4. 简单线性回归的python实现
点击标题即可获取源代码和笔记
4.1 导入相关包
import numpy as np
import pandas as pd
import random
import matplotlib as mpl
import matplotlib.pyplot as pltplt.rcParams['font.sans-serif'] = ['simhei'] # 显示中文
plt.rcParams['axes.unicode_minus'] = False # 用来正常显示负号%matplotlib inline # 将图片嵌套在输出框中显示，而不是单独跳出一张图片
4.2 导入数据集并探索数据
ex0 = pd.read_table("./datas/ex0.txt",header=None)
ex0.head()
| 0 | 1 | 2 | |
|---|---|---|---|
| 0 | 1.0 | 0.067732 | 3.176513 |
| 1 | 1.0 | 0.427810 | 3.816464 |
| 2 | 1.0 | 0.995731 | 4.550095 |
| 3 | 1.0 | 0.738336 | 4.256571 |
| 4 | 1.0 | 0.981083 | 4.560815 |
ex0.shape
(200, 3)
ex0.describe()
| 0 | 1 | 2 | |
|---|---|---|---|
| count | 200.0 | 200.000000 | 200.000000 |
| mean | 1.0 | 0.488319 | 3.835601 |
| std | 0.0 | 0.292943 | 0.503443 |
| min | 1.0 | 0.014855 | 3.078132 |
| 25% | 1.0 | 0.234368 | 3.452775 |
| 50% | 1.0 | 0.466573 | 3.839350 |
| 75% | 1.0 | 0.730712 | 4.247613 |
| max | 1.0 | 0.995731 | 4.692514 |
4.3 构建辅助函数
ex0.iloc[:,-1].values
array([3.176513, 3.816464, 4.550095, 4.256571, 4.560815, 3.929515,3.52617 , 3.156393, 3.110301, 3.149813, 3.476346, 4.119688,4.282233, 3.486582, 4.655492, 3.965162, 3.5149 , 3.125947,4.094115, 3.476039, 3.21061 , 3.190612, 4.631504, 4.29589 ,3.085028, 3.44808 , 3.16744 , 3.364266, 3.993482, 3.891471,3.143259, 3.114204, 3.851484, 4.621899, 4.580768, 3.620992,3.580501, 4.618706, 3.676867, 4.641845, 3.175939, 4.26498 ,3.558448, 3.436632, 3.831052, 3.182853, 3.498906, 3.946833,3.900583, 4.238522, 4.23308 , 3.521557, 3.203344, 4.278105,3.555705, 3.502661, 3.859776, 4.275956, 3.916191, 3.587961,3.183004, 4.225236, 4.231083, 4.240544, 3.222372, 4.021445,3.567479, 3.56258 , 4.262059, 3.208813, 3.169825, 4.193949,3.491678, 4.533306, 3.550108, 4.636427, 3.557078, 3.552874,3.494159, 3.206828, 3.195266, 4.221292, 4.413372, 4.184347,3.742878, 3.201878, 4.648964, 3.510117, 3.274434, 3.579622,3.489244, 4.237386, 3.913749, 3.22899 , 4.286286, 4.628614,3.239536, 4.457997, 3.513384, 3.729674, 3.834274, 3.811155,3.598316, 4.692514, 4.604859, 3.864912, 3.184236, 3.500796,3.743365, 3.622905, 4.310796, 3.583357, 3.901852, 3.233521,3.105266, 3.865544, 4.628625, 4.231213, 3.791149, 3.968271,4.25391 , 3.19471 , 3.996503, 3.904358, 3.503976, 4.557545,3.699876, 4.613614, 3.140401, 4.206717, 3.969524, 4.476096,3.136528, 4.279071, 3.200603, 3.299012, 3.209873, 3.632942,3.248361, 3.995783, 3.563262, 3.649712, 3.951845, 3.145031,3.181577, 4.637087, 3.404964, 3.873188, 4.633648, 3.154768,4.623637, 3.078132, 3.913596, 3.221817, 3.938071, 3.880822,4.176436, 4.648161, 3.332312, 4.240614, 4.532224, 4.557105,4.610072, 4.636569, 4.229813, 3.50086 , 4.245514, 4.605182,3.45434 , 3.180775, 3.38082 , 4.56502 , 3.279973, 4.554241,4.63352 , 4.281037, 3.844426, 3.891601, 3.849728, 3.492215,4.592374, 4.632025, 3.75675 , 3.133555, 3.567919, 4.363382,3.560165, 4.564305, 4.215055, 4.174999, 4.58664 , 3.960008,3.529963, 4.213412, 3.908685, 3.585821, 4.374394, 3.213817,3.952681, 3.129283])
ex0.iloc[:,-1].values.shape
(200,)
(ex0.iloc[:,-1].values).T
array([3.176513, 3.816464, 4.550095, 4.256571, 4.560815, 3.929515,3.52617 , 3.156393, 3.110301, 3.149813, 3.476346, 4.119688,4.282233, 3.486582, 4.655492, 3.965162, 3.5149 , 3.125947,4.094115, 3.476039, 3.21061 , 3.190612, 4.631504, 4.29589 ,3.085028, 3.44808 , 3.16744 , 3.364266, 3.993482, 3.891471,3.143259, 3.114204, 3.851484, 4.621899, 4.580768, 3.620992,3.580501, 4.618706, 3.676867, 4.641845, 3.175939, 4.26498 ,3.558448, 3.436632, 3.831052, 3.182853, 3.498906, 3.946833,3.900583, 4.238522, 4.23308 , 3.521557, 3.203344, 4.278105,3.555705, 3.502661, 3.859776, 4.275956, 3.916191, 3.587961,3.183004, 4.225236, 4.231083, 4.240544, 3.222372, 4.021445,3.567479, 3.56258 , 4.262059, 3.208813, 3.169825, 4.193949,3.491678, 4.533306, 3.550108, 4.636427, 3.557078, 3.552874,3.494159, 3.206828, 3.195266, 4.221292, 4.413372, 4.184347,3.742878, 