什么是决策树？
决策树是一种基本的分类和回归方法。以分类决策树为例：
决策树通常包含哪三个步骤？
特征选择、决策树的生成和决策树的修剪
决策树与if-then规则？
直接以一个例子看看数如何构建决策树的：
根据不同的特征可以有不同的决策树：
那么如何从根节点开始选择特征进行决策树的构建呢？
最基础的是使用信息增益来表示。
首先得了解熵和条件熵的定义。
熵：用于表示随机变量不确定性的度量 。假设X是一个取值有限的随机变量，其概率分布为：
那么随机变量的熵的定义是：
什么是信息增益？
信息增益有什么作用呢？
计算步骤？
这里以上述表格中的数据为例：
我们最终需要的是分为是否会申请贷款，针对于是否需要申请贷款（即经验熵）为：
然后我们分别计算每一个特征的条件经验熵（也就是每一个特征对于数据集D的信息增益） ，分别以A1,A2,A3,A4标识年龄、有工作、有自己方法、信贷情况4个特征，则有：
信息增益存在的问题？
那么什么是信息增益比？
提到决策树就需要了解到ID3、C4.5和CART三种。其中ID3就是使用信息增益来进行特征选择，而C4.5使用的是信息增益比进行选择。
ID3生成的决策树如下：
由于ID3只有决策树的生成过程，因此容易过拟合。
CART算法？
以分类为例，CART使用基尼指数来进行特征选择：
还是以上述的数据集进行计算：
还有其剪枝算法，就不列举了。
上述解释摘自：统计学习方法。
下面是代码实现，代码来源： https://github.com/eriklindernoren/ML-From-Scratch
代码语言：javascript复制from __future__ import division, print_function
import numpy as np
from mlfromscratch.utils import divide_on_feature, train_test_split, standardize, mean_squared_error
from mlfromscratch.utils import calculate_entropy, accuracy_score, calculate_variance
class DecisionNode():
"""Class that represents a decision node or leaf in the decision tree
Parameters:
-----------
feature_i: int
Feature index which we want to use as the threshold measure.
threshold: float
The value that we will compare feature values at feature_i against to
determine the prediction.
value: float
The class prediction if classification tree, or float value if regression tree.
true_branch: DecisionNode
Next decision node for samples where features value met the threshold.
false_branch: DecisionNode
Next decision node for samples where features value did not meet the threshold.
"""
def __init__(self, feature_i=None, threshold=None,
value=None, true_branch=None, false_branch=None):
self.feature_i = feature_i # Index for the feature that is tested
self.threshold = threshold # Threshold value for feature
self.value = value # Value if the node is a leaf in the tree
self.true_branch = true_branch # 'Left' subtree
self.false_branch = false_branch # 'Right' subtree
# Super class of RegressionTree and ClassificationTree
class DecisionTree(object):
"""Super class of RegressionTree and ClassificationTree.
Parameters:
-----------
min_samples_split: int
The minimum number of samples needed to make a split when building a tree.
min_impurity: float
The minimum impurity required to split the tree further.
max_depth: int
The maximum depth of a tree.
loss: function
Loss function that is used for Gradient Boosting models to calculate impurity.
"""
def __init__(self, min_samples_split=2, min_impurity=1e-7,
max_depth=float("inf"), loss=None):
