树回归

输入数据与目标变量之间为非线性关系时，可用树回归，使用树对预测值分段，包括分段常数、分段直线，前者为回归树，后者为模型树。若数据过拟合，需剪枝。

#!/usr/bin/python
# -*- coding: utf-8 -*-
#coding=utf-8

from numpy import *

#导入数据
def loadDataSet(fileName):
datMat = []
fr = open(fileName)
for line in fr.readlines():
curLine = line.strip().split('\t')
frLine = map(float, curLine)
datMat.append(frLine)
return datMat

#输入参数：数据集合，待切分的特征，该特征的某个某个值
def binSplitDataSet(dataSet, feature, value):
mat0 = dataSet[nonzero(dataSet[:, feature] > value)[0], :][0]
mat1 = dataSet[nonzero(dataSet[:, feature] <= value)[0], :][0]
return mat0, mat1

#生成叶节点，为目标变量的均值
def regLeaf(dataSet):
return mean(dataSet[:, -1])

#误差估计函数，总方差
def regErr(dataSet):
return var(dataSet[:, -1]) * shape(dataSet)[0]

#找到数据的最佳二元切分方式
#如果找不到一个“好”的二元切分，返回None并同时调用createTree()产生叶结点
def chooseBestSplit(dataSet, leafType = regLeaf, errType = regErr, ops=(1,4)):
tolS = ops[0]  #容许的误差下降值
tolN = ops[1]  #切分的最少样本数，如果为1，直接返回
if len(set(dataSet[:, -1].T.tolist()[0])) == 1:
return None, leafType(dataSet)
m, n = shape(dataSet)
S = errType(dataSet)  #误差
bestS = inf
bestIndex = 0
bestValue = 0
for featIndex in range(n-1):  #每个特征
for splitVal in set(dataSet[:, featIndex]):  #该特征的所有取值
mat0, mat1 = binSplitDataSet(dataSet, featIndex, splitVal)
if (shape(mat0)[0] < tolN) or (shape(mat1)[0] < tolN):
continue
newS = errType(mat0) + errType(mat1)  #新误差
if newS < bestS:
bestIndex = featIndex
bestValue = splitVal
bestS = newS
if (S - bestS) < tolS:  #如果误差减少不大则退出
return None, leafType(dataSet)
mat0, mat1 = binSplitDataSet(dataSet, bestIndex, bestValue)
if (shape(mat0)[0] < tolN) or (shape(mat1)[0] < tolN):  #如果切分出的数据集很小则退出
return None, leafType(dataSet)
return bestIndex, bestValue

#SART 分类回归树
#输入参数：数据集合，建立叶结点函数，误差计算函数，构建树所需的其它参数的元组
def createTree(dataSet, leafType=regLeaf, errType=regErr, ops=(1,4)):
feat, val = chooseBestSplit(dataSet, leafType, errType, ops)  #将数据集切分成2部分
if feat == None:  #满足停止条件时，返回叶节点值
return val
retTree = {}
retTree['spInd'] = feat
retTree['spVal'] = val
lSet, rSet = binSplitDataSet(dataSet, feat, val)
retTree['left'] = createTree(lSet, leafType, errType, ops)
retTree['right'] = createTree(rSet, leafType, errType, ops)
return retTree

#回归树剪枝函数
#判断输入数据是否为一棵树
def isTree(obj):
return (type(obj).__name__ == 'dict')

#计算2个叶结点的平均值。对树进行塌陷处理
def getMean(tree):
if isTree(tree['right']):
tree['right'] = getMean(tree['right'])
if isTree(tree['left']):
tree['left'] = getMean(tree['left'])
return (tree['right'] + tree['left']) / 2.0

#输入参数：待剪枝的树，待测试的数据
def prune(tree, testData):
if shape(testData)[0] == 0:
return getMean(tree)
if (isTree(tree['right']) or isTree(tree['left'])):
lSet, rSet = binSplitDataSet(testData, tree['spInd'], tree['spVal'])
#对左右子树剪枝
if isTree(tree['left']):
tree['left'] = prune(tree['left'], lSet)
if isTree(tree['right']):
tree['right'] = prune(tree['right'], rSet)
#检查剪枝后的左右子树是否是树，如果不是，可以进行合并
#与合并前的误差进行比较，如果合并后的误差小，则合并，否则不合并
if not isTree(tree['left']) and not isTree(tree['right']):
lSet, rSet = binSplitDataSet(testData, tree['spInd'], tree['spVal'])
errorNoMerge = sum(power(lSet[:,-1] - tree['left'], 2)) + sum(power(rSet[:,-1] - tree['right'], 2))
treeMean = (tree['left'] + tree['right']) / 2.0
errorMerge = sum(power(testData[:, -1] - treeMean, 2))
if errorMerge < errorNoMerge:
print "merging"
return treeMean
else:
return tree
else:
return tree

#模型树
#模型树的叶结点生成函数
#将数据集格式化成目标变量Y和自变量X
def linearSolve(dataSet):
m, n = shape(dataSet)
X = mat(ones((m, n)))
Y = mat(ones((m, 1)))
X[:, 1:n] = dataSet[:, 0:n-1]
Y = dataSet[:, -1]
xTx = X.T * X
if linalg.det(xTx) == 0.0:
raise NameError('This matrix is singular, cannot do inverse, \n try increaseing the second value of ops')
ws = xTx.I * (X.T * Y)
return ws, X, Y