3.201878, 4.648964, 3.510117, 3.274434, 3.579622,3.489244, 4.237386, 3.913749, 3.22899 , 4.286286, 4.628614,3.239536, 4.457997, 3.513384, 3.729674, 3.834274, 3.811155,3.598316, 4.692514, 4.604859, 3.864912, 3.184236, 3.500796,3.743365, 3.622905, 4.310796, 3.583357, 3.901852, 3.233521,3.105266, 3.865544, 4.628625, 4.231213, 3.791149, 3.968271,4.25391 , 3.19471 , 3.996503, 3.904358, 3.503976, 4.557545,3.699876, 4.613614, 3.140401, 4.206717, 3.969524, 4.476096,3.136528, 4.279071, 3.200603, 3.299012, 3.209873, 3.632942,3.248361, 3.995783, 3.563262, 3.649712, 3.951845, 3.145031,3.181577, 4.637087, 3.404964, 3.873188, 4.633648, 3.154768,4.623637, 3.078132, 3.913596, 3.221817, 3.938071, 3.880822,4.176436, 4.648161, 3.332312, 4.240614, 4.532224, 4.557105,4.610072, 4.636569, 4.229813, 3.50086 , 4.245514, 4.605182,3.45434 , 3.180775, 3.38082 , 4.56502 , 3.279973, 4.554241,4.63352 , 4.281037, 3.844426, 3.891601, 3.849728, 3.492215,4.592374, 4.632025, 3.75675 , 3.133555, 3.567919, 4.363382,3.560165, 4.564305, 4.215055, 4.174999, 4.58664 , 3.960008,3.529963, 4.213412, 3.908685, 3.585821, 4.374394, 3.213817,3.952681, 3.129283])
(ex0.iloc[:,-1].values).T.shape
(200,)
'''
函数功能：输入DF数据集（最后一列为标签），返回特征矩阵和标签矩阵
'''
def get_Mat(dataSet):xMat = np.mat(dataSet.iloc[:,:-1].values)yMat = np.mat(dataSet.iloc[:,-1].values).Treturn xMat,yMat# 查看函数运行结果
xMat,yMat = get_Mat(ex0)
xMat.shape
(200, 2)
xMat
matrix([[1. , 0.067732],[1. , 0.42781 ],[1. , 0.995731],[1. , 0.738336],[1. , 0.981083],[1. , 0.526171],[1. , 0.378887],[1. , 0.033859],[1. , 0.132791],[1. , 0.138306],[1. , 0.247809],[1. , 0.64827 ],[1. , 0.731209],[1. , 0.236833],[1. , 0.969788],[1. , 0.607492],[1. , 0.358622],[1. , 0.147846],[1. , 0.63782 ],[1. , 0.230372],[1. , 0.070237],[1. , 0.067154],[1. , 0.925577],[1. , 0.717733],[1. , 0.015371],[1. , 0.33507 ],[1. , 0.040486],[1. , 0.212575],[1. , 0.617218],[1. , 0.541196],[1. , 0.045353],[1. , 0.126762],[1. , 0.556486],[1. , 0.901144],[1. , 0.958476],[1. , 0.274561],[1. , 0.394396],[1. , 0.87248 ],[1. , 0.409932],[1. , 0.908969],[1. , 0.166819],[1. , 0.665016],[1. , 0.263727],[1. , 0.231214],[1. , 0.552928],[1. , 0.047744],[1. , 0.365746],[1. , 0.495002],[1. , 0.493466],[1. , 0.792101],[1. , 0.76966 ],[1. , 0.251821],[1. , 0.181951],[1. , 0.808177],[1. , 0.334116],[1. , 0.33863 ],[1. , 0.452584],[1. , 0.69477 ],[1. , 0.590902],[1. , 0.307928],[1. , 0.148364],[1. , 0.70218 ],[1. , 0.721544],[1. , 0.666886],[1. , 0.124931],[1. , 0.618286],[1. , 0.381086],[1. , 0.385643],[1. , 0.777175],[1. , 0.116089],[1. , 0.115487],[1. , 0.66351 ],[1. , 0.254884],[1. , 0.993888],[1. , 0.295434],[1. , 0.952523],[1. , 0.307047],[1. , 0.277261],[1. , 0.279101],[1. , 0.175724],[1. , 0.156383],[1. , 0.733165],[1. , 0.848142],[1. , 0.771184],[1. , 0.429492],[1. , 0.162176],[1. , 0.917064],[1. , 0.315044],[1. , 0.201473],[1. , 0.297038],[1. , 0.336647],[1. , 0.666109],[1. , 0.583888],[1. , 0.085031],[1. , 0.687006],[1. , 0.949655],[1. , 0.189912],[1. , 0.844027],[1. , 0.333288],[1. , 0.427035],[1. , 0.466369],[1. , 0.550659],[1. , 0.278213],[1. , 0.918769],[1. , 0.886555],[1. , 0.569488],[1. , 0.066379],[1. , 0.335751],[1. , 0.426863],[1. , 0.395746],[1. , 0.694221],[1. , 0.27276 ],[1. , 0.503495],[1. , 0.067119],[1. , 0.038326],[1. , 0.599122],[1. , 0.947054],[1. , 0.671279],[1. , 0.434811],[1. , 0.509381],[1. , 0.749442],[1. , 0.058014],[1. , 0.482978],[1. , 0.466776],[1. , 0.357767],[1. , 0.949123],[1. , 0.41732 ],[1. , 0.920461],[1. , 0.156433],[1. , 0.656662],[1. , 0.616418],[1. , 0.853428],[1. , 0.133295],[1. , 0.693007],[1. , 0.178449],[1. , 0.199526],[1. , 0.073224],[1. , 0.286515],[1. , 0.182026],[1. , 0.621523],[1. , 0.344584],[1. , 0.398556],[1. , 0.480369],[1. , 0.15335 ],[1. , 0.171846],[1. , 0.867082],[1. , 0.223855],[1. , 0.528301],[1. , 0.890192],[1. , 0.106352],[1. , 0.917886],[1. , 0.014855],[1. , 0.567682],[1. , 