簡介
CART樹即分類迴歸樹(classification and regression tree)，顧名思義，它即能用作分類任務又能用作迴歸任務，它的應用比較廣泛，通常會用作整合學習的基分類器，總得來說，它與ID3/C4.5有如下不同：
（1）它是一顆二叉樹；
（2）特徵選擇的方法不一樣，CART分類樹利用基尼係數做特徵選擇，CART迴歸樹利用平方誤差做特徵選擇；
接下來，分別對CART分類樹和迴歸樹做介紹
CART分類樹
首先介紹特徵選擇方法，基尼係數：
\[Gini(p)=\sum_{k=1}^Kp_k(1-p_k)=1-\sum_{k=1}^Kp_k^2
\]
所以，對於給定的樣本集合\(D\)，其基尼指數：
\[Gini(D)=1-\sum_{k=1}^K(\frac{\mid C_k \mid}{\mid D \mid})^2
\]
這裡，\(C_k\)是\(D\)中屬於第\(k\)類的樣本子集，\(K\)是類的個數，由於CART樹是二叉樹，所以對於某特徵\(A\)，判斷其對分類標籤的貢獻時，只需要判斷該特徵是否等於某個取值\(a\)的情況，將當前資料集分割成\(D_1\)和\(D_2\)兩部分：
\[D_1=\{(x,y)\in D\mid A(x)=a\},D_2=D-D_1
\]
所以在特徵\(A(x)=a\)的條件下，集合\(D\)的基尼指數可以定義為：
\[Gini(D,A,a)=\frac{\mid D_1 \mid}{\mid D \mid}Gini(D_1)+\frac{\mid D_2 \mid}{\mid D \mid}Gini(D_2),這裡D_1=\{(x,y)\in D\mid A(x)=a\},D_2=D-D_1
\]
程式碼實現
接下來進行CART分類樹的程式碼實現，這裡與ID3/C4.5最大的不同就是每次對當前結點僅進行二分處理
"""
定義計算gini係數相關的函式,程式碼封裝到ml_models.utils
"""
import numpy as np
def gini(x, sample_weight=None):
"""
計算基尼係數 Gini(D)
:param x:
:param sample_weight:
:return:
"""
x_num = len(x)
# 如果sample_weight為None設均設定一樣
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
x_counter = {}
weight_counter = {}
# 統計各x取值出現的次數以及其對應的sample_weight列表
for index in range(0, x_num):
x_value = x[index]
if x_counter.get(x_value) is None:
x_counter[x_value] = 0
weight_counter[x_value] = []
x_counter[x_value] += 1
weight_counter[x_value].append(sample_weight[index])
# 計算gini係數
gini_value = 1.0
for key, value in x_counter.items():
p_i = 1.0 * value * np.mean(weight_counter.get(key)) / x_num
gini_value -= p_i * p_i
return gini_value
def cond_gini(x, y, sample_weight=None):
"""
計算條件gini係數:Gini(y,x)
"""
x = np.asarray(x)
y = np.asarray(y)
# x中元素個數
x_num = len(x)
# 如果sample_weight為None設均設定一樣
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
# 計算
gini_value = .0
for x_value in set(x):
x_index = np.where(x == x_value)
new_x = x[x_index]
new_y = y[x_index]
new_sample_weight = sample_weight[x_index]
p_i = 1.0 * len(new_x) / x_num
gini_value += p_i * gini(new_y, new_sample_weight)
return gini_value
def gini_gain(x, y, sample_weight=None):
"""
gini值的增益
"""
x_num = len(x)
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
return gini(y, sample_weight) - cond_gini(x, y, sample_weight)
import os
os.chdir('../')
from ml_models import utils
from ml_models.wrapper_models import DataBinWrapper
"""
CART分類樹的實現，程式碼封裝到ml_models.tree模組
"""
class CARTClassifier(object):
class Node(object):
"""
樹節點，用於儲存節點資訊以及關聯子節點
"""
def __init__(self, feature_index: int = None, feature_value=None, target_distribute: dict = None,
weight_distribute: dict = None,
left_child_node=None, right_child_node=None, num_sample: int = None):
"""
:param feature_index: 特徵id
:param feature_value: 特徵取值
:param target_distribute: 目標分佈
:param weight_distribute:權重分佈
:param left_child_node: 左孩子結點
:param right_child_node: 右孩子結點
:param num_sample:樣本量
"""
self.feature_index = feature_index
self.feature_value = feature_value
self.target_distribute = target_distribute
self.weight_distribute = weight_distribute
self.left_child_node = left_child_node
self.right_child_node = right_child_node
self.num_sample = num_sample
def __init__(self, criterion='gini', max_depth=None, min_samples_split=2, min_samples_leaf=1,
