机器学习：决策树的预剪枝和后剪枝
述概：
剪枝：在机器学习的决策树算法中，为防止过拟合现象和过度开销，而采用剪枝的方法，主要有预剪枝和后剪枝两种常见方法。
预剪枝：在决策树生成的过程中，预先估计对结点进行划分能否提升决策树泛化性能。如果能提升，则对此结点进行划分，否则不划分。
优点：
1、使用预剪枝，决策树中很多分支未展开，可以很好的防止过拟合。
2、因为是在构造决策树的过程中进行的，所以时间开销比较小。
缺点：
1、预剪枝是基于贪心的策略。虽然一个结点进行划分不能带来泛化性能的提升，但很可能其后续结点能够带来泛化性能的提升。所以这种贪心策略放弃了一些泛化性能提升的可能性。
2、由于贪心策略，预剪枝决策树欠拟合的风险会比较大。
后剪枝：后剪枝是在决策树构建完成之后，自底向上地对每一个非叶结点进行考察，如果将此结点地子树替换为叶结点能够带来决策树模型泛化性能地提升，那么就将此非叶结点地子树替换为叶结点，否则不替换。
优点：
1、与预剪枝相比，保留了更多的分支，欠拟合风险比较小
2、泛化性能一般情况下也比预剪枝得到的决策树泛化性能好
缺点：
剪枝发生在决策树构建完成之后，而且要自底向上的检查每个非叶结点，时间开销会比较大。
Python实现：
以西瓜书中的”西瓜数据集“为例实现剪枝。
import math
import matplotlib
import numpy as np
# # 创建西瓜书数据集2.0
def createDataXG20():
data = np.array([['青绿', '蜷缩', '浊响', '清晰', '凹陷', '硬滑']
, ['乌黑', '蜷缩', '沉闷', '清晰', '凹陷', '硬滑']
, ['乌黑', '蜷缩', '浊响', '清晰', '凹陷', '硬滑']
, ['青绿', '蜷缩', '沉闷', '清晰', '凹陷', '硬滑']
, ['浅白', '蜷缩', '浊响', '清晰', '凹陷', '硬滑']
, ['青绿', '稍蜷', '浊响', '清晰', '稍凹', '软粘']
, ['乌黑', '稍蜷', '浊响', '稍糊', '稍凹', '软粘']
, ['乌黑', '稍蜷', '浊响', '清晰', '稍凹', '硬滑']
, ['乌黑', '稍蜷', '沉闷', '稍糊', '稍凹', '硬滑']
, ['青绿', '硬挺', '清脆', '清晰', '平坦', '软粘']
, ['浅白', '硬挺', '清脆', '模糊', '平坦', '硬滑']
, ['浅白', '蜷缩', '浊响', '模糊', '平坦', '软粘']
, ['青绿', '稍蜷', '浊响', '稍糊', '凹陷', '硬滑']
, ['浅白', '稍蜷', '沉闷', '稍糊', '凹陷', '硬滑']
, ['乌黑', '稍蜷', '浊响', '清晰', '稍凹', '软粘']
, ['浅白', '蜷缩', '浊响', '模糊', '平坦', '硬滑']
, ['青绿', '蜷缩', '沉闷', '稍糊', '稍凹', '硬滑']])
label = np.array(['是', '是', '是', '是', '是', '是', '是', '是', '否', '否', '否', '否', '否', '否', '否', '否', '否'])
name = np.array(['色泽', '根蒂', '敲声', '纹理', '脐部', '触感'])
return data, label, name
#划分数据集
def splitXgData20(xgData, xgLabel):
xgDataTrain = xgData[[0, 1, 2, 5, 6, 9, 13, 14, 15, 16], :]
xgDataTest = xgData[[3, 4, 7, 8, 10, 11, 12], :]
xgLabelTrain = xgLabel[[0, 1, 2, 5, 6, 9, 13, 14, 15, 16]]
xgLabelTest = xgLabel[[3, 4, 7, 8, 10, 11, 12]]
return xgDataTrain, xgLabelTrain, xgDataTest, xgLabelTest
# 定义一个常用函数 用来求numpy array中数值等于某值的元素数量
equalNums = lambda x, y: 0 if x is None else x[x == y].size
# 定义计算信息熵的函数
def singleEntropy(x):
"""计算一个输入序列的信息熵"""
# 转换为 numpy 矩阵
x = np.asarray(x)
# 取所有不同值
xValues = set(x)
# 计算熵值
entropy = 0
for xValue in xValues:
p = equalNums(x, xValue) / x.size
entropy -= p * math.log(p, 2)
return entropy
# 定义计算条件信息熵的函数
def conditionnalEntropy(feature, y):
"""计算 某特征feature 条件下y的信息熵"""
# 转换为numpy
feature = np.asarray(feature)
y = np.asarray(y)
# 取特征的不同值
featureValues = set(feature)
# 计算熵值
entropy = 0
for feat in featureValues:
p = equalNums(feature, feat) / feature.size
entropy += p * singleEntropy(y[feature == feat])
return entropy
