()

# left: no regularization (default params): overfitting
# right: more reasonable.

Instability

• DTs strongly favor orthogonal decision boundaries. They are sensitive to training set rotations.
• More generally: DTs are sensitive to training data variations.

rnd.seed(6)
Xs = rnd.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")
plt.show()

# left: std linearly separable dataset
# right: dataset rotated by 45degrees.