0.068854],[1. , 0.603535],[1. , 0.53205 ],[1. , 0.651362],[1. , 0.901225],[1. , 0.204337],[1. , 0.696081],[1. , 0.963924],[1. , 0.98139 ],[1. , 0.987911],[1. , 0.990947],[1. , 0.736021],[1. , 0.253574],[1. , 0.674722],[1. , 0.939368],[1. , 0.235419],[1. , 0.110521],[1. , 0.218023],[1. , 0.869778],[1. , 0.19683 ],[1. , 0.958178],[1. , 0.972673],[1. , 0.745797],[1. , 0.445674],[1. , 0.470557],[1. , 0.549236],[1. , 0.335691],[1. , 0.884739],[1. , 0.918916],[1. , 0.441815],[1. , 0.116598],[1. , 0.359274],[1. , 0.814811],[1. , 0.387125],[1. , 0.982243],[1. , 0.78088 ],[1. , 0.652565],[1. , 0.87003 ],[1. , 0.604755],[1. , 0.255212],[1. , 0.730546],[1. , 0.493829],[1. , 0.257017],[1. , 0.833735],[1. , 0.070095],[1. , 0.52707 ],[1. , 0.116163]])
# xMat.A ,把matrix变为array类型
xMat.A[:,1]
array([0.067732, 0.42781 , 0.995731, 0.738336, 0.981083, 0.526171,0.378887, 0.033859, 0.132791, 0.138306, 0.247809, 0.64827 ,0.731209, 0.236833, 0.969788, 0.607492, 0.358622, 0.147846,0.63782 , 0.230372, 0.070237, 0.067154, 0.925577, 0.717733,0.015371, 0.33507 , 0.040486, 0.212575, 0.617218, 0.541196,0.045353, 0.126762, 0.556486, 0.901144, 0.958476, 0.274561,0.394396, 0.87248 , 0.409932, 0.908969, 0.166819, 0.665016,0.263727, 0.231214, 0.552928, 0.047744, 0.365746, 0.495002,0.493466, 0.792101, 0.76966 , 0.251821, 0.181951, 0.808177,0.334116, 0.33863 , 0.452584, 0.69477 , 0.590902, 0.307928,0.148364, 0.70218 , 0.721544, 0.666886, 0.124931, 0.618286,0.381086, 0.385643, 0.777175, 0.116089, 0.115487, 0.66351 ,0.254884, 0.993888, 0.295434, 0.952523, 0.307047, 0.277261,0.279101, 0.175724, 0.156383, 0.733165, 0.848142, 0.771184,0.429492, 0.162176, 0.917064, 0.315044, 0.201473, 0.297038,0.336647, 0.666109, 0.583888, 0.085031, 0.687006, 0.949655,0.189912, 0.844027, 0.333288, 0.427035, 0.466369, 0.550659,0.278213, 0.918769, 0.886555, 0.569488, 0.066379, 0.335751,0.426863, 0.395746, 0.694221, 0.27276 , 0.503495, 0.067119,0.038326, 0.599122, 0.947054, 0.671279, 0.434811, 0.509381,0.749442, 0.058014, 0.482978, 0.466776, 0.357767, 0.949123,0.41732 , 0.920461, 0.156433, 0.656662, 0.616418, 0.853428,0.133295, 0.693007, 0.178449, 0.199526, 0.073224, 0.286515,0.182026, 0.621523, 0.344584, 0.398556, 0.480369, 0.15335 ,0.171846, 0.867082, 0.223855, 0.528301, 0.890192, 0.106352,0.917886, 0.014855, 0.567682, 0.068854, 0.603535, 0.53205 ,0.651362, 0.901225, 0.204337, 0.696081, 0.963924, 0.98139 ,0.987911, 0.990947, 0.736021, 0.253574, 0.674722, 0.939368,0.235419, 0.110521, 0.218023, 0.869778, 0.19683 , 0.958178,0.972673, 0.745797, 0.445674, 0.470557, 0.549236, 0.335691,0.884739, 0.918916, 0.441815, 0.116598, 0.359274, 0.814811,0.387125, 0.982243, 0.78088 , 0.652565, 0.87003 , 0.604755,0.255212, 0.730546, 0.493829, 0.257017, 0.833735, 0.070095,0.52707 , 0.116163])
xMat.A[:,1].shape
(200,)
yMat
matrix([[3.176513],[3.816464],[4.550095],[4.256571],[4.560815],[3.929515],[3.52617 ],[3.156393],[3.110301],[3.149813],[3.476346],[4.119688],[4.282233],[3.486582],[4.655492],[3.965162],[3.5149 ],[3.125947],[4.094115],[3.476039],[3.21061 ],[3.190612],[4.631504],[4.29589 ],[3.085028],[3.44808 ],[3.16744 ],[3.364266],[3.993482],[3.891471],[3.143259],[3.114204],[3.851484],[4.621899],[4.580768],[3.620992],[3.580501],[4.618706],[3.676867],[4.641845],[3.175939],[4.26498 ],[3.558448],[3.436632],[3.831052],[3.182853],[3.498906],[3.946833],[3.900583],[4.238522],[4.23308 ],[3.521557],[3.203344],[4.278105],[3.555705],[3.502661],[3.859776],[4.275956],[3.916191],[3.587961],[3.183004],[4.225236],[4.231083],[4.240544],[3.222372],[4.021445],[3.567479],[3.56258 ],[4.262059],[3.208813],[3.169825],[4.193949],[3.491678],[4.533306],[3.550108],[4.636427],[3.557078],[3.552874],[3.494159],[3.206828],[3.195266],[4.221292],[4.413372],[4.184347],[3.742878],[3.201878],[4.648964],[3.510117],[3.274434],[3.579622],[3.489244],[4.237386],[3.913749],[3.22899 ],[4.286286],[4.628614],[3.239536],[4.457997],[3.513384],[3.729674],[3.834274],[3.811155],[3.598316],[4.692514],[4.604859],[3.864912],[3.184236],[3.500796],[3.743365],[3.622905],[4.310796],[3.583357],[3.901852],[3.233521],[3.105266],[3.865544],[4.628625],[4.231213],[3.791149],[3.968271],[4.25391 ],[3.19471 ],[3.996503],[3.904358],[3.503976],[4.557545],[3.699876],[4.613614],[3.140401],[4.206717],[3.969524],[4.476096],[3.136528],[4.279071],[3.200603],[3.299012],[3.209873],[3.632942],[3.248361],[3.995783],[3.563262],[3.649712],[3.951845],[3.145031],[3.181577],[4.637087],[3.404964],[3.873188],[4.633648],[3.154768],[4.623637],[3.078132],[3.913596],[3.221817],[3.938071],[3.880822],[4.176436],[4.648161],[3.332312],[4.240614],[4.532224],[4.557105],[4.610072],[4.636569],[4.229813],[3.50086 ],[4.245514],[4.605182],[3.45434 ],[3.180775],[3.38082 ],[4.56502 ],[3.279973],[4.554241],[4.63352 ],[4.281037],[3.844426],[3.891601],[3.849728],[3.492215],[4.592374],[4.632025],[3.75675 ],[3.133555],[3.567919],[4.363382],[3.560165],[4.564305],[4.215055],[4.174999],[4.58664 ],[3.960008],[3.529963],[4.213412],[3.908685],[3.585821],[4.374394],[3.213817],[3.952681],[3.129283]])
'''
函数功能：数据集可视化
'''
def plotShow(dataSet):xMat,yMat = get_Mat(dataSet)plt.scatter(xMat.A[:,1],yMat.A,c='b',s=5)plt.show()plotShow(ex0)
![[外链图片转存失败,源站可能有防盗链机制,建议将图片保存下来直接上传(img-ywlKXwSV-1593053028237)(output_19_0.png)]](https://img-blog.csdnimg.cn/2020062510443121.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDgyNzQxOA==,size_16,color_FFFFFF,t_70#pic_center)
4.5 计算回归系数
'''
函数功能：计算回归系数
参数说明：dataSet:原始数据集
返回：ws:回归系数
'''
def standRegres(dataSet):xMat,yMat = get_Mat(dataSet)xTx = xMat.T * xMatif np.linalg.det(xTx) == 0:print('矩阵为奇异矩阵，无法求逆！')returnws = xTx.I*(xMat.T*yMat) # xTx.I ，用来求逆矩阵return ws
说明：det(A)指的是矩阵A的行列式（determinant），如果det(A)=0，则说明矩阵A是奇异矩阵，不可逆。
ws = standRegres(ex0)
ws
matrix([[3.00774324],[1.69532264]])
4.6 绘制最佳拟合直线
'''
函数功能：绘制散点图和最佳拟合直线
'''def plotReg(dataSet):xMat,yMat = get_Mat(dataSet)plt.scatter(xMat.A[:,1],yMat.A,c='b',s=5)ws = standRegres(dataSet)yHat = xMat*wsplt.plot(xMat[:,1],yHat,c='r')plt.xlabel("第2列特征的数值：xMat[:,1]")plt.ylabel("预测值：yHat")plt.title('简单线性回归')plt.show()
# 绘制ex0数据集的散点图和最佳拟合直线
plotReg(ex0)
![[外链图片转存失败,源站可能有防盗链机制,建议将图片保存下来直接上传(img-fToTr6Zb-1593053028241)(output_26_0.png)]](https://img-blog.csdnimg.cn/20200625104442624.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDgyNzQxOA==,size_16,color_FFFFFF,t_70#pic_center)
4.7 计算相关系数
xMat,yMat = get_Mat(ex0)
ws = standRegres(ex0)
yHat = xMat*ws
np.corrcoef(yHat.T,yMat.T) # 参数需要保证两个都是行向量
array([[1. , 0.98647356],[0.98647356, 1. ]])
该矩阵包含所有两两组合的相关系数。可以看到，对角线上全部为1.0，因为自身匹配肯定是最完美的，而yHat和yMat的相关系数为0.98。看起来似乎是一个不错的结果。但是仔细观察数据集，会发现数据呈现有规律的波动，但是直线似乎没有很好的捕捉到这些波动。
局部加权线性回归
#此段代码供大家参考
xMat,yMat = get_Mat(ex0)
x=0.5
xi = np.arange(0,1.0,0.01)
k1,k2,k3=0.5,0.1,0.01
w1 = np.exp((xi-x)**2/(-2*k1**2))
w2 = np.exp((xi-x)**2/(-2*k2**2))
w3 = np.exp((xi-x)**2/(-2*k3**2))#创建画布
fig = plt.figure(figsize=(6,8),dpi=100)
#子画布1，原始数据集
fig1 = fig.add_subplot(411)
plt.scatter(xMat.A[:,1],yMat.A,c='b',s=5) #子画布2，k=0.5
fig2 = fig.add_subplot(412)
plt.plot(xi,w1,color='r')
plt.legend(['k = 0.5'])#子画布3，k=0.1
fig3 = fig.add_subplot(413)
plt.plot(xi,w2,color='g')
plt.legend(['k = 0.1'])#子画布4，k=0.01
fig4 = fig.add_subplot(414)
plt.plot(xi,w3,color='orange')
plt.legend(['k = 0.01'])
plt.show()
![[外链图片转存失败,源站可能有防盗链机制,建议将图片保存下来直接上传(img-IxntBFAQ-1593053028243)(output_31_0.png)]](https://img-blog.csdnimg.cn/20200625104454335.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDgyNzQxOA==,size_16,color_FFFFFF,t_70#pic_center)
这里假定我们预测的点是x=0.5，最上面的图是原始数据集，从下面三张图可以看出随着k的减小，被用于训练模型的数据点越来越少。
1. 构建LWLR函数
这个过程与简单线性函数的基本一致，唯一不同的是加入了权重weights，这里我将权重参数求解和预测yHat放在了一个函数里面。
# np.eye(5) 单位矩阵
a_eye = np.eye(5)
a_eye[0,2]=55
a_eye
array([[ 1., 0., 55., 0., 0.],[ 0., 1., 0., 0., 0.],[ 0., 0., 1., 0., 0.],[ 0., 0., 0., 1., 0.],[ 0., 0., 0., 0., 1.]])