# 定义信息增益
def infoGain(feature, y):
return singleEntropy(y) - conditionnalEntropy(feature, y)
# 定义信息增益率
def infoGainRatio(feature, y):
return 0 if singleEntropy(feature) == 0 else infoGain(feature, y) / singleEntropy(feature)
# 使用西瓜数据测试函数
xgData, xgLabel, xgName = createDataXG20()
print("Ent(D)为：" + str(round(singleEntropy(xgLabel), 4)))
print("Gain(D, 色泽)为：" + str(round(infoGain(xgData[:,0] ,xgLabel), 4)))
print("Gain(D, 根蒂)为：" + str(round(infoGain(xgData[:,1] ,xgLabel), 4)))
print("Gain(D, 敲声)为：" + str(round(infoGain(xgData[:,2] ,xgLabel), 4)))
print("Gain(D, 纹理)为：" + str(round(infoGain(xgData[:,3] ,xgLabel), 4)))
print("Gain(D, 脐部)为：" + str(round(infoGain(xgData[:,4] ,xgLabel), 4)))
print("Gain(D, 触感)为：" + str(round(infoGain(xgData[:,5] ,xgLabel), 4)))
# 特征选取
def bestFeature(data, labels, method='id3'):
assert method in ['id3', 'c45']
data = np.asarray(data)
labels = np.asarray(labels)
# 根据输入的method选取 评估特征的方法：id3 -> 信息增益; c45 -> 信息增益率
def calcEnt(feature, labels):
if method == 'id3':
return infoGain(feature, labels)
elif method == 'c45':
return infoGainRatio(feature, labels)
# 特征数量 即 data 的列数量
featureNum = data.shape[1]
# 计算最佳特征
bestEnt = 0
bestFeat = -1
for feature in range(featureNum):
ent = calcEnt(data[:, feature], labels)
if ent >= bestEnt:
bestEnt = ent
bestFeat = feature
# print("feature " + str(feature + 1) + " ent: " + str(ent)+ "t bestEnt: " + str(bestEnt))
return bestFeat, bestEnt
# 根据特征及特征值分割原数据集 删除data中的feature列，并根据feature列中的值分割 data和label
def splitFeatureData(data, labels, feature):
"""feature 为特征列的索引"""
# 取特征列
features = np.asarray(data)[:, feature]
# 数据集中删除特征列
data = np.delete(np.asarray(data), feature, axis=1)
# 标签
labels = np.asarray(labels)
uniqFeatures = set(features)
dataSet = {}
labelSet = {}
for feat in uniqFeatures:
dataSet[feat] = data[features == feat]
labelSet[feat] = labels[features == feat]
return dataSet, labelSet
# 多数投票
def voteLabel(labels):
uniqLabels = list(set(labels))
labels = np.asarray(labels)
finalLabel = 0
labelNum = []
for label in uniqLabels:
# 统计每个标签值得数量
labelNum.append(equalNums(labels, label))
# 返回数量最大的标签
return uniqLabels[labelNum.index(max(labelNum))]
# 创建决策树
def createTree(data, labels, names, method='id3'):
data = np.asarray(data)
labels = np.asarray(labels)
names = np.asarray(names)
# 如果结果为单一结果
if len(set(labels)) == 1:
return labels[0]
# 如果没有待分类特征
elif data.size == 0:
return voteLabel(labels)
# 其他情况则选取特征
bestFeat, bestEnt = bestFeature(data, labels, method=method)
