一、平方误差的计算
square_error_utils.py
python
import numpy as np
class SquareErrorUtils:
"""
平方误差最小化准则，选择其中最优的一个作为切分点
对特征属性进行分箱处理
"""
@staticmethod
def _set_sample_weight(sample_weight, n_samples):
"""
扩展到集成学习，此处为样本权重的设置
:param sample_weight: 各样本的权重
:param n_samples: 样本量
:return:
"""
if sample_weight is None:
sample_weight = np.asarray([1.0] * n_samples)
return sample_weight
@staticmethod
def square_error(y, sample_weight):
"""
平方误差
:param y: 当前划分区域的目标值集合
:param sample_weight: 当前样本的权重
:return:
"""
y = np.asarray(y)
return np.sum((y - y.mean()) ** 2 * sample_weight)
def cond_square_error(self, x, y, sample_weight):
"""
计算根据某个特征x划分的区域中y的误差值
:param x: 某个特征划分区域所包含的样本
:param y: x对应的目标值
:param sample_weight: 当前x的权重
:return:
"""
x, y = np.asarray(x), np.asarray(y)
error = 0.0
for x_val in set(x):
x_idx = np.where(x == x_val) # 按区域计算误差
new_y = y[x_idx] # 对应区域的目标值
new_sample_weight = sample_weight[x_idx]
error += self.square_error(new_y, new_sample_weight)
return error
def square_error_gain(self, x, y, sample_weight=None):
"""
平方误差带来的增益值
:param x: 某个特征变量
:param y: 对应的目标值
:param sample_weight: 样本权重
:return:
"""
sample_weight = self._set_sample_weight(sample_weight, len(x))
return self.square_error(y, sample_weight) - self.cond_square_error(x, y, sample_weight)
二、树的结点信息封装
python
class TreeNode_R:
"""
决策树回归算法，树的结点信息封装，实体类：setXXX()、getXXX()
"""
def __init__(self, feature_idx: int = None, feature_val=None, y_hat=None, square_error: float = None,
criterion_val=None, n_samples: int = None, left_child_Node=None, right_child_Node=None):
"""
决策树结点信息封装
:param feature_idx: 特征索引，如果指定特征属性的名称，可以按照索引取值
:param feature_val: 特征取值
:param square_error: 划分结点的标准：当前结点的平方误差
:param n_samples: 当前结点所包含的样本量
:param y_hat: 当前结点的预测值：Ci
:param left_child_Node: 左子树
:param right_child_Node: 右子树
"""
self.feature_idx = feature_idx
self.feature_val = feature_val
self.criterion_val = criterion_val
self.square_error = square_error
self.n_samples = n_samples
self.y_hat = y_hat
self.left_child_Node = left_child_Node # 递归
self.right_child_Node = right_child_Node # 递归
def level_order(self):
"""
按层次遍历树...
:return:
"""
pass
# def get_feature_idx(self):
# return self.get_feature_idx()
#
# def set_feature_idx(self, feature_idx):
# self.feature_idx = feature_idx三、回归决策树CART算法实现
python
import numpy as np
from utils.square_error_utils import SquareErrorUtils
from utils.tree_node_R import TreeNode_R
from utils.data_bin_wrapper import DataBinsWrapper
class DecisionTreeRegression:
"""
回归决策树CART算法实现：按照二叉树构造
1. 划分标准：平方误差最小化
2. 创建决策树fit()，递归算法实现，注意出口条件
3. 预测predict_proba()、predict() --> 对树的搜索
4. 数据的预处理操作，尤其是连续数据的离散化，分箱
5. 剪枝处理
"""
def __init__(self, criterion="mse", max_depth=None, min_sample_split=2, min_sample_leaf=1,
min_target_std=1e-3, min_impurity_decrease=0, max_bins=10):
self.utils = SquareErrorUtils() # 结点划分类
self.criterion = criterion # 结点的划分标准
if criterion.lower() == "mse":
self.criterion_func = self.utils.square_error_gain # 平方误差增益
else:
raise ValueError("参数criterion仅限mse...")