a_eye[0]
array([ 1., 0., 55., 0., 0.])
a_eye.T
array([[ 1., 0., 0., 0., 0.],[ 0., 1., 0., 0., 0.],[55., 0., 1., 0., 0.],[ 0., 0., 0., 1., 0.],[ 0., 0., 0., 0., 1.]])
a_eye.T[0]
array([1., 0., 0., 0., 0.])
'''
函数功能：计算局部加权线性回归的预测值
参数说明：testMat:测试集xMat:训练集的特征矩阵yMat:训练集的标签矩阵返回：yHat：函数预测值
'''
def LWLR(testMat,xMat,yMat,k=1.0):n = testMat.shape[0] # 测试数据集行数m = xMat.shape[0] # 训练集特征矩阵行数weights = np.mat(np.eye(m)) # 用单位矩阵来初始化权重矩阵，yHat = np.zeros(n) # 用0矩阵来初始化预测值矩阵for i in range(n):for j in range(m):diffMat = testMat[i] - xMat[j]weights[j,j] = np.exp(diffMat*diffMat.T / (-2*k**2))xTx = xMat.T*(weights*xMat)if np.linalg.det(xTx) == 0:print('矩阵为奇异矩阵，无法求逆')returnws = xTx.I*(xMat.T*(weights*yMat))yHat[i] = testMat[i] * wsreturn ws,yHat
xMat
matrix([[1. , 0.067732],[1. , 0.42781 ],[1. , 0.995731],[1. , 0.738336],[1. , 0.981083],[1. , 0.526171],[1. , 0.378887],[1. , 0.033859],[1. , 0.132791],[1. , 0.138306],[1. , 0.247809],[1. , 0.64827 ],[1. , 0.731209],[1. , 0.236833],[1. , 0.969788],[1. , 0.607492],[1. , 0.358622],[1. , 0.147846],[1. , 0.63782 ],[1. , 0.230372],[1. , 0.070237],[1. , 0.067154],[1. , 0.925577],[1. , 0.717733],[1. , 0.015371],[1. , 0.33507 ],[1. , 0.040486],[1. , 0.212575],[1. , 0.617218],[1. , 0.541196],[1. , 0.045353],[1. , 0.126762],[1. , 0.556486],[1. , 0.901144],[1. , 0.958476],[1. , 0.274561],[1. , 0.394396],[1. , 0.87248 ],[1. , 0.409932],[1. , 0.908969],[1. , 0.166819],[1. , 0.665016],[1. , 0.263727],[1. , 0.231214],[1. , 0.552928],[1. , 0.047744],[1. , 0.365746],[1. , 0.495002],[1. , 0.493466],[1. , 0.792101],[1. , 0.76966 ],[1. , 0.251821],[1. , 0.181951],[1. , 0.808177],[1. , 0.334116],[1. , 0.33863 ],[1. , 0.452584],[1. , 0.69477 ],[1. , 0.590902],[1. , 0.307928],[1. , 0.148364],[1. , 0.70218 ],[1. , 0.721544],[1. , 0.666886],[1. , 0.124931],[1. , 0.618286],[1. , 0.381086],[1. , 0.385643],[1. , 0.777175],[1. , 0.116089],[1. , 0.115487],[1. , 0.66351 ],[1. , 0.254884],[1. , 0.993888],[1. , 0.295434],[1. , 0.952523],[1. , 0.307047],[1. , 0.277261],[1. , 0.279101],[1. , 0.175724],[1. , 0.156383],[1. , 0.733165],[1. , 0.848142],[1. , 0.771184],[1. , 0.429492],[1. , 0.162176],[1. , 0.917064],[1. , 0.315044],[1. , 0.201473],[1. , 0.297038],[1. , 0.336647],[1. , 0.666109],[1. , 0.583888],[1. , 0.085031],[1. , 0.687006],[1. , 0.949655],[1. , 0.189912],[1. , 0.844027],[1. , 0.333288],[1. , 0.427035],[1. , 0.466369],[1. , 0.550659],[1. , 0.278213],[1. , 0.918769],[1. , 0.886555],[1. , 0.569488],[1. , 0.066379],[1. , 0.335751],[1. , 0.426863],[1. , 0.395746],[1. , 0.694221],[1. , 0.27276 ],[1. , 0.503495],[1. , 0.067119],[1. , 0.038326],[1. , 0.599122],[1. , 0.947054],[1. , 0.671279],[1. , 0.434811],[1. , 0.509381],[1. , 0.749442],[1. , 0.058014],[1. , 0.482978],[1. , 0.466776],[1. , 0.357767],[1. , 0.949123],[1. , 0.41732 ],[1. , 0.920461],[1. , 0.156433],[1. , 0.656662],[1. , 0.616418],[1. , 0.853428],[1. , 0.133295],[1. , 0.693007],[1. , 0.178449],[1. , 0.199526],[1. , 0.073224],[1. , 0.286515],[1. , 0.182026],[1. , 0.621523],[1. , 0.344584],[1. , 0.398556],[1. , 0.480369],[1. , 0.15335 ],[1. , 0.171846],[1. , 0.867082],[1. , 0.223855],[1. , 0.528301],[1. , 0.890192],[1. , 0.106352],[1. , 0.917886],[1. , 0.014855],[1. , 0.567682],[1. , 0.068854],[1. , 0.603535],[1. , 0.53205 ],[1. , 0.651362],[1. , 0.901225],[1. , 0.204337],[1. , 0.696081],[1. , 0.963924],[1. , 0.98139 ],[1. , 0.987911],[1. , 0.990947],[1. , 0.736021],[1. , 0.253574],[1. , 0.674722],[1. , 0.939368],[1. , 0.235419],[1. , 0.110521],[1. , 0.218023],[1. , 0.869778],[1. , 0.19683 ],[1. , 0.958178],[1. , 0.972673],[1. , 0.745797],[1. , 0.445674],[1. , 0.470557],[1. , 0.549236],[1. , 0.335691],[1. , 0.884739],[1. , 0.918916],[1. , 0.441815],[1. , 0.116598],[1. , 0.359274],[1. , 0.814811],[1. , 0.387125],[1. , 0.982243],[1. , 0.78088 ],[1. , 0.652565],[1. , 0.87003 ],[1. , 0.604755],[1. , 0.255212],[1. , 0.730546],[1. , 0.493829],[1. , 0.257017],[1. , 0.833735],[1. , 0.070095],[1. , 0.52707 ],[1. , 0.116163]])