# 取特征名称
bestFeatName = names[bestFeat]
# 从特征名称列表删除已取得特征名称
names = np.delete(names, [bestFeat])
# 根据选取的特征名称创建树节点
decisionTree = {bestFeatName: {}}
# 根据最优特征进行分割
dataSet, labelSet = splitFeatureData(data, labels, bestFeat)
# 对最优特征的每个特征值所分的数据子集进行计算
for featValue in dataSet.keys():
decisionTree[bestFeatName][featValue] = createTree(dataSet.get(featValue), labelSet.get(featValue), names,
method)
return decisionTree
# 树信息统计 叶子节点数量 和 树深度
def getTreeSize(decisionTree):
nodeName = list(decisionTree.keys())[0]
nodeValue = decisionTree[nodeName]
leafNum = 0
treeDepth = 0
leafDepth = 0
for val in nodeValue.keys():
if type(nodeValue[val]) == dict:
leafNum += getTreeSize(nodeValue[val])[0]
leafDepth = 1 + getTreeSize(nodeValue[val])[1]
else:
leafNum += 1
leafDepth = 1
treeDepth = max(treeDepth, leafDepth)
return leafNum, treeDepth
# 使用模型对其他数据分类
def dtClassify(decisionTree, rowData, names):
names = list(names)
# 获取特征
feature = list(decisionTree.keys())[0]
# 决策树对于该特征的值的判断字段
featDict = decisionTree[feature]
# 获取特征的列
feat = names.index(feature)
# 获取数据该特征的值
featVal = rowData[feat]
# 根据特征值查找结果，如果结果是字典说明是子树，调用本函数递归
if featVal in featDict.keys():
if type(featDict[featVal]) == dict:
classLabel = dtClassify(featDict[featVal], rowData, names)
else:
classLabel = featDict[featVal]
return classLabel
# 可视化
import matplotlib.pyplot as plt
matplotlib.rc("font", family='Microsoft YaHei')
decisionNodeStyle = dict(boxstyle="sawtooth", fc="0.8")
leafNodeStyle = {"boxstyle": "round4", "fc": "0.8"}
arrowArgs = {"arrowstyle": "<-"}
# 画节点
def plotNode(nodeText, centerPt, parentPt, nodeStyle):
createPlot.ax1.annotate(nodeText, xy=parentPt, xycoords="axes fraction", xytext=centerPt
, textcoords="axes fraction", va="center", ha="center", bbox=nodeStyle,
arrowprops=arrowArgs)
# 添加箭头上的标注文字
def plotMidText(centerPt, parentPt, lineText):
xMid = (centerPt[0] + parentPt[0]) / 2.0
yMid = (centerPt[1] + parentPt[1]) / 2.0
createPlot.ax1.text(xMid, yMid, lineText)
# 画树
def plotTree(decisionTree, parentPt, parentValue):
# 计算宽与高
leafNum, treeDepth = getTreeSize(decisionTree)
# 在 1 * 1 的范围内画图，因此分母为 1
# 每个叶节点之间的偏移量
plotTree.xOff = plotTree.figSize / (plotTree.totalLeaf - 1)
# 每一层的高度偏移量
plotTree.yOff = plotTree.figSize / plotTree.totalDepth
# 节点名称
nodeName = list(decisionTree.keys())[0]
# 根节点的起止点相同，可避免画线；如果是中间节点，则从当前叶节点的位置开始，
# 然后加上本次子树的宽度的一半，则为决策节点的横向位置
centerPt = (plotTree.x + (leafNum - 1) * plotTree.xOff / 2.0, plotTree.y)
# 画出该决策节点
plotNode(nodeName, centerPt, parentPt, decisionNodeStyle)
