树回归

上篇主要讲了线性回归的一些方法，为全局模型。当数据拥有总舵特征并且特征关系复杂时，全局模型会出现较大的偏差。实际情况中很多问题都是非线性的，全局线性模型手段不利于分析。树回归主要有数值型回归树和模型树两种

CART回归树

决策树主要用于数据的分类，一般用于处理离散型的数据，利用香浓熵来度量集合的无组织程度，选用其他方法来替代香浓熵，就可以使用树构建算法来完成回归。CART算法是应用较广的回归树方法。

创建树伪代码

找到最佳的待切分特征:    如果该节点不能再分，该节点设置为叶节点    执行二元切分    在左子树调用创建树方法(递归)    在右子树调用创建树方法(递归)

CART树中最佳切分的选择为：

每个特征：    对应每个特征值：        将数据集切分为两份        计算切分的误差        如果当前误差小于最小误差，将当前切分设定为最佳切分更新最小误差返回最佳切分的体征和阈值

示例代码

def binSplitDataSet(dataSet, feature, value):    mat0 = dataSet[nonzero(dataSet[:, feature] > value)[0], :]    mat1 = dataSet[nonzero(dataSet[:, feature] <= value)[0], :]    return mat0, mat1# 回归树模型def regLeaf(dataSet):#returns the value used for each leaf    return mean(dataSet[:,-1])def regErr(dataSet):    return var(dataSet[:,-1]) * shape(dataSet)[0]# 创建树以及剪枝操作################################################################################## 选择最优节点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.A.tolist())[0])) == 1: #exit cond 1        return None, leafType(dataSet)    m, n = shape(dataSet)    #the choice of the best feature is driven by Reduction in RSS error from mean    S = errType(dataSet)    bestS = inf; bestIndex = 0; bestValue = 0    for featIndex in range(n-1):        for splitVal in set((dataSet[:,featIndex].T.A.tolist())[0]):            mat0, mat1 = binSplitDataSet(dataSet, featIndex, splitVal)            if (shape(mat0)[0] < tolN) or (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 (shape(mat0)[0] < tolN) or (shape(mat1)[0] < tolN):  #exit cond 3        return None, leafType(dataSet)    return bestIndex,bestValue#returns the best feature to split on                              #and the value used for that splitdef 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 retTree

树剪枝

一棵树如果节点过多，说明该模型可能对数据进行了“过拟合”。通过降低树的复杂度来避免过拟合的过程成为剪枝。剪枝分为预剪枝和后剪枝，预剪枝体现于上面示例代码中的tolS和tolN。预剪枝参数需要人为设定，不同场景差别很大。后剪枝是利用测试集来对树进行剪枝，由于不需要用户制定参数，后剪枝是更理想的方法。

后剪枝伪代码:

基于已有的树切分测试数据：    入股偶存在任一子集树一棵树，则在该子集递归剪枝过程    计算将当前两个叶子节点合并后的误差    计算不合并的误差    如果不合并会降低误差，将叶子节点合并

示例代码

def isTree(obj):    return (type(obj).__name__ == 'dict')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'])/2.0def 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'])    if isTree(tree['left']):        tree['left'] = prune(tree['left'], lSet)    if isTree(tree['right']):        tree['right'] = prune(tree['right'], rSet)    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

将叶节点设定为分段线性函数，模型是有多个线性片段组成的。对前面的稍加修改，就可以在叶节点生成线性模型而不是常数值。

示例代码如下

# 模型树def linearSolve(dataSet):    m, n = shape(dataSet)    X = mat(ones((m, n))); Y = mat(ones((m, 1)))    X[:, 1:n] = dataSet[0, 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 increasing the second value of ops')    ws = xTx.I*(X.T*Y)    return ws, X, Ydef modelLeaf(dataSet):    ws, X, Y = linearSolve(dataSet)    return wsdef modelErr(dataSet):    ws,X,Y = linearSolve(dataSet)    yHat = X*ws    return sum(power((Y-yHat), 2))

数据预测(都适用)
示例代码：

def regTreeEval(model, inDat):    return float(model)def modelTreeEval(model, inDat):    n = shape(inDat)[1]    X = mat(ones((1, n+1)))    X[:,1:n+1] = inDat    return float(X*model)def treeForceCast(tree, inData, modelEval = regTreeEval):    if not isTree(tree): return modelEval(tree, inData)    if inData[tree['spInd']] > tree['spVal']:        if isTree(tree['left']):            return treeForceCast(tree['left'], inData, modelEval)        else:            return modelEval(tree['left'], inData)    else:        if isTree(tree['right']):            return treeForceCast(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] = treeForceCast(tree, mat(testData[i]), modelEval)    return yHat

算法特点

优点： 可以对复杂和非线性数据建模
缺点： 结果不易理解
适用数据类型： 数值型和标称型数据