xMat[0]
matrix([[1. , 0.067732]])
xMat[0] - xMat[1]
matrix([[ 0. , -0.360078]])
2. 不同k值的结果图
我们调整k值，然后查看不同k值对模型的影响
xMat,yMat = get_Mat(ex0)
#将数据点排列（argsort()默认升序排列，返回索引）
srtInd = xMat[:,1].argsort(0)
srtInd
matrix([[151],[ 24],[ 7],[114],[ 26],[ 30],[ 45],[121],[106],[113],[ 21],[ 0],[153],[197],[ 20],[136],[ 93],[149],[169],[ 70],[ 69],[199],[183],[ 64],[ 31],[ 8],[132],[ 9],[ 17],[ 60],[143],[ 80],[128],[ 85],[ 40],[144],[ 79],[134],[ 52],[138],[ 96],[172],[135],[ 88],[158],[ 27],[170],[146],[ 19],[ 43],[168],[ 13],[ 10],[ 51],[165],[ 72],[192],[195],[ 42],[111],[ 35],[ 77],[102],[ 78],[137],[ 74],[ 89],[ 76],[ 59],[ 87],[ 98],[ 54],[ 25],[179],[107],[ 90],[ 55],[140],[124],[ 16],[184],[ 46],[ 6],[ 66],[ 67],[186],[ 36],[109],[141],[ 38],[126],[108],[ 99],[ 1],[ 84],[118],[182],[176],[ 56],[100],[123],[177],[142],[122],[ 48],[194],[ 47],[112],[119],[ 5],[198],[147],[155],[ 29],[178],[101],[ 44],[ 32],[152],[105],[ 92],[ 58],[115],[154],[191],[ 15],[130],[ 28],[ 65],[139],[ 18],[ 11],[156],[189],[129],[ 71],[ 41],[ 91],[ 63],[117],[166],[ 94],[133],[110],[ 57],[159],[ 61],[ 23],[ 62],[193],[ 12],[ 81],[164],[ 3],[175],[120],[ 50],[ 83],[ 68],[188],[ 49],[ 53],[185],[196],[ 97],[ 82],[131],[145],[171],[190],[ 37],[180],[104],[148],[ 33],[157],[ 39],[ 86],[150],[103],[181],[127],[ 22],[167],[116],[125],[ 95],[ 75],[173],[ 34],[160],[ 14],[174],[ 4],[161],[187],[162],[163],[ 73],[ 2]], dtype=int64)
xMat[srtInd]
matrix([[[1. , 0.014855]],[[1. , 0.015371]],[[1. , 0.033859]],[[1. , 0.038326]],[[1. , 0.040486]],[[1. , 0.045353]],[[1. , 0.047744]],[[1. , 0.058014]],[[1. , 0.066379]],[[1. , 0.067119]],[[1. , 0.067154]],[[1. , 0.067732]],[[1. , 0.068854]],[[1. , 0.070095]],[[1. , 0.070237]],[[1. , 0.073224]],[[1. , 0.085031]],[[1. , 0.106352]],[[1. , 0.110521]],[[1. , 0.115487]],[[1. , 0.116089]],[[1. , 0.116163]],[[1. , 0.116598]],[[1. , 0.124931]],[[1. , 0.126762]],[[1. , 0.132791]],[[1. , 0.133295]],[[1. , 0.138306]],[[1. , 0.147846]],[[1. , 0.148364]],[[1. , 0.15335 ]],[[1. , 0.156383]],[[1. , 0.156433]],[[1. , 0.162176]],[[1. , 0.166819]],[[1. , 0.171846]],[[1. , 0.175724]],[[1. , 0.178449]],[[1. , 0.181951]],[[1. , 0.182026]],[[1. , 0.189912]],[[1. , 0.19683 ]],[[1. , 0.199526]],[[1. , 0.201473]],[[1. , 0.204337]],[[1. , 0.212575]],[[1. , 0.218023]],[[1. , 0.223855]],[[1. , 0.230372]],[[1. , 0.231214]],[[1. , 0.235419]],[[1. , 0.236833]],[[1. , 0.247809]],[[1. , 0.251821]],[[1. , 0.253574]],[[1. , 0.254884]],[[1. , 0.255212]],[[1. , 0.257017]],[[1. , 0.263727]],[[1. , 0.27276 ]],[[1. , 0.274561]],[[1. , 0.277261]],[[1. , 0.278213]],[[1. , 0.279101]],[[1. , 0.286515]],[[1. , 0.295434]],[[1. , 0.297038]],[[1. , 0.307047]],[[1. , 0.307928]],[[1. , 0.315044]],[[1. , 0.333288]],[[1. , 0.334116]],[[1. , 0.33507 ]],[[1. , 0.335691]],[[1. , 0.335751]],[[1. , 0.336647]],[[1. , 0.33863 ]],[[1. , 0.344584]],[[1. , 0.357767]],[[1. , 0.358622]],[[1. , 0.359274]],[[1. , 0.365746]],[[1. , 