# 标记本节点对应父节点的属性值
plotMidText(centerPt, parentPt, parentValue)
# 取本节点的属性值
treeValue = decisionTree[nodeName]
# 下一层各节点的高度
plotTree.y = plotTree.y - plotTree.yOff
# 绘制下一层
for val in treeValue.keys():
# 如果属性值对应的是字典，说明是子树，进行递归调用； 否则则为叶子节点
if type(treeValue[val]) == dict:
plotTree(treeValue[val], centerPt, str(val))
else:
plotNode(treeValue[val], (plotTree.x, plotTree.y), centerPt, leafNodeStyle)
plotMidText((plotTree.x, plotTree.y), centerPt, str(val))
# 移到下一个叶子节点
plotTree.x = plotTree.x + plotTree.xOff
# 递归完成后返回上一层
plotTree.y = plotTree.y + plotTree.yOff
# 画出决策树
def createPlot(decisionTree):
fig = plt.figure(1, facecolor="white")
fig.clf()
axprops = {"xticks": [], "yticks": []}
createPlot.ax1 = plt.subplot(111, frameon=False, **axprops)
# 定义画图的图形尺寸
plotTree.figSize = 1.5
# 初始化树的总大小
plotTree.totalLeaf, plotTree.totalDepth = getTreeSize(decisionTree)
# 叶子节点的初始位置x 和 根节点的初始层高度y
plotTree.x = 0
plotTree.y = plotTree.figSize
plotTree(decisionTree, (plotTree.figSize / 2.0, plotTree.y), "")
plt.show()
# 使用西瓜数据测试函数
xgData, xgLabel, xgName = createDataXG20()
xgTree = createTree(xgData, xgLabel, xgName, method = 'id3')
print(xgTree)
createPlot(xgTree)
初始决策树：
# 创建预剪枝决策树
def createTreePrePruning(dataTrain, labelTrain, dataTest, labelTest, names, method='id3'):
trainData = np.asarray(dataTrain)
labelTrain = np.asarray(labelTrain)
testData = np.asarray(dataTest)
labelTest = np.asarray(labelTest)
names = np.asarray(names)
# 如果结果为单一结果
if len(set(labelTrain)) == 1:
return labelTrain[0]
# 如果没有待分类特征
elif trainData.size == 0:
return voteLabel(labelTrain)
# 其他情况则选取特征
bestFeat, bestEnt = bestFeature(dataTrain, labelTrain, method=method)
# 取特征名称
bestFeatName = names[bestFeat]
# 从特征名称列表删除已取得特征名称
names = np.delete(names, [bestFeat])
# 根据最优特征进行分割
dataTrainSet, labelTrainSet = splitFeatureData(dataTrain, labelTrain, bestFeat)
# 预剪枝评估
# 划分前的分类标签
labelTrainLabelPre = voteLabel(labelTrain)
labelTrainRatioPre = equalNums(labelTrain, labelTrainLabelPre) / labelTrain.size
# 划分后的精度计算
if dataTest is not None:
dataTestSet, labelTestSet = splitFeatureData(dataTest, labelTest, bestFeat)
# 划分前的测试标签正确比例
labelTestRatioPre = equalNums(labelTest, labelTrainLabelPre) / labelTest.size
# 划分后 每个特征值的分类标签正确的数量
labelTrainEqNumPost = 0
for val in labelTrainSet.keys():
labelTrainEqNumPost += equalNums(labelTestSet.get(val), voteLabel(labelTrainSet.get(val))) + 0.0
# 划分后 正确的比例
labelTestRatioPost = labelTrainEqNumPost / labelTest.size
# 如果没有评估数据 但划分前的精度等于最小值0.5 则继续划分
if dataTest is None and labelTrainRatioPre == 0.5:
