python机器学习笔记（8）—— 树回归
0x00 什么是树回归？
这里用到的是叫做CART(Classification And Regression Trees)的算法。
简单说就是把散点的数据聚成一团一团（跟聚类是两码事），然后用树的方式记录它们的位置。如果在树的构建时加入线性回归能得到更好的结果。
- 优点：可以对复杂和非线性的数据建模。
- 缺点：结果不易理解。
- 适用数据范围：数值型和标量型。
0x01 构建树
def binSplitDataSet(dataSet, feature, value):
#nonzero()会返回两个array，分别记录非零的行与列
mat0 = dataSet[nonzero(dataSet[:,feature] > value)[0],:]
mat1 = dataSet[nonzero(dataSet[:,feature] <= value)[0],:]
return mat0,mat1
def regLeaf(dataSet):#returns the value used for each leaf
return mean(dataSet[:,-1])
def regErr(dataSet):
return var(dataSet[:,-1]) * shape(dataSet)[0]
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 = 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]):
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 split
def createTree(dataSet, leafType=regLeaf, errType=regErr, ops=(1,4)):#assume dataSet is NumPy Mat so we can array filtering
feat, val = chooseBestSplit(dataSet, leafType, errType, ops)#choose the best split
if feat == None: return val #if the splitting hit a stop condition 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
关键在于chooseBestSplit()函数。该函数计算了在何种分类情况下分类结果与训练数据的误差最小。
0x02 树剪枝
当数据值差距较大时，可能出现分支过多的情况，也就是过拟合。对此，能使用预剪枝和后剪枝的方式来处理。
预剪枝在上面的代码中已经用到了，其实就是设置每个分支最少样本数和最大允许误差的过程。
def chooseBestSplit(dataSet, leafType=regLeaf, errType=regErr, ops=(1,4))
如果设置样本数过少，那么就容易造成过拟合。
然后是后剪枝的操作：
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'])/2.0
def prune(tree, testData):
if shape(testData)[0] == 0: return getMean(tree) #if we have no test data collapse the tree
if (isTree(tree['right']) or isTree(tree['left'])):#if the branches are not trees try to prune them
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 they are now both leafs, see if we can merge them
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
主要思想是如果两个分支合并后误差比未合并小，那么久合并两个分支。
0x03 模型树
现在对代码做部分改动，把每个节点由原来的平局值，换成线性回归的回归系数，达到一个更好的拟合效果。
def linearSolve(dataSet): #helper function used in two places
m,n = shape(dataSet)
X = mat(ones((m,n))); Y = mat(ones((m,1)))#create a copy of data with 1 in 0th postion
X[:,1:n] = dataSet[:,0:n-1]; Y = dataSet[:,-1]#and strip out Y
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,Y
def modelLeaf(dataSet):#create linear model and return coeficients
ws,X,Y = linearSolve(dataSet)
return ws
def modelErr(dataSet):
ws,X,Y = linearSolve(dataSet)
yHat = X * ws
return sum(power(Y - yHat,2))
0x04 预测代码
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 treeForeCast(tree, inData, modelEval=regTreeEval):
if not isTree(tree): 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