0.378887]],[[1. , 0.381086]],[[1. , 0.385643]],[[1. , 0.387125]],[[1. , 0.394396]],[[1. , 0.395746]],[[1. , 0.398556]],[[1. , 0.409932]],[[1. , 0.41732 ]],[[1. , 0.426863]],[[1. , 0.427035]],[[1. , 0.42781 ]],[[1. , 0.429492]],[[1. , 0.434811]],[[1. , 0.441815]],[[1. , 0.445674]],[[1. , 0.452584]],[[1. , 0.466369]],[[1. , 0.466776]],[[1. , 0.470557]],[[1. , 0.480369]],[[1. , 0.482978]],[[1. , 0.493466]],[[1. , 0.493829]],[[1. , 0.495002]],[[1. , 0.503495]],[[1. , 0.509381]],[[1. , 0.526171]],[[1. , 0.52707 ]],[[1. , 0.528301]],[[1. , 0.53205 ]],[[1. , 0.541196]],[[1. , 0.549236]],[[1. , 0.550659]],[[1. , 0.552928]],[[1. , 0.556486]],[[1. , 0.567682]],[[1. , 0.569488]],[[1. , 0.583888]],[[1. , 0.590902]],[[1. , 0.599122]],[[1. , 0.603535]],[[1. , 0.604755]],[[1. , 0.607492]],[[1. , 0.616418]],[[1. , 0.617218]],[[1. , 0.618286]],[[1. , 0.621523]],[[1. , 0.63782 ]],[[1. , 0.64827 ]],[[1. , 0.651362]],[[1. , 0.652565]],[[1. , 0.656662]],[[1. , 0.66351 ]],[[1. , 0.665016]],[[1. , 0.666109]],[[1. , 0.666886]],[[1. , 0.671279]],[[1. , 0.674722]],[[1. , 0.687006]],[[1. , 0.693007]],[[1. , 0.694221]],[[1. , 0.69477 ]],[[1. , 0.696081]],[[1. , 0.70218 ]],[[1. , 0.717733]],[[1. , 0.721544]],[[1. , 0.730546]],[[1. , 0.731209]],[[1. , 0.733165]],[[1. , 0.736021]],[[1. , 0.738336]],[[1. , 0.745797]],[[1. , 0.749442]],[[1. , 0.76966 ]],[[1. , 0.771184]],[[1. , 0.777175]],[[1. , 0.78088 ]],[[1. , 0.792101]],[[1. , 0.808177]],[[1. , 0.814811]],[[1. , 0.833735]],[[1. , 0.844027]],[[1. , 0.848142]],[[1. , 0.853428]],[[1. , 0.867082]],[[1. , 0.869778]],[[1. , 0.87003 ]],[[1. , 0.87248 ]],[[1. , 0.884739]],[[1. , 0.886555]],[[1. , 0.890192]],[[1. , 0.901144]],[[1. , 0.901225]],[[1. , 0.908969]],[[1. , 0.917064]],[[1. , 0.917886]],[[1. , 0.918769]],[[1. , 0.918916]],[[1. , 0.920461]],[[1. , 0.925577]],[[1. , 0.939368]],[[1. , 0.947054]],[[1. , 0.949123]],[[1. , 0.949655]],[[1. , 0.952523]],[[1. , 0.958178]],[[1. , 0.958476]],[[1. , 0.963924]],[[1. , 0.969788]],[[1. , 0.972673]],[[1. , 0.981083]],[[1. , 0.98139 ]],[[1. , 0.982243]],[[1. , 0.987911]],[[1. , 0.990947]],[[1. , 0.993888]],[[1. , 0.995731]]])
xSort=xMat[srtInd][:,0]
xSort
matrix([[1. , 0.014855],[1. , 0.015371],[1. , 0.033859],[1. , 0.038326],[1. , 0.040486],[1. , 0.045353],[1. , 0.047744],[1. , 0.058014],[1. , 0.066379],[1. , 0.067119],[1. , 0.067154],[1. , 0.067732],[1. , 0.068854],[1. , 0.070095],[1. , 0.070237],[1. , 0.073224],[1. , 0.085031],[1. , 0.106352],[1. , 0.110521],[1. , 0.115487],[1. , 0.116089],[1. , 0.116163],[1. , 0.116598],[1. , 0.124931],[1. , 0.126762],[1. , 0.132791],[1. , 0.133295],[1. , 0.138306],[1. , 0.147846],[1. , 0.148364],[1. , 0.15335 ],[1. , 0.156383],[1. , 0.156433],[1. , 0.162176],[1. , 0.166819],[1. , 0.171846],[1. , 0.175724],[1. , 0.178449],[1. , 0.181951],[1. , 0.182026],[1. , 0.189912],[1. , 0.19683 ],[1. , 0.199526],[1. , 0.201473],[1. , 0.204337],[1. , 0.212575],[1. , 0.218023],[1. , 0.223855],[1. , 0.230372],[1. , 0.231214],[1. , 0.235419],[1. , 0.236833],[1. , 0.247809],[1. , 0.251821],[1. , 0.253574],[1. , 0.254884],[1. , 0.255212],[1. , 0.257017],[1. , 0.263727],[1. , 0.27276 ],[1. , 0.274561],[1. , 0.277261],[1. , 0.278213],[1. , 0.279101],[1. , 0.286515],[1. , 0.295434],[1. , 0.297038],[1. , 0.307047],[1. , 0.307928],[1. , 0.315044],[1. , 0.333288],[1. , 0.334116],[1. , 0.33507 ],[1. , 0.335691],[1. , 0.335751],[1. , 0.336647],[1. , 0.33863 ],[1. , 0.344584],[1. , 0.357767],[1. , 