decisionTree = {bestFeatName: {}}
for featValue in dataTrainSet.keys():
decisionTree[bestFeatName][featValue] = createTreePrePruning(dataTrainSet.get(featValue),
labelTrainSet.get(featValue)
, None, None, names, method)
elif dataTest is None:
return labelTrainLabelPre
# 如果划分后的精度相比划分前的精度下降, 则直接作为叶子节点返回
elif labelTestRatioPost < labelTestRatioPre:
return labelTrainLabelPre
else:
# 根据选取的特征名称创建树节点
decisionTree = {bestFeatName: {}}
# 对最优特征的每个特征值所分的数据子集进行计算
for featValue in dataTrainSet.keys():
decisionTree[bestFeatName][featValue] = createTreePrePruning(dataTrainSet.get(featValue),
labelTrainSet.get(featValue)
, dataTestSet.get(featValue),
labelTestSet.get(featValue)
, names, method)
return decisionTree
# 将西瓜数据分割为测试集和训练集
xgDataTrain, xgLabelTrain, xgDataTest, xgLabelTest = splitXgData20(xgData, xgLabel)
# 生成不剪枝的树
xgTreeTrain = createTree(xgDataTrain, xgLabelTrain, xgName, method = 'id3')
# 生成预剪枝的树
xgTreePrePruning = createTreePrePruning(xgDataTrain, xgLabelTrain, xgDataTest, xgLabelTest, xgName, method = 'id3')
# 画剪枝前的树
print("剪枝前的树")
createPlot(xgTreeTrain)
# 画剪枝后的树
print("剪枝后的树")
createPlot(xgTreePrePruning)
预剪枝前决策树：
预剪枝后决策树：
# 创建决策树 带预划分标签
def createTreeWithLabel(data, labels, names, method='id3'):
data = np.asarray(data)
labels = np.asarray(labels)
names = np.asarray(names)
# 如果不划分的标签为
votedLabel = voteLabel(labels)
# 如果结果为单一结果
if len(set(labels)) == 1:
return votedLabel
# 如果没有待分类特征
elif data.size == 0:
return votedLabel
# 其他情况则选取特征
bestFeat, bestEnt = bestFeature(data, labels, method=method)
# 取特征名称
bestFeatName = names[bestFeat]
# 从特征名称列表删除已取得特征名称
names = np.delete(names, [bestFeat])
# 根据选取的特征名称创建树节点 划分前的标签votedPreDivisionLabel=_vpdl
decisionTree = {bestFeatName: {"_vpdl": votedLabel}}
# 根据最优特征进行分割
dataSet, labelSet = splitFeatureData(data, labels, bestFeat)
# 对最优特征的每个特征值所分的数据子集进行计算
for featValue in dataSet.keys():
decisionTree[bestFeatName][featValue] = createTreeWithLabel(dataSet.get(featValue), labelSet.get(featValue),
names, method)
return decisionTree
# 将带预划分标签的tree转化为常规的tree
# 函数中进行的copy操作
def convertTree(labeledTree):
labeledTreeNew = labeledTree.copy()
nodeName = list(labeledTree.keys())[0]
labeledTreeNew[nodeName] = labeledTree[nodeName].copy()
for val in list(labeledTree[nodeName].keys()):
if val == "_vpdl":
labeledTreeNew[nodeName].pop(val)
elif type(labeledTree[nodeName][val]) == dict:
labeledTreeNew[nodeName][val] = convertTree(labeledTree[nodeName][val])
return labeledTreeNew
