线性回归

1.标准回归

from numpy import *
import matplotlib.pyplot as plt

#标准回归函数和数据导入函数
def loadDataSet(filename):
    # f = open(filename)
    # dataSet = []
    # data=f.readlines()
    # for line in data:
    #     line=line.strip().split('	')
    #     dataSet.append(line)
    # print(dataSet)
    numFeat = len(open(filename).readline().split('	'))-1
    dataMat = []
    labelMat = []
    f = open(filename)
    for line in f.readlines():
        lineArr = []
        curline = line.strip().split('	')
        for i in range(numFeat):
            lineArr.append(float(curline[i]))
        labelMat.append(float(line.strip().split('	')[-1]))
        dataMat.append(lineArr)
    return dataMat,labelMat
    # print("dataMat",dataMat)
    # print("labelMat",labelMat)

#计算回归系数
def standRegres(xArr,yArr):
    xMat = mat(xArr)
    yMat = mat(yArr).T
    xTx = xMat.T*xMat
    if linalg.det(xTx) == 0.0:
        print("该矩阵是奇异矩阵，不能求逆矩阵")
        return
    ws = xTx.I * (xMat.T * yMat)#回归系数
    return ws

#绘标准回归
def fig_regre(xMat,yMat,ws):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xMat[:,1].flatten().A[0],yMat.T[:,0].flatten().A[0])#flatten用于array和mat对象，flatten是深拷贝，不改变
    #xMat[:,1].flatten()==>matrix[[]]         xMat[:,1].flatten().A==>array[[]]           xMat[:,1].flatten().A[0]==>array[]
    #plt.show()

    xCopy = xMat.copy()#浅拷贝，改变
    xCopy.sort(0)
    #print(xCopy.sort(0))
    print("ws",ws)
    yHat = xCopy * ws
    ax.plot(xCopy[:,1],yHat,c='r')
    plt.show()

2.局部加权线性回归

局部加权回归是基于非参数学习算法的思想，使得特征的选择更好。赋予预测点附近每一个点以一定的权值，在这上面基于波长函数来进行普通的线性回归.可以实现对临近点的精确拟合同时忽略那些距离较远的点的贡献，即近点的权值大，远点的权值小，k为波长参数，控制了权值随距离下降的速度，越大下降的越快。越小越精确并且太小可能出现过拟合的问题。(红色地方加上W权重，W跟距离有关，距离越近，权重越大)

#局部加权线性回归函数
def lwlr(testPoint,xArr,yArr,k=1.0):
    #创建对角矩阵
    xMat = mat(xArr)
    yMat = mat(yArr).T
    m = list(shape(xMat))[0]
    weights = mat(eye(m))

    #权重值大小以指数级衰减
    for j in range(m):
        diffMat = testPoint - xMat[j,:]
        weights[j,j] = exp(diffMat*diffMat.T/(-2.0 * k**2))
        # weights[j, j] = exp (linalg.det(diffMat) / (-2.0 * k ** 2))
    xTx = xMat.T * (weights * xMat)
    if linalg.det(xTx) == 0:
        print("非奇异，不能求逆")
    ws = xTx.I * (xMat.T * (weights * yMat))
    return testPoint * ws

#测试,得到y的估计值
def lwlrText(testArr,xArr,yArr,k=1.0):
    m = list(shape(testArr))[0]
    yHat = zeros(m)
    for i in range(m):
        yHat[i] = lwlr(testArr[i],xArr,yArr,k)
    return yHat

测试

if __name__ == '__main__':
    xArr1, yArr1 = loadDataSet("ex0.txt")
    # ws1 = standRegres(xArr1,yArr1)
    xMat = mat(xArr1)
    yMat = mat(yArr1)
    # yHat1 = xMat1 * ws1#估测值
    #fig_regre(xMat1, yMat1, ws1)
    # p1 = corrcoef(yHat1.T, yMat1)
    yHat1 = lwlrText (xArr1, xArr1, yArr1, k=1.0)
    yHat2 = lwlrText (xArr1, xArr1, yArr1, k=0.01)
    yHat3 = lwlrText (xArr1, xArr1, yArr1, k=0.003)

    strInd = xMat[:, 1].argsort (0)  # argsort函数返回的是数组值从小到大的索引值
    xSort = xMat[strInd][:, 0, :]

    fig = plt.figure ()
    ax = fig.add_subplot(221)
    ax.plot (xSort[:, 1], yHat1[strInd])
    ax.scatter (xMat[:, 1].flatten ().A[0], yMat.T.flatten ().A[0], s=2, c='blue')

    ax = fig.add_subplot (222)
    ax.plot (xSort[:, 1], yHat2[strInd])
    ax.scatter (xMat[:, 1].flatten ().A[0], yMat.T.flatten ().A[0], s=2, c='green')

    ax = fig.add_subplot (223)
    ax.plot (xSort[:, 1], yHat3[strInd])
    ax.scatter (xMat[:, 1].flatten ().A[0], yMat.T.flatten ().A[0], s=2, c='red')
    plt.show ()
    # xArr2, yArr2 = loadDataSet ("ex1.txt")
    # ws2 = standRegres (xArr2, yArr2)
    # xMat2 = mat (xArr2)
    # yMat2 = mat (yArr2)
    # yHat2 = xMat2 * ws2  # 估测值
    #fig_regre (xMat2, yMat2, ws2)
    # p2 = corrcoef (yHat2.T, yMat2)#计算预测值和真实值之间的相关性
    # print ("p2", p2)

结果如下：图1 欠拟合 k=1.0 与最小二乘法差不多，

图2 k=0.001 可以挖出数据的潜在规律

图3 k=0.003 过拟合