一 概述

通常情况下的线性拟合不能很好地预测所有的值，因为它容易导致欠拟合（under fitting），比如数据集是
一个钟形的曲线。而多项式拟合能拟合所有数据，但是在预测新样本的时候又会变得很糟糕，因为它导致数据的
过拟合（overfitting），不符合数据真实的模型。

局部加权回归（LWR）是非参数学习方法。 首先参数学习方法是这样一种方法：在训练完成所有数据后得到一系列训练参数，然后根据训练参数来预测新样本的值，这时不再依赖之前的训练数据了，参数值是确定的。而非参数学习方法是这样一种算法：在预测新样本值时候每次都会重新训练数据得到新的参数值，也就是说每次预测新样本都会依赖训练数据集合，所以每次得到的参数值是不确定的。

具体的介绍在我之前的斯坦福机器学习笔记（一）中有详细介绍。链接为:http://blog.csdn.net/qq_30091945/article/details/61615501

这次运用的数据是来自《机器学习实战》这本书中第八章ex0文件中的数据。

二 代码

import numpy as np
import matplotlib.pyplot as pltdef LoadDataSet(file):""":param file:文件路径:return: 返回测试数据与标签"""ftrain = open(file)trainingset = []labelset = []for line in ftrain.readlines():LineArr = line.strip().split('\t')trainingset.append([float(LineArr[0]),float(LineArr[1])])labelset.append(float(LineArr[2]))return trainingset, labelsetdef LWRTest(traingxarr,xarr,yarr,k):Xmat = np.mat(xarr)Ymat = np.mat(yarr)# m是输入数据的个数,weights是单位矩阵m = np.shape(Xmat)[0]weights = np.eye(m)for i in range(m):diffmat = traingxarr - Xmat[i,:]weights[i,i] = np.exp(-(diffmat*diffmat.transpose())/(2.0*k**2))XT = Xmat.transpose()*(weights*Xmat)if np.linalg.det(XT) == 0:              #矩阵的行列式为0时不能进行之后的计算print('This Matrix is singular, cannot do inverse')return#theta是回归系数theta = XT.I*Xmat.transpose()*(weights*Ymat.transpose())ytest= traingxarr*thetareturn ytestdef LWR(trainingset,xarr,yarr,k):""":param trianingset:训练数据集:param xarr: 输入样本的横坐标:param yarr: 输入样本的纵坐标:param k: 带宽参数:return:返回预测值"""# m是输入数据的个数,weights是单位矩阵m = np.shape(trainingset)[0]ytest = np.zeros(m)for i in range(m):ytest[i] = LWRTest(trainingset[i],xarr,yarr,k)return  ytestdef Show(xarr,yarr,ytest,k):""":param xarr: 样本数据特征值:param ytest: LWR回归得到的预测:param yarr: 样本数据标记值"""xmat = np.mat(xarr)strInd = xmat[:, 1].argsort(0)xSort = xmat[strInd][:, 0, :]fig = plt.figure()ax = fig.add_subplot(1,1,1)ax.plot(xSort[:, 1], ytest[strInd])ax.scatter(xmat[:, 1].flatten().A[0], np.mat(yarr).T.flatten().A[0], s=2, c='red')title = "k = "title += str(k)ax.set_title(title)plt.savefig("D:\\Program Files (x86)\\机器学习\\斯坦福大学机器学习\\中文笔记\\Linear Regression\\LWR\\"+title+".jpg",dpi=400,bbox_inches='tight')plt.show()def run_main():"""这是主函数"""file = 'D:\\Program Files (x86)\\机器学习\\机器学习实战\\源代码\\Ch08\\ex0.txt'xarr,yarr = LoadDataSet(file)k = [1.0,0.01,0.003]for i in k:ytest = LWR(xarr,xarr,yarr,i)Show(xarr,yarr,ytest,i)if __name__ == '__main__':run_main()

---

下面是将上述代码利用面向对象程序设计进行重写的Python代码：

#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Time    : 2018/8/1614:07
# @Author  : DaiPuWei
# E-Mail   : 771830171@qq.com
# @Site    : 
# @File    : LWR.py
# @Software: PyCharmimport numpy as np
import matplotlib as mpl
import matplotlib.pyplot as pltclass LWR:def __init__(self,train_data,train_predict):"""这是局部加权回归的构造函数:param train_data: 训练数据集:param train_predict: 训练数据集对应的预测值"""self.Train_Data = train_dataself.Train_Predict = train_predictdef Gaussian_Weight(self,data,k):"""这是计算测试数据data高斯核函数:param data: 预测数据:param k: 带宽系数"""# data的数据类型是np.array，那么利用dot方法进行矩阵运算的结果是矩阵，哪怕只有一个元素sum = np.sum(data*data)return np.exp(sum/(-2*k*k))def predict(self,test_data,k):"""这是预测测试数据结果的函数:param test_data:测试数据:param k: 带宽系数"""size = np.shape(self.Train_Data)[0]weights = np.zeros((size,size))for i in range(size):diff = test_data-self.Train_Data[i]weights[i,i] = self.Gaussian_Weight(diff,k)XT = self.Train_Data.T.dot(weights)inv = np.linalg.inv(XT.dot(self.Train_Data))theta = inv.T.dot(XT.dot(self.Train_Predict))test_predict = np.sum(theta.T*test_data)return test_predictdef Test(self,Test_Data,k):Test_Predict = []for test_data in Test_Data:test_predict = self.predict(test_data,k)Test_Predict.append(test_predict)Test_Predict = np.array(Test_Predict)return Test_Predictdef LoadDataSet(file):'''导入西瓜数据'''ftrain = open(file)trainingset = []labelset = []for line in ftrain.readlines():LineArr = line.strip().split('\t')trainingset.append([float(LineArr[0]),float(LineArr[1])])labelset.append(float(LineArr[2]))trainingset = np.array(trainingset)labelset = np.array(labelset).reshape((len(labelset),1))return trainingset, labelsetdef run_main():"""这是主函数"""path = "./ex0.txt"trainingset, labelset = LoadDataSet(path)# 解决画图是的中文乱码问题mpl.rcParams['font.sans-serif'] = [u'simHei']mpl.rcParams['axes.unicode_minus'] = FalseK = [1.0,0.1,0.01,0.003]for k in K:lwr = LWR(trainingset,labelset)testpredict = lwr.Test(trainingset,k)plt.scatter(trainingset[:,1],labelset,s =2,c='red')index = trainingset[:, 1].argsort(0)xdata = trainingset[index][:,1]plt.plot(xdata,testpredict[index],'b')plt.title("k=%f"%(k))plt.savefig("k=%f.jpg"%(k))plt.show()plt.close()if __name__ == '__main__':run_main()

下面是模型运行结果如下:
k = 1.0 (欠拟合)

k = 0.01(最佳拟合)

k = 0.003(过拟合)