线性回归、梯度下降、正规方程组

线性回归、梯度下降、正规方程组

这个系列的公开课看了有段时间了，原来也想着一直记录下去，还是懒病的问题；中间停了这么长时间，最近又想着复习一下，就写点随笔。最前方关于Ng介绍机器学习的相关概念部分的几张截图，我用了张雨石大佬公开课笔记的截图(实在是懒的不想打字了)，在这里我要感谢张雨石大佬！话不多说，直接进入正题！

附录证明

代码实现

批量梯度下降算法：

import matplotlib.pyplot as plt
import csv
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
# 从文件中取出x1,x2,y
file_name = "practise.csv"
with open(file_name) as f:reader = ader(f)header_row = next(reader)print(header_row)x1 = []x2 = []y = []for row in reader:x1.append(int(row[1]))x2.append(int(row[2]))y.append(int(row[3]))# 定义学习率
learning_rate = 0.00001
# 计算样本个数
m = len(y)
# 初始化参数theta
theta = [0]*3
print(theta)
# 误差
error0 = 0
error1 = 0
# 退出迭代的条件
epsilon = 0.0001def h(x1,x2):return theta[0]+theta[1]*x1+theta[2]*x2
# 开始迭代
cnt = 0
while True:cnt += 1diff = [0]*3for i in range(m):diff[0] += h(x1[i], x2[i]) - y[i]diff[1] += (h(x1[i], x2[i]) - y[i])*x1[i]diff[2] += (h(x1[i], x2[i]) - y[i])*x2[i]theta[0] -= learning_rate /m * diff[0]theta[1] -= learning_rate /m * diff[1]theta[2] -= learning_rate /m * diff[2]error1 = 0for i in range(m):error1 += (h(x1[i], x2[i])-y[i])**2#print("error1: %d" %(error1))if abs(error1 - error0)<epsilon:print("cnt: %d" % cnt)breakelse:error0 = error1print(theta)
y_cal = []
for i in range(m):y_cal.append(theta[0]+theta[1]*x1[i]+theta[2]*x2[i])
print(y)
print(y_cal)

使用正规方程组法求解参数：

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# 房屋大小(13组数据）
x = [[100,74], [80,98], [120,50], [75,32], [60,22], [43,21], [140,77], [132,61], [63,30], [55,25], [74,37],[44,20], [88,44]]
# 房屋价格
y = [[120], [92], [143], [87], [60], [50], [167], [147], [80], [60], [90], [57], [99]]# 使用最小二乘法求解误差函数最优解
def least_square_regress(x,y):# 在每组输入特征前面插入X0 = 1for xs in x:xs.insert(0, 1)# 将原始数据转换为矩阵形式x_array = np.array(x)y_array = np.array(y)x_mat = np.mat(x_array)y_mat = np.mat(y_array)print(x_mat.shape)xT_x = np.matmul(x_mat.T, x_mat)print(xT_x)if(np.linalg.det(xT_x == 0)):return ("无法求解")para = xT_x.I * x_mat.T * y_matreturn para
# 得到线性回归模型的参数值，即输入特征前面的系数
para = least_square_regress(x,y)
print(para)
x_array = np.array(x)
x_mat = np.mat(x_array)
result = x_mat * para
print(result.T)
print(y)