self.min_target_std = min_target_std # 最小的样本目标值方差，小于阈值不划分
self.max_depth = max_depth # 树的最大深度，不传参，则一直划分下去
self.min_sample_split = min_sample_split # 最小的划分结点的样本量，小于则不划分
self.min_sample_leaf = min_sample_leaf # 叶子结点所包含的最小样本量，剩余的样本小于这个值，标记叶子结点
self.min_impurity_decrease = min_impurity_decrease # 最小结点不纯度减少值，小于这个值，不足以划分
self.max_bins = max_bins # 连续数据的分箱数，越大，则划分越细
self.root_node: TreeNode_R() = None # 回归决策树的根节点
self.dbw = DataBinsWrapper(max_bins=max_bins) # 连续数据离散化对象
self.dbw_XrangeMap = {} # 存储训练样本连续特征分箱的端点
def fit(self, x_train, y_train, sample_weight=None):
"""
回归决策树的创建，递归操作前的必要信息处理（分箱）
:param x_train: 训练样本：ndarray，n * k
:param y_train: 目标集：ndarray，（n， ）
:param sample_weight: 各样本的权重，（n， ）
:return:
"""
x_train, y_train = np.asarray(x_train), np.asarray(y_train)
self.class_values = np.unique(y_train) # 样本的类别取值
n_samples, n_features = x_train.shape # 训练样本的样本量和特征属性数目
if sample_weight is None:
sample_weight = np.asarray([1.0] * n_samples)
self.root_node = TreeNode_R() # 创建一个空树
self.dbw.fit(x_train)
x_train = self.dbw.transform(x_train)
self._build_tree(1, self.root_node, x_train, y_train, sample_weight)
def _build_tree(self, cur_depth, cur_node: TreeNode_R, x_train, y_train, sample_weight):
"""
递归创建回归决策树算法，核心算法。按先序（中序、后序）创建的
:param cur_depth: 递归划分后的树的深度
:param cur_node: 递归划分后的当前根结点
:param x_train: 递归划分后的训练样本
:param y_train: 递归划分后的目标集合
:param sample_weight: 递归划分后的各样本权重
:return:
"""
n_samples, n_features = x_train.shape # 当前样本子集中的样本量和特征属性数目
# 计算当前数结点的预测值，即加权平均值，
cur_node.y_hat = np.dot(sample_weight / np.sum(sample_weight), y_train)
cur_node.n_samples = n_samples
# 递归出口判断
cur_node.square_error = ((y_train - y_train.mean()) ** 2).sum()
# 所有的样本目标值较为集中，样本方差非常小，不足以划分
if cur_node.square_error <= self.min_target_std:
# 如果为0，则表示当前样本集合为空，递归出口3
return
if n_samples < self.min_sample_split: # 当前结点所包含的样本量不足以划分
return
if self.max_depth is not None and cur_depth > self.max_depth: # 树的深度达到最大深度
return
# 划分标准，选择最佳的划分特征及其取值
best_idx, best_val, best_criterion_val = None, None, 0.0
for k in range(n_features): # 对当前样本集合中每个特征计算划分标准
for f_val in sorted(np.unique(x_train[:, k])): # 当前特征的不同取值
region_x = (x_train[:, k] <= f_val).astype(int) # 是当前取值f_val就是1，否则就是0
criterion_val = self.criterion_func(region_x, y_train, sample_weight)
if criterion_val > best_criterion_val:
best_criterion_val = criterion_val # 最佳的划分标准值
best_idx, best_val = k, f_val # 当前最佳特征索引以及取值
# 递归出口的判断
if best_idx is None: # 当前属性为空，或者所有样本在所有属性上取值相同，无法划分
return
if best_criterion_val <= self.min_impurity_decrease: # 小于最小不纯度阈值，不划分
return
cur_node.criterion_val = best_criterion_val
cur_node.feature_idx = best_idx
cur_node.feature_val = best_val
# print("当前划分的特征索引：", best_idx, "取值：", best_val, "最佳标准值：", best_criterion_val)
# print("当前结点的类别分布：", target_dist)
# 创建左子树，并递归创建以当前结点为子树根节点的左子树
left_idx = np.where(x_train[:, best_idx] <= best_val) # 左子树所包含的样本子集索引
if len(left_idx) >= self.min_sample_leaf: # 小于叶子结点所包含的最少样本量，则标记为叶子结点
left_child_node = TreeNode_R() # 创建左子树空结点
# 以当前结点为子树根结点，递归创建
cur_node.left_child_Node = left_child_node
self._build_tree(cur_depth + 1, left_child_node, x_train[left_idx],
y_train[left_idx], sample_weight[left_idx])
right_idx = np.where(x_train[:, best_idx] > best_val) # 右子树所包含的样本子集索引
if len(right_idx) >= self.min_sample_leaf: # 小于叶子结点所包含的最少样本量，则标记为叶子结点
right_child_node = TreeNode_R() # 创建右子树空结点
# 以当前结点为子树根结点，递归创建
cur_node.right_child_Node = right_child_node
self._build_tree(cur_depth + 1, right_child_node, x_train[right_idx],
y_train[right_idx], sample_weight[right_idx])
def _search_tree_predict(self, cur_node: TreeNode_R, x_test):
"""
根据测试样本从根结点到叶子结点搜索路径，判定所属区域（叶子结点）
搜索：按照后续遍历
:param x_test: 单个测试样本
:return:
"""
if cur_node.left_child_Node and x_test[cur_node.feature_idx] <= cur_node.feature_val:
return self._search_tree_predict(cur_node.left_child_Node, x_test)
elif cur_node.right_child_Node and x_test[cur_node.feature_idx] > cur_node.feature_val:
return self._search_tree_predict(cur_node.right_child_Node, x_test)
else:
# 叶子结点，类别，包含有类别分布
return cur_node.y_hat
def predict(self, x_test):
"""
预测测试样本x_test的预测值
:param x_test: 测试样本ndarray、numpy数值运算
:return:
"""
x_test = np.asarray(x_test) # 避免传递DataFrame、list...
if self.dbw.XrangeMap is None:
raise ValueError("请先进行回归决策树的创建，然后预测...")