0.358622],[1. , 0.359274],[1. , 0.365746],[1. , 0.378887],[1. , 0.381086],[1. , 0.385643],[1. , 0.387125],[1. , 0.394396],[1. , 0.395746],[1. , 0.398556],[1. , 0.409932],[1. , 0.41732 ],[1. , 0.426863],[1. , 0.427035],[1. , 0.42781 ],[1. , 0.429492],[1. , 0.434811],[1. , 0.441815],[1. , 0.445674],[1. , 0.452584],[1. , 0.466369],[1. , 0.466776],[1. , 0.470557],[1. , 0.480369],[1. , 0.482978],[1. , 0.493466],[1. , 0.493829],[1. , 0.495002],[1. , 0.503495],[1. , 0.509381],[1. , 0.526171],[1. , 0.52707 ],[1. , 0.528301],[1. , 0.53205 ],[1. , 0.541196],[1. , 0.549236],[1. , 0.550659],[1. , 0.552928],[1. , 0.556486],[1. , 0.567682],[1. , 0.569488],[1. , 0.583888],[1. , 0.590902],[1. , 0.599122],[1. , 0.603535],[1. , 0.604755],[1. , 0.607492],[1. , 0.616418],[1. , 0.617218],[1. , 0.618286],[1. , 0.621523],[1. , 0.63782 ],[1. , 0.64827 ],[1. , 0.651362],[1. , 0.652565],[1. , 0.656662],[1. , 0.66351 ],[1. , 0.665016],[1. , 0.666109],[1. , 0.666886],[1. , 0.671279],[1. , 0.674722],[1. , 0.687006],[1. , 0.693007],[1. , 0.694221],[1. , 0.69477 ],[1. , 0.696081],[1. , 0.70218 ],[1. , 0.717733],[1. , 0.721544],[1. , 0.730546],[1. , 0.731209],[1. , 0.733165],[1. , 0.736021],[1. , 0.738336],[1. , 0.745797],[1. , 0.749442],[1. , 0.76966 ],[1. , 0.771184],[1. , 0.777175],[1. , 0.78088 ],[1. , 0.792101],[1. , 0.808177],[1. , 0.814811],[1. , 0.833735],[1. , 0.844027],[1. , 0.848142],[1. , 0.853428],[1. , 0.867082],[1. , 0.869778],[1. , 0.87003 ],[1. , 0.87248 ],[1. , 0.884739],[1. , 0.886555],[1. , 0.890192],[1. , 0.901144],[1. , 0.901225],[1. , 0.908969],[1. , 0.917064],[1. , 0.917886],[1. , 0.918769],[1. , 0.918916],[1. , 0.920461],[1. , 0.925577],[1. , 0.939368],[1. , 0.947054],[1. , 0.949123],[1. , 0.949655],[1. , 0.952523],[1. , 0.958178],[1. , 0.958476],[1. , 0.963924],[1. , 0.969788],[1. , 0.972673],[1. , 0.981083],[1. , 0.98139 ],[1. , 0.982243],[1. , 0.987911],[1. , 0.990947],[1. , 0.993888],[1. , 0.995731]])
#计算不同k取值下的y估计值yHat
ws1,yHat1 = LWLR(xMat,xMat,yMat,k=1.0)
ws2,yHat2 = LWLR(xMat,xMat,yMat,k=0.01)
ws3,yHat3 = LWLR(xMat,xMat,yMat,k=0.003)
#创建画布
fig = plt.figure(figsize=(6,8),dpi=100)#子图1绘制k=1.0的曲线
fig1=fig.add_subplot(311)
plt.scatter(xMat[:,1].A,yMat.A,c='b',s=2)
plt.plot(xSort[:,1],yHat1[srtInd],linewidth=1,color='r')
plt.title('局部加权回归曲线，k=1.0',size=10,color='r')#子图2绘制k=0.01的曲线
fig2=fig.add_subplot(312)
plt.scatter(xMat[:,1].A,yMat.A,c='b',s=2)
plt.plot(xSort[:,1],yHat2[srtInd],linewidth=1,color='r')
plt.title('局部加权回归曲线，k=0.01',size=10,color='r')#子图3绘制k=0.003的曲线
fig3=fig.add_subplot(313)
plt.scatter(xMat[:,1].A,yMat.A,c='b',s=2)
plt.plot(xSort[:,1],yHat3[srtInd],linewidth=1,color='r')
plt.title('局部加权回归曲线，k=0.003',size=10,color='r')#调整子图的间距
plt.tight_layout(pad=1.2)
plt.show()
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这三个图是不同平滑值绘出的局部加权线性回归结果。当k=1.0时，模型的效果与最小二乘法差不多；k=0.01时，该模型基本上已经挖出了数据的潜在规律，当继续减小到k=0.003时，会发现模型考虑了太多的噪音，进而导致了过拟合现象。
#四种模型相关系数比较
np.corrcoef(yHat.T,yMat.T) # 最小二乘法
array([[1. , 0.98647356],[0.98647356, 1. ]])
np.corrcoef(yHat1,yMat.T) # k=1.0模型
array([[1. , 0.98647703],[0.98647703, 1. ]])
np.corrcoef(yHat2,yMat.T) # k=0.01模型
array([[1. , 0.9985249],[0.9985249, 1. ]])
np.corrcoef(yHat3,yMat.T) # k=0.003模型
array([[1. , 0.99931945],[0.99931945, 1. ]])
局部加权线性回归也存在一个问题——增加了计算量，因为它对每个点预测都要使用整个数据集。从不同k值的结果图中可以看出，当k=0.01时模型可以很好地拟合数据潜在规律，但是同时看一下，k值与权重关系图，可以发现，当k=0.01时，大部分数据点的权重都接近0，也就是说他们基本上可以不用带入计算。所以如果一开始就能去掉这些数据点的计算，那么就可以大大减少程序的运行时间了，从而缓解计算量增加带来的问题。后面我们会讲解这个操作。