self.root = None # Root node in dec. tree
# Minimum n of samples to justify split
self.min_samples_split = min_samples_split
# The minimum impurity to justify split
self.min_impurity = min_impurity
# The maximum depth to grow the tree to
self.max_depth = max_depth
# Function to calculate impurity (classif.=>info gain, regr=>variance reduct.)
self._impurity_calculation = None
# Function to determine prediction of y at leaf
self._leaf_value_calculation = None
# If y is one-hot encoded (multi-dim) or not (one-dim)
self.one_dim = None
# If Gradient Boost
self.loss = loss
def fit(self, X, y, loss=None):
""" Build decision tree """
self.one_dim = len(np.shape(y)) == 1
self.root = self._build_tree(X, y)
self.loss=None
def _build_tree(self, X, y, current_depth=0):
""" Recursive method which builds out the decision tree and splits X and respective y
on the feature of X which (based on impurity) best separates the data"""
largest_impurity = 0
best_criteria = None # Feature index and threshold
best_sets = None # Subsets of the data
# Check if expansion of y is needed
if len(np.shape(y)) == 1:
y = np.expand_dims(y, axis=1)
# Add y as last column of X
Xy = np.concatenate((X, y), axis=1)
n_samples, n_features = np.shape(X)
if n_samples >= self.min_samples_split and current_depth <= self.max_depth:
# Calculate the impurity for each feature
for feature_i in range(n_features):
# All values of feature_i
feature_values = np.expand_dims(X[:, feature_i], axis=1)
unique_values = np.unique(feature_values)
# Iterate through all unique values of feature column i and
# calculate the impurity
for threshold in unique_values:
# Divide X and y depending on if the feature value of X at index feature_i
# meets the threshold
Xy1, Xy2 = divide_on_feature(Xy, feature_i, threshold)
if len(Xy1) > 0 and len(Xy2) > 0:
# Select the y-values of the two sets
y1 = Xy1[:, n_features:]
y2 = Xy2[:, n_features:]
# Calculate impurity
impurity = self._impurity_calculation(y, y1, y2)
# If this threshold resulted in a higher information gain than previously
# recorded save the threshold value and the feature
# index
if impurity > largest_impurity:
largest_impurity = impurity
best_criteria = {"feature_i": feature_i, "threshold": threshold}
best_sets = {
"leftX": Xy1[:, :n_features], # X of left subtree
"lefty": Xy1[:, n_features:], # y of left subtree
"rightX": Xy2[:, :n_features], # X of right subtree
"righty": Xy2[:, n_features:] # y of right subtree
}
if largest_impurity > self.min_impurity:
# Build subtrees for the right and left branches
true_branch = self._build_tree(best_sets["leftX"], best_sets["lefty"], current_depth 1)
false_branch = self._build_tree(best_sets["rightX"], best_sets["righty"], current_depth 1)
return DecisionNode(feature_i=best_criteria["feature_i"], threshold=best_criteria[
"threshold"], true_branch=true_branch, false_branch=false_branch)
# We're at leaf => determine value
leaf_value = self._leaf_value_calculation(y)
return DecisionNode(value=leaf_value)
def predict_value(self, x, tree=None):
""" Do a recursive search down the tree and make a prediction of the data sample by the
value of the leaf that we end up at """
if tree is None:
tree = self.root
# If we have a value (i.e we're at a leaf) => return value as the prediction
if tree.value is not None:
return tree.value
# Choose the feature that we will test
feature_value = x[tree.feature_i]
# Determine if we will follow left or right branch
branch = tree.false_branch
if isinstance(feature_value, int) or isinstance(feature_value, float):
if feature_value >= tree.threshold:
branch = tree.true_branch
elif feature_value == tree.threshold:
branch = tree.true_branch
# Test subtree
return self.predict_value(x, branch)
def predict(self, X):
""" Classify samples one by one and return the set of labels """
y_pred = [self.predict_value(sample) for sample in X]
return y_pred
def print_tree(self, tree=None, indent=" "):
""" Recursively print the decision tree """
if not tree:
tree = self.root
# If we're at leaf => print the label
if tree.value is not None:
print (tree.value)
# Go deeper down the tree
else:
# Print test
print ("%s:%s? " % (tree.feature_i, tree.threshold))
# Print the true scenario
print ("%sT->" % (indent), end="")
self.print_tree(tree.true_branch, indent indent)
# Print the false scenario
print ("%sF->" % (indent), end="")
self.print_tree(tree.false_branch, indent indent)
class XGBoostRegressionTree(DecisionTree):
"""
Regression tree for XGBoost
- Reference -
http://xgboost.readthedocs.io/en/latest/model.html
"""
def _split(self, y):
""" y contains y_true in left half of the middle column and
y_pred in the right half. Split and return the two matrices """
col = int(np.shape(y)[1]/2)
y, y_pred = y[:, :col], y[:, col:]
return y, y_pred
def _gain(self, y, y_pred):
