CART算法的树回归：
返回的每个节点最后是一个最终确定的平均值。
#coding:utf-8import numpy as np# 加载文件数据def loadDataSet(fileName): #general function to parse tab -delimited floats dataMat = [] #assume last column is target value fr = open(fileName) for line in fr.readlines(): curLine = line.strip().split('\t') fltLine = map(float,curLine) #map all elements to float() dataMat.append(fltLine) return dataMat#在dataset中选择特征为feature的这一列，以value值分成两部分def binSplitDataSet(dataSet, feature, value): mat0 = dataSet[np.nonzero(dataSet[:,feature] > value)[0],:][0] mat1 = dataSet[np.nonzero(dataSet[:,feature] <= value)[0],:][0] return mat0,mat1#计算此矩阵的最后一列结果的平均值，用平均值来当做最后的返回结果，后面的模型树返回的是一个 线性模型def regLeaf(dataSet): return np.mean(dataSet[:,-1])#计算dataset结果的混乱程度，用方差反应，因为是连续数据def regErr(dataSet): return np.var(dataSet[:,-1]) * np.shape(dataSet)[0]#选择最佳的分离特征和该特征的分离点#这里的ops是预先的给定值，1是差别太小就不分了，4是分开后的各自样本数，太小就舍去，这是一 种预剪枝方法def chooseBestSplit(dataSet, leafType=regLeaf, errType=regErr, ops=(1,4)): tolS = ops[0]; tolN = ops[1] #if all the target variables are the same value: quit and return value if len(set(dataSet[:,-1].T.tolist()[0])) == 1: #exit cond 1 return None, leafType(dataSet) m,n = np.shape(dataSet) #the choice of the best feature is driven by Reduction in RSS error from mean S = errType(dataSet) bestS = np.inf; bestIndex = 0; bestValue = 0 #循环所有的特征 for featIndex in range(n-1): #循环该特征下的所有特征值 for splitVal in set(dataSet[:,featIndex]): mat0, mat1 = binSplitDataSet(dataSet, featIndex, splitVal) #如果更具这个特征值分成的两类有一个小与预先给定值，说明分类太偏，则不考虑 if (np.shape(mat0)[0] < tolN) or (np.shape(mat1)[0] < tolN): continue newS = errType(mat0) + errType(mat1) if newS < bestS: bestIndex = featIndex bestValue = splitVal bestS = newS #if the decrease (S-bestS) is less than a threshold don't do the split if (S - bestS) < tolS: return None, leafType(dataSet) #exit cond 2 mat0, mat1 = binSplitDataSet(dataSet, bestIndex, bestValue) if (np.shape(mat0)[0] < tolN) or (np.shape(mat1)[0] < tolN): #exit cond 3 return None, leafType(dataSet) return bestIndex,bestValue #创建树def createTree(dataSet, leafType=regLeaf, errType=regErr, ops=(1,4)): feat, val = chooseBestSplit(dataSet, leafType, errType, ops) if feat == None: 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 retTreemyDat = loadDataSet('ex0.txt')myMat = np.mat(myDat)result = createTree(myMat)print result 结果：
{'spInd': 1, 'spVal': matrix(` 0`.`39435`), 'right': {'spInd': 1, 'spVal': matrix(` 0`.`197834`), 'right': -0.023838155555555553, 'left': 1.0289583666666666}, 'left': {'spInd': 1, 'spVal': matrix(` 0`.`582002`), 'right': 1.980035071428571, 'left': {'spInd': 1, 'spVal': matrix(` 0`.`797583`), 'right': 2.9836209534883724, 'left': 3.9871631999999999}}}
结果的意思是：第几个特征，以多大作为特征值分开，分成左右，依次分下去。
这个算法很好，但是对数据的分类太过于高，容易造成过拟合。因此要采用剪枝技术。
通过降低决策树的复杂度来避免过拟合的过程称为剪枝。
#判断obj是否是一个子树def isTree(obj): return (type(obj).__name__=='dict')#用于坍塌处理，当测试数据集是空是，则取整个树的平均值def getMean(tree): if isTree(tree['right']): tree['right'] = getMean(tree['right']) if isTree(tree['left']): tree['left'] = getMean(tree['left']) return (tree['left']+tree['right'])/2.0#剪枝函数def prune(tree, testData): #如果测试数据集为空，则坍塌处理 if np.shape(testData)[0] == 0: return getMean(tree) #如果左或者右是树，则把测试数据集根据决策树进行分割 if (isTree(tree['right']) or isTree(tree['left'])): lSet, rSet = binSplitDataSet(testData, tree['spInd'], tree['spVal']) #如果左侧是树，则把数据集和子树带入继续找 if isTree(tree['left']): tree['left'] = prune(tree['left'], lSet) #同理 if isTree(tree['right']): tree['right'] = prune(tree['right'], rSet) #if they are now both leafs, see if we can merge them #如果左右都是节点，则计算节点误差 if not isTree(tree['left']) and not isTree(tree['right']): lSet, rSet = binSplitDataSet(testData, tree['spInd'], tree['spVal']) #计算不合并的误差 errorNoMerge = sum(np.power(lSet[:,-1] - tree['left'],2)) + sum(np.power(rSet[:,-1] - tree['right'],2)) treeMean = (tree['left']+tree['right'])/2.0 #计算将当前两个叶子节点合并后的误差 errorMerge = sum(np.power(testData[:,-1] - treeMean,2)) if errorMerge < errorNoMerge: print "merging" #可以合并就返回平均值 return treeMean #不可以合并就返回树，不变 else: return tree else: return tree
一般来说都是预剪枝和后剪枝合并使用
模型树
每个节点是一个线性模型
其他基本一样：
#对数据集进行线性回归def linearSolve(dataSet): m,n = np.shape(dataSet) X = np.mat(np.ones((m,n))); Y = np.mat(np.ones((m,1))) #有一列是常数项，因此要多出一列放置常数项 X[:,1:n] = dataSet[:,0:n-1]; Y = dataSet[:,-1] xTx = X.T*X if np.linalg.det(xTx) == 0.0: raise NameError('This matrix is singular, cannot do inverse,\n\ try increasing the second value of ops') ws = xTx.I * (X.T * Y) return ws,X,Y#产生针对该数据集的线性模型#相当于上面的regLeaf函数def modelLeaf(dataSet): ws,X,Y = linearSolve(dataSet) return ws#产生针对该数据集的线性模型，并计算误差返回#相当于上面的regErr函数，计算模型的误差，如果分后和不分的误差差不多则选择不分def modelErr(dataSet): ws,X,Y = linearSolve(dataSet) yHat = X * ws return sum(np.power(Y - yHat,2)) 模型树回归很好，而且可以用作预测