min_impurity_decrease=0, max_bins=10):
"""
:param criterion:劃分標準，預設為gini,另外entropy表示用資訊增益比
:param max_depth:樹的最大深度
:param min_samples_split:當對一個內部結點劃分時，要求該結點上的最小樣本數，預設為2
:param min_samples_leaf:設定葉子結點上的最小樣本數，預設為1
:param min_impurity_decrease:打算劃分一個內部結點時，只有當劃分後不純度(可以用criterion引數指定的度量來描述)減少值不小於該引數指定的值，才會對該結點進行劃分，預設值為0
"""
self.criterion = criterion
if criterion == 'gini':
self.criterion_func = utils.gini_gain
else:
self.criterion_func = utils.info_gain_rate
self.max_depth = max_depth
self.min_samples_split = min_samples_split
self.min_samples_leaf = min_samples_leaf
self.min_impurity_decrease = min_impurity_decrease
self.root_node: self.Node = None
self.dbw = DataBinWrapper(max_bins=max_bins)
def _build_tree(self, current_depth, current_node: Node, x, y, sample_weight):
"""
遞迴進行特徵選擇，構建樹
:param x:
:param y:
:param sample_weight:
:return:
"""
rows, cols = x.shape
# 計算y分佈以及其權重分佈
target_distribute = {}
weight_distribute = {}
for index, tmp_value in enumerate(y):
if tmp_value not in target_distribute:
target_distribute[tmp_value] = 0.0
weight_distribute[tmp_value] = []
target_distribute[tmp_value] += 1.0
weight_distribute[tmp_value].append(sample_weight[index])
for key, value in target_distribute.items():
target_distribute[key] = value / rows
weight_distribute[key] = np.mean(weight_distribute[key])
current_node.target_distribute = target_distribute
current_node.weight_distribute = weight_distribute
current_node.num_sample = rows
# 判斷停止切分的條件
if len(target_distribute) <= 1:
return
if rows < self.min_samples_split:
return
if self.max_depth is not None and current_depth > self.max_depth:
return
# 尋找最佳的特徵以及取值
best_index = None
best_index_value = None
best_criterion_value = 0
for index in range(0, cols):
for index_value in set(x[:, index]):
criterion_value = self.criterion_func((x[:, index] == index_value).astype(int), y, sample_weight)
if criterion_value > best_criterion_value:
best_criterion_value = criterion_value
best_index = index
best_index_value = index_value
# 如果criterion_value減少不夠則停止
if best_index is None:
return
if best_criterion_value <= self.min_impurity_decrease:
return
# 切分
current_node.feature_index = best_index
current_node.feature_value = best_index_value
selected_x = x[:, best_index]
# 建立左孩子結點
left_selected_index = np.where(selected_x == best_index_value)
# 如果切分後的點太少，以至於都不能做葉子節點，則停止分割
if len(left_selected_index[0]) >= self.min_samples_leaf:
left_child_node = self.Node()
current_node.left_child_node = left_child_node
self._build_tree(current_depth + 1, left_child_node, x[left_selected_index], y[left_selected_index],
sample_weight[left_selected_index])
# 建立右孩子結點
right_selected_index = np.where(selected_x != best_index_value)
# 如果切分後的點太少，以至於都不能做葉子節點，則停止分割
if len(right_selected_index[0]) >= self.min_samples_leaf:
right_child_node = self.Node()
current_node.right_child_node = right_child_node
self._build_tree(current_depth + 1, right_child_node, x[right_selected_index], y[right_selected_index],
sample_weight[right_selected_index])
def fit(self, x, y, sample_weight=None):
# check sample_weight
n_sample = x.shape[0]
if sample_weight is None:
sample_weight = np.asarray([1.0] * n_sample)