x_test = self.dbw.transform(x_test)
y_test_pred = [] # 用于存储测试样本的预测值
for i in range(x_test.shape[0]):
y_test_pred.append(self._search_tree_predict(self.root_node, x_test[i]))
return np.asarray(y_test_pred)
@staticmethod
def cal_mse_r2(y_test, y_pred):
"""
模型预测的均方误差MSE和判决系数R2
:param y_test: 测试样本的真值
:param y_pred: 测试样本的预测值
:return:
"""
y_test, y_pred = y_test.reshape(-1), y_pred.reshape(-1)
mse = ((y_pred - y_test) ** 2).mean() # 均方误差
r2 = 1 - ((y_pred - y_test) ** 2).sum() / ((y_test - y_test.mean()) ** 2).sum()
return mse, r2
def _prune_node(self, cur_node: TreeNode_R, alpha):
"""
递归剪枝，针对决策树中的内部结点，自底向上，逐个考察
方法：后序遍历
:param cur_node: 当前递归的决策树的内部结点
:param alpha: 剪枝阈值
:return:
"""
# 若左子树存在，递归左子树进行剪枝
if cur_node.left_child_Node:
self._prune_node(cur_node.left_child_Node, alpha)
# 若右子树存在，递归右子树进行剪枝
if cur_node.right_child_Node:
self._prune_node(cur_node.right_child_Node, alpha)
# 针对决策树的内部结点剪枝，非叶结点
if cur_node.left_child_Node is not None or cur_node.right_child_Node is not None:
for child_node in [cur_node.left_child_Node, cur_node.right_child_Node]:
if child_node is None:
# 可能存在左右子树之一为空的情况，当左右子树划分的样本子集数小于min_samples_leaf
continue
if child_node.left_child_Node is not None or child_node.right_child_Node is not None:
return
# 计算剪枝前的损失值（平方误差），2表示当前结点包含两个叶子结点
pre_prune_value = 2 * alpha
if cur_node and cur_node.left_child_Node is not None:
pre_prune_value += (0.0 if cur_node.left_child_Node.square_error is None
else cur_node.left_child_Node.square_error)
if cur_node and cur_node.right_child_Node is not None:
pre_prune_value += (0.0 if cur_node.right_child_Node.square_error is None
else cur_node.right_child_Node.square_error)
# 计算剪枝后的损失值，当前结点即是叶子结点
after_prune_value = alpha + cur_node.square_error
if after_prune_value <= pre_prune_value: # 进行剪枝操作
cur_node.left_child_Node = None
cur_node.right_child_Node = None
cur_node.feature_idx, cur_node.feature_val = None, None
cur_node.square_error = None
def prune(self, alpha=0.01):
"""
决策树后剪枝算法（李航）C(T) + alpha * |T|
:param alpha: 剪枝阈值，权衡模型对训练数据的拟合程度与模型的复杂度
:return:
"""
self._prune_node(self.root_node, alpha)
return self.root_node四、回归决策树算法的测试
test_decision_tree_R.py
python
import numpy as np
import matplotlib.pyplot as plt
from decision_tree_R import DecisionTreeRegression
from sklearn.tree import DecisionTreeClassifier, DecisionTreeRegressor
obj_fun = lambda x: np.sin(x)
np.random.seed(0)
n = 100
x = np.linspace(0, 10, n)
target = obj_fun(x) + 0.3 * np.random.randn(n)
data = x[:, np.newaxis] # 二维数组
tree = DecisionTreeRegression(max_bins=50, max_depth=10)
tree.fit(data, target)
x_test = np.linspace(0, 10, 200)
y_test_pred = tree.predict(x_test[:, np.newaxis])
mse, r2 = tree.cal_mse_r2(obj_fun(x_test), y_test_pred)
plt.figure(figsize=(14, 5))
plt.subplot(121)
plt.scatter(data, target, s=15, c="k", label="Raw Data")
plt.plot(x_test, y_test_pred, "r-", lw=1.5, label="Fit Model")
plt.xlabel("x", fontdict={"fontsize": 12, "color": "b"})
plt.ylabel("y", fontdict={"fontsize": 12, "color": "b"})
plt.grid(ls=":")
plt.legend(frameon=False)
plt.title("Regression Decision Tree(UnPrune) and MSE = %.5f R2 = %.5f" % (mse, r2))
plt.subplot(122)
tree.prune(0.5)
y_test_pred = tree.predict(x_test[:, np.newaxis])
mse, r2 = tree.cal_mse_r2(obj_fun(x_test), y_test_pred)
plt.scatter(data, target, s=15, c="k", label="Raw Data")
plt.plot(x_test, y_test_pred, "r-", lw=1.5, label="Fit Model")
plt.xlabel("x", fontdict={"fontsize": 12, "color": "b"})
plt.ylabel("y", fontdict={"fontsize": 12, "color": "b"})
plt.grid(ls=":")
plt.legend(frameon=False)
plt.title("Regression Decision Tree(Prune) and MSE = %.5f R2 = %.5f" % (mse, r2))
plt.show()