#当数据不再需要切分时，生成叶结点的模型
def modelLeaf(dataSet):
ws, X, Y = linearSolve(dataSet)
return ws

#在给定的数据集上计算误差
def modelErr(dataSet):
ws, X, Y = linearSolve(dataSet)
yHat = X * ws
return sum(power(Y-yHat, 2))

#利用树回归进行预测
#对回归树叶结点进行预测
def regTreeEval(model, inDat):
return float(model)

#对模型树结点进行预测
def modelTreeEval(model, inDat):
n = shape(inDat)[1]
X = mat(ones((1, n+1)))
X[:, 1:n+1] = inDat  #增加第0列
return float(X * model)

#对于输入的单个数据点或者行向量，返回一个浮点值
def treeForeCast(tree, inData, modelEval=regTreeEval):
if not isTree(tree):  #如果是叶结点
return modelEval(tree, inData)
if inData[tree['spInd']] > tree['spVal']:
if isTree(tree['left']):
return treeForeCast(tree['left'], inData, modelEval)
else:
return modelEval(tree['left'], inData)
else:
if isTree(tree['right']):
return treeForeCast(tree['right'], inData, modelEval)
else:
return modelEval(tree['right'], inData)

def createForeCast(tree, testData, modelEval=regTreeEval):
m = len(testData)
yHat = mat(zeros((m, 1)))
for i in range(m):
yHat[i, 0] = treeForeCast(tree, mat(testData[i]), modelEval)
return yHat

测试：回归树

>>> import regTree
>>> myDatl = loadDataSet('ex0.txt')
>>> myDatl = mat(myDatl)
>>> createTree(myDatl)
{'spInd': 1, 'spVal': matrix([[ 0.39435]]), 'right': {'spInd': 1, 'spVal': matrix([[ 0.197834]]), 'right': -0.023838155555555553, 'left': 1.0289583666666664}, 'left': {'spInd': 1, 'spVal': matrix([[ 0.582002]]), 'right': 1.9800350714285717, 'left': {'spInd': 1, 'spVal': matrix([[ 0.797583]]), 'right': 2.9836209534883724, 'left': 3.9871632000000004}}}

测试：剪枝

>>> myMat2 = loadDataSet('ex2.txt')
>>> myMat2 = mat(myMat2)
>>> myTree = createTree(myMat2, ops=(0,1))
>>> myDatTest = loadDataSet('ex2test.txt')
>>> myMat2Test = mat(myDatTest)
>>> prune(myTree, myMat2Test)
merging
merging
... ...
850000001}}, 'left': {'spInd': 0, 'spVal': matrix([[ 0.948822]]), 'right': 69.318648999999994, 'left': 96.41885225}}}}}}}}}}}, 'left': {'spInd': 0, 'spVal': matrix([[ 0.965969]]), 'right': {'spInd': 0, 'spVal': matrix([[ 0.956951]]), 'right': 111.2013225, 'left': {'spInd': 0, 'spVal': matrix([[ 0.958512]]), 'right': 135.83701300000001, 'left': {'spInd': 0, 'spVal': matrix([[ 0.960398]]), 'right': 123.559747, 'left': 112.386764}}}, 'left': 92.523991499999994}}}}

测试：模型树

>>> import regTree
>>> myMat2 = mat(loadDataSet('exp2.txt'))
>>> createTree(myMat2, modelLeaf, modelErr, (1,10))
{'spInd': 0, 'spVal': matrix([[ 0.285477]]),
'right': matrix([[ 3.46877936],[ 1.18521743]]),
'left': matrix([[  1.69855694e-03],[1.19647739e+01]])}

测试：树回归与标准回归比较

R^2 越接近1越好

#回归树
>>> trainMat = mat(loadDataSet('bikeSpeedVsIq_train.txt'))
>>> testMat = mat(loadDataSet('bikeSpeedVsIq_test.txt'))
>>> myTree = createTree(trainMat, ops=(1,20))
>>> yHat = createForeCast(myTree, testMat[:,0])
>>> corrcoef(yHat, testMat[:,1], rowvar=0)[0,1]
0.96408523182221306

#模型树
>>> import regTree
>>> trainMat = mat(loadDataSet('bikeSpeedVsIq_train.txt'))
>>> testMat = mat(loadDataSet('bikeSpeedVsIq_test.txt'))
>>> myTree = createTree(trainMat, modelLeaf, modelErr, ops=(1,20))
>>> yHat = createForeCast(myTree, testMat[:, 0], modelTreeEval)
>>> corrcoef(yHat, testMat[:,1], rowvar=0)[0,1]
0.97604121913806285

#标准回归
>>> ws, X, Y = linearSolve(trainMat)
>>> ws
matrix([[ 37.58916794], [6.18978355]])
>>> yHat = testMat[:,0] * ws[1,0] + ws[0,0]
>>> corrcoef(yHat, testMat[:,1], rowvar=0)[0,1]
0.94346842356747584

可以看到，树回归要由于标准回归