# check sample_weight
if len(sample_weight) != n_sample:
raise Exception('sample_weight size error:', len(sample_weight))
# 構建空的根節點
self.root_node = self.Node()
# 對x分箱
self.dbw.fit(x)
# 遞迴構建樹
self._build_tree(1, self.root_node, self.dbw.transform(x), y, sample_weight)
# 檢索葉子節點的結果
def _search_node(self, current_node: Node, x, class_num):
if current_node.left_child_node is not None and x[current_node.feature_index] == current_node.feature_value:
return self._search_node(current_node.left_child_node, x, class_num)
elif current_node.right_child_node is not None and x[current_node.feature_index] != current_node.feature_value:
return self._search_node(current_node.right_child_node, x, class_num)
else:
result = []
total_value = 0.0
for index in range(0, class_num):
value = current_node.target_distribute.get(index, 0) * current_node.weight_distribute.get(index, 1.0)
result.append(value)
total_value += value
# 歸一化
for index in range(0, class_num):
result[index] = result[index] / total_value
return result
def predict_proba(self, x):
# 計算結果概率分佈
x = self.dbw.transform(x)
rows = x.shape[0]
results = []
class_num = len(self.root_node.target_distribute)
for row in range(0, rows):
results.append(self._search_node(self.root_node, x[row], class_num))
return np.asarray(results)
def predict(self, x):
return np.argmax(self.predict_proba(x), axis=1)
def _prune_node(self, current_node: Node, alpha):
# 如果有子結點,先對子結點部分剪枝
if current_node.left_child_node is not None:
self._prune_node(current_node.left_child_node, alpha)
if current_node.right_child_node is not None:
self._prune_node(current_node.right_child_node, alpha)
# 再嘗試對當前結點剪枝
if current_node.left_child_node is not None or current_node.right_child_node is not None:
# 避免跳層剪枝
for child_node in [current_node.left_child_node, current_node.right_child_node]:
# 當前剪枝的層必須是葉子結點的層
if child_node.left_child_node is not None or child_node.right_child_node is not None:
return
# 計算剪枝的前的損失值
pre_prune_value = alpha * 2
for child_node in [current_node.left_child_node, current_node.right_child_node]:
for key, value in child_node.target_distribute.items():
pre_prune_value += -1 * child_node.num_sample * value * np.log(
value) * child_node.weight_distribute.get(key, 1.0)
# 計算剪枝後的損失值
after_prune_value = alpha
for key, value in current_node.target_distribute.items():
after_prune_value += -1 * current_node.num_sample * value * np.log(
value) * current_node.weight_distribute.get(key, 1.0)
if after_prune_value <= pre_prune_value:
# 剪枝操作
current_node.left_child_node = None
current_node.right_child_node = None
current_node.feature_index = None
current_node.feature_value = None
def prune(self, alpha=0.01):
"""
決策樹剪枝 C(T)+alpha*|T|
:param alpha:
:return:
"""
# 遞迴剪枝
self._prune_node(self.root_node, alpha)
#造偽資料
from sklearn.datasets import make_classification
data, target = make_classification(n_samples=100, n_features=2, n_classes=2, n_informative=1, n_redundant=0,
n_repeated=0, n_clusters_per_class=1, class_sep=.5,random_state=21)
#訓練並檢視效果
tree = CARTClassifier()
tree.fit(data, target)
utils.plot_decision_function(data, target, tree)
一樣的，如果不加以限制，同樣會存在過擬合現象，所以可以剪枝...