# 后剪枝 训练完成后决策节点进行替换评估 这里可以直接对xgTreeTrain进行操作
def treePostPruning(labeledTree, dataTest, labelTest, names):
newTree = labeledTree.copy()
dataTest = np.asarray(dataTest)
labelTest = np.asarray(labelTest)
names = np.asarray(names)
# 取决策节点的名称 即特征的名称
featName = list(labeledTree.keys())[0]
# print("n当前节点：" + featName)
# 取特征的列
featCol = np.argwhere(names == featName)[0][0]
names = np.delete(names, [featCol])
newTree[featName] = labeledTree[featName].copy()
featValueDict = newTree[featName]
featPreLabel = featValueDict.pop("_vpdl")
# print("当前节点预划分标签：" + featPreLabel)
# 是否为子树的标记
subTreeFlag = 0
# 分割测试数据 如果有数据 则进行测试或递归调用 np的array我不知道怎么判断是否None, 用is None是错的
dataFlag = 1 if sum(dataTest.shape) > 0 else 0
if dataFlag == 1:
# print("当前节点有划分数据！")
dataTestSet, labelTestSet = splitFeatureData(dataTest, labelTest, featCol)
for featValue in featValueDict.keys():
# print("当前节点属性 {0} 的子节点：{1}".format(featValue ,str(featValueDict[featValue])))
if dataFlag == 1 and type(featValueDict[featValue]) == dict:
subTreeFlag = 1
# 如果是子树则递归
newTree[featName][featValue] = treePostPruning(featValueDict[featValue], dataTestSet.get(featValue),
labelTestSet.get(featValue), names)
# 如果递归后为叶子 则后续进行评估
if type(featValueDict[featValue]) != dict:
subTreeFlag = 0
# 如果没有数据 则转换子树
if dataFlag == 0 and type(featValueDict[featValue]) == dict:
subTreeFlag = 1
# print("当前节点无划分数据！直接转换树："+str(featValueDict[featValue]))
newTree[featName][featValue] = convertTree(featValueDict[featValue])
# print("转换结果：" + str(convertTree(featValueDict[featValue])))
if subTreeFlag == 0:
ratioPreDivision = equalNums(labelTest, featPreLabel) / labelTest.size
equalNum = 0
for val in labelTestSet.keys():
equalNum += equalNums(labelTestSet[val], featValueDict[val])
ratioAfterDivision = equalNum / labelTest.size
# 如果划分后的测试数据准确率低于划分前的，则划分无效，进行剪枝，即使节点等于预划分标签
# 注意这里取的是小于，如果有需要 也可以取 小于等于
if ratioAfterDivision < ratioPreDivision:
newTree = featPreLabel
return newTree
# 生成的树的结构
xgTreeBeforePostPruning = {"脐部": {"_vpdl": "是"
, '凹陷': {'色泽':{"_vpdl": "是", '青绿': '是', '乌黑': '是', '浅白': '否'}}
, '稍凹': {'根蒂':{"_vpdl": "是"
, '稍蜷': {'色泽': {"_vpdl": "是"
, '青绿': '是'
, '乌黑': {'纹理': {"_vpdl": "是"
, '稍糊': '是', '清晰': '否', '模糊': '是'}}
, '浅白': '是'}}
, '蜷缩': '否'
, '硬挺': '是'}}
, '平坦': '否'}}
xgTreePostPruning = treePostPruning(xgTreeBeforePostPruning, xgDataTest, xgLabelTest, xgName)
createPlot(convertTree(xgTreeBeforePostPruning))
createPlot(xgTreePostPruning)
后剪枝前决策树：
后剪枝后决策树：
总结：一棵过拟合的树常常十分复杂，剪枝技术的数显就是为了解决这个问题。两种简直方法分别是预剪枝（在书的构建过程中就进行剪枝）和后剪枝（当属构建完成后再剪枝），预剪枝更有效但需要定义一些参数。