nominator = np.power((y * self.loss.gradient(y, y_pred)).sum(), 2)
denominator = self.loss.hess(y, y_pred).sum()
return 0.5 * (nominator / denominator)
def _gain_by_taylor(self, y, y1, y2):
# Split
y, y_pred = self._split(y)
y1, y1_pred = self._split(y1)
y2, y2_pred = self._split(y2)
true_gain = self._gain(y1, y1_pred)
false_gain = self._gain(y2, y2_pred)
gain = self._gain(y, y_pred)
return true_gain false_gain - gain
def _approximate_update(self, y):
# y split into y, y_pred
y, y_pred = self._split(y)
# Newton's Method
gradient = np.sum(y * self.loss.gradient(y, y_pred), axis=0)
hessian = np.sum(self.loss.hess(y, y_pred), axis=0)
update_approximation = gradient / hessian
return update_approximation
def fit(self, X, y):
self._impurity_calculation = self._gain_by_taylor
self._leaf_value_calculation = self._approximate_update
super(XGBoostRegressionTree, self).fit(X, y)
class RegressionTree(DecisionTree):
def _calculate_variance_reduction(self, y, y1, y2):
var_tot = calculate_variance(y)
var_1 = calculate_variance(y1)
var_2 = calculate_variance(y2)
frac_1 = len(y1) / len(y)
frac_2 = len(y2) / len(y)
# Calculate the variance reduction
variance_reduction = var_tot - (frac_1 * var_1 frac_2 * var_2)
return sum(variance_reduction)
def _mean_of_y(self, y):
value = np.mean(y, axis=0)
return value if len(value) > 1 else value[0]
def fit(self, X, y):
self._impurity_calculation = self._calculate_variance_reduction
self._leaf_value_calculation = self._mean_of_y
super(RegressionTree, self).fit(X, y)
class ClassificationTree(DecisionTree):
def _calculate_information_gain(self, y, y1, y2):
# Calculate information gain
p = len(y1) / len(y)
entropy = calculate_entropy(y)
info_gain = entropy - p *
calculate_entropy(y1) - (1 - p) *
calculate_entropy(y2)
return info_gain
def _majority_vote(self, y):
most_common = None
max_count = 0
for label in np.unique(y):
# Count number of occurences of samples with label
count = len(y[y == label])
if count > max_count:
most_common = label
max_count = count
return most_common
def fit(self, X, y):
self._impurity_calculation = self._calculate_information_gain
self._leaf_value_calculation = self._majority_vote
super(ClassificationTree, self).fit(X, y)运行主函数：
代码语言：javascript复制from __future__ import division, print_function
import numpy as np
from sklearn import datasets
import matplotlib.pyplot as plt
import sys
import os
import sys
sys.path.append("/content/drive/My Drive/learn/ML-From-Scratch/")
# Import helper functions
from mlfromscratch.utils import train_test_split, standardize, accuracy_score
from mlfromscratch.utils import mean_squared_error, calculate_variance, Plot
from mlfromscratch.supervised_learning import ClassificationTree
def main():
print ("-- Classification Tree --")
data = datasets.load_iris()
X = data.data
y = data.target
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4)
clf = ClassificationTree()
clf.fit(X_train, y_train)
y_pred = clf.predict(X_test)
accuracy = accuracy_score(y_test, y_pred)
print ("Accuracy:", accuracy)
Plot().plot_in_2d(X_test, y_pred,
title="Decision Tree",
accuracy=accuracy,
legend_labels=data.target_names)
if __name__ == "__main__":
main()运行结果：
-- Classification Tree --
Accuracy: 0.9
回归主函数：
代码语言：javascript复制from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sys
sys.path.append("/content/drive/My Drive/learn/ML-From-Scratch/")
from mlfromscratch.utils import train_test_split, standardize, accuracy_score
from mlfromscratch.utils import mean_squared_error, calculate_variance, Plot
from mlfromscratch.supervised_learning import RegressionTree
def main():
print ("-- Regression Tree --")
# Load temperature data
data = pd.read_csv('mlfromscratch/data/TempLinkoping2016.txt', sep="t")
time = np.atleast_2d(data["time"].values).T
temp = np.atleast_2d(data["temp"].values).T
X = standardize(time) # Time. Fraction of the year [0, 1]
y = temp[:, 0] # Temperature. Reduce to one-dim
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3)
model = RegressionTree()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)
y_pred_line = model.predict(X)
# Color map
cmap = plt.get_cmap('viridis')
mse = mean_squared_error(y_test, y_pred)
print ("Mean Squared Error:", mse)
# Plot the results
# Plot the results
m1 = plt.scatter(366 * X_train, y_train, color=cmap(0.9), s=10)
m2 = plt.scatter(366 * X_test, y_test, color=cmap(0.5), s=10)
m3 = plt.scatter(366 * X_test, y_pred, color='black', s=10)
plt.suptitle("Regression Tree")
plt.title("MSE: %.2f" % mse, fontsize=10)
plt.xlabel('Day')
plt.ylabel('Temperature in Celcius')
plt.legend((m1, m2, m3), ("Training data", "Test data", "Prediction"), loc='lower right')
plt.savefig("test2.png")
plt.show()
if __name__ == "__main__":
main()结果：
-- Regression Tree --
Mean Squared Error: 9.445229357798167