#剪枝
tree.prune(5)
utils.plot_decision_function(data, target, tree)
CART迴歸樹
迴歸樹的特徵選擇是使用的平方誤差，即選擇一個特徵\(j\)和一個取值\(s\),將訓練集按\(X^j\leq s\)和\(X^j>s\)分為兩部分，尋找使這兩部分的誤差平方之和下降最多的\(j,s\)，這個過程可以描述如下：
\[\min_{j,s}[\min_{c_1}\sum_{x_i\in R_1(j,s)}(y_i-c_1)^2+\min_{c_2}\sum_{x_i\in R_2(j,s)}(y_i-c_2)^2]
\]
這裡\(R_1(j,s)=\{x\mid x^j\leq s\},R_2(j,s)=\{x\mid x^j> s\},c_1=ave(y_i\mid x_i\in R_1(j,s)),c_2=ave(y_i\mid x_i\in R_2(j,s))\)
程式碼實現：
"""
平方誤差相關函式，封裝到ml_models.utils
"""
def square_error(x, sample_weight=None):
"""
平方誤差
:param x:
:param sample_weight:
:return:
"""
x = np.asarray(x)
x_mean = np.mean(x)
x_num = len(x)
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
error = 0.0
for index in range(0, x_num):
error += (x[index] - x_mean) * (x[index] - x_mean) * sample_weight[index]
return error
def cond_square_error(x, y, sample_weight=None):
"""
計算按x分組的y的誤差值
:param x:
:param y:
:param sample_weight:
:return:
"""
x = np.asarray(x)
y = np.asarray(y)
# x中元素個數
x_num = len(x)
# 如果sample_weight為None設均設定一樣
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
# 計算
error = .0
for x_value in set(x):
x_index = np.where(x == x_value)
new_y = y[x_index]
new_sample_weight = sample_weight[x_index]
error += square_error(new_y, new_sample_weight)
return error
def square_error_gain(x, y, sample_weight=None):
"""
平方誤差帶來的增益值
:param x:
:param y:
:param sample_weight:
:return:
"""
x_num = len(x)
if sample_weight is None:
sample_weight = np.asarray([1.0] * x_num)
return square_error(y, sample_weight) - cond_square_error(x, y, sample_weight)
"""
CART迴歸樹實現，封裝到ml_models.tree
"""
class CARTRegressor(object):
class Node(object):
"""
樹節點，用於儲存節點資訊以及關聯子節點
"""
def __init__(self, feature_index: int = None, feature_value=None, y_hat=None, square_error=None,
left_child_node=None, right_child_node=None, num_sample: int = None):
"""
:param feature_index: 特徵id
:param feature_value: 特徵取值
:param y_hat: 預測值
:param square_error: 當前結點的平方誤差
:param left_child_node: 左孩子結點
:param right_child_node: 右孩子結點
:param num_sample:樣本量
"""
self.feature_index = feature_index
self.feature_value = feature_value
self.y_hat = y_hat
self.square_error = square_error
self.left_child_node = left_child_node
self.right_child_node = right_child_node
self.num_sample = num_sample
def __init__(self, criterion='mse', max_depth=None, min_samples_split=2, min_samples_leaf=1, min_std=1e-3,
min_impurity_decrease=0, max_bins=10):
"""
:param criterion:劃分標準，目前僅有平方誤差
:param max_depth:樹的最大深度
:param min_samples_split:當對一個內部結點劃分時，要求該結點上的最小樣本數，預設為2
:param min_std:最小的標準差
:param min_samples_leaf:設定葉子結點上的最小樣本數，預設為1
:param min_impurity_decrease:打算劃分一個內部結點時，只有當劃分後不純度(可以用criterion引數指定的度量來描述)減少值不小於該引數指定的值，才會對該結點進行劃分，預設值為0
"""
self.criterion = criterion
if criterion == 'mse':
self.criterion_func = utils.square_error_gain
self.max_depth = max_depth
self.min_samples_split = min_samples_split
self.min_samples_leaf = min_samples_leaf
self.min_std = min_std
self.min_impurity_decrease = min_impurity_decrease
self.root_node: self.Node = None
self.dbw = DataBinWrapper(max_bins=max_bins)
def _build_tree(self, current_depth, current_node: Node, x, y, sample_weight):
"""
遞迴進行特徵選擇，構建樹
:param x:
:param y:
:param sample_weight:
:return:
"""
rows, cols = x.shape
# 計算當前y的加權平均值
current_node.y_hat = np.dot(sample_weight / np.sum(sample_weight), y)
current_node.num_sample = rows
# 判斷停止切分的條件
current_node.square_error = np.dot(y - np.mean(y), y - np.mean(y))
if np.sqrt(current_node.square_error / rows) <= self.min_std:
return
if rows < self.min_samples_split:
return
if self.max_depth is not None and current_depth > self.max_depth:
return
# 尋找最佳的特徵以及取值
best_index = None
best_index_value = None
best_criterion_value = 0
for index in range(0, cols):
for index_value in sorted(set(x[:, index])):
criterion_value = self.criterion_func((x[:, index] <= index_value).astype(int), y, sample_weight)
if criterion_value > best_criterion_value:
best_criterion_value = criterion_value
best_index = index
best_index_value = index_value
# 如果criterion_value減少不夠則停止
if best_index is None:
return
if best_criterion_value <= self.min_impurity_decrease:
return
# 切分
current_node.feature_index = best_index
current_node.feature_value = best_index_value
selected_x = x[:, best_index]
# 建立左孩子結點
left_selected_index = np.where(selected_x <= best_index_value)
# 如果切分後的點太少，以至於都不能做葉子節點，則停止分割
if len(left_selected_index[0]) >= self.min_samples_leaf:
left_child_node = self.Node()
current_node.left_child_node = left_child_node
self._build_tree(current_depth + 1, left_child_node, x[left_selected_index], y[left_selected_index],
sample_weight[left_selected_index])
# 建立右孩子結點
right_selected_index = np.where(selected_x > best_index_value)
# 如果切分後的點太少，以至於都不能做葉子節點，則停止分割
if len(right_selected_index[0]) >= self.min_samples_leaf:
right_child_node = self.Node()
current_node.right_child_node = right_child_node
self._build_tree(current_depth + 1, right_child_node, x[right_selected_index], y[right_selected_index],
sample_weight[right_selected_index])
def fit(self, x, y, sample_weight=None):
# check sample_weight
n_sample = x.shape[0]
if sample_weight is None:
sample_weight = np.asarray([1.0] * n_sample)
# check sample_weight
if len(sample_weight) != n_sample:
raise Exception('sample_weight size error:', len(sample_weight))
# 構建空的根節點
self.root_node = self.Node()
# 對x分箱
self.dbw.fit(x)
# 遞迴構建樹
self._build_tree(1, self.root_node, self.dbw.transform(x), y, sample_weight)
# 檢索葉子節點的結果
def _search_node(self, current_node: Node, x):
if current_node.left_child_node is not None and x[current_node.feature_index] <= current_node.feature_value:
return self._search_node(current_node.left_child_node, x)
elif current_node.right_child_node is not None and x[current_node.feature_index] > current_node.feature_value:
return self._search_node(current_node.right_child_node, x)
else:
return current_node.y_hat
def predict(self, x):
# 計算結果概率分佈
x = self.dbw.transform(x)
rows = x.shape[0]
results = []
for row in range(0, rows):
results.append(self._search_node(self.root_node, x[row]))
return np.asarray(results)
def _prune_node(self, current_node: Node, alpha):
# 如果有子結點,先對子結點部分剪枝
if current_node.left_child_node is not None:
self._prune_node(current_node.left_child_node, alpha)
if current_node.right_child_node is not None:
self._prune_node(current_node.right_child_node, alpha)
# 再嘗試對當前結點剪枝
if current_node.left_child_node is not None or current_node.right_child_node is not None:
# 避免跳層剪枝
for child_node in [current_node.left_child_node, current_node.right_child_node]:
# 當前剪枝的層必須是葉子結點的層
if child_node.left_child_node is not None or child_node.right_child_node is not None:
return
# 計算剪枝的前的損失值
pre_prune_value = alpha * 2 + \
(0.0 if current_node.left_child_node.square_error is None else current_node.left_child_node.square_error) + \
(0.0 if current_node.right_child_node.square_error is None else current_node.right_child_node.square_error)
# 計算剪枝後的損失值
after_prune_value = alpha + current_node.square_error
if after_prune_value <= pre_prune_value:
# 剪枝操作
current_node.left_child_node = None
current_node.right_child_node = None
current_node.feature_index = None
current_node.feature_value = None
current_node.square_error = None
def prune(self, alpha=0.01):
"""
決策樹剪枝 C(T)+alpha*|T|
:param alpha:
:return:
"""
# 遞迴剪枝
self._prune_node(self.root_node, alpha)
#構造資料
data = np.linspace(1, 10, num=100)
target = np.sin(data) + np.random.random(size=100)#新增噪聲
data = data.reshape((-1, 1))
tree = CARTRegressor(max_bins=50)
tree.fit(data, target)
import matplotlib.pyplot as plt
plt.scatter(data, target)
plt.plot(data, tree.predict(data), color='r')
[<matplotlib.lines.Line2D at 0x221783ed9b0>]
#剪枝
tree.prune(1)
plt.scatter(data, target)
plt.plot(data, tree.predict(data), color='r')
[<matplotlib.lines.Line2D at 0x221783fcb70>]
