数据挖掘实践（4）：基础理论（四）数学基础（四）向量与矩阵（四）最⼩⼆乘法（二）代码

import numpy as np
import matplotlib.pyplot as plt
# 在直线 y = 5x+3 附近⽣成随机点
X = np.arange(0, 5, 0.1) Z = [5 * x + 3 for x in X] Y = [np.random.normal(z, 0.5) for z in Z]
plt.plot(X, Y, 'ro')
plt.show()

from scipy.optimize import leastsq
# 需要拟合的函数func :指定函数的形状
def func(p,x):
 k,b=p
 return k*x+b
# 误差函数：
def error(p,x,y):
 return func(p,x)-y
p0=[1,20]
Para=leastsq(error,p0,args=(X,Y))
k,b=Para[0]
_X = [0, 5] 
_Y = [b + k * x for x in _X]
plt.plot(X, Y, 'ro', _X, _Y, 'b', linewidth=2) 
plt.title("y = {}x + {}".format(k, b)) 
plt.show()

# 线性回归解正则⽅程
def linear_regression(x, y): 
 N = len(x)
 sumx = sum(x)
 sumy = sum(y)
 sumx2 = sum(x**2)
 sumxy = sum(x*y)
 
 A = np.mat([[N, sumx], [sumx, sumx2]])
 b = np.array([sumy, sumxy])
 
 return np.linalg.solve(A, b)
 
a0, a1 = linear_regression(X, Y)
# ⽣成拟合直线的绘制点
_X = [0, 5] 
_Y = [a0 + a1 * x for x in _X]
 
plt.plot(X, Y, 'ro', _X, _Y, 'b', linewidth=2) 
plt.title("y = {} + {}x".format(a0, a1)) 
plt.show()

import numpy as np 
import matplotlib.pyplot as plt
 
# y = 2 + 3x + 4x^2
X = np.arange(0, 5, 0.1) 
Z = [2 + 3 * x + 4 * x ** 2 for x in X] 
Y = np.array([np.random.normal(z,3) for z in Z])
 
plt.plot(X, Y, 'ro') 
plt.show()

from scipy.optimize import leastsq
# 需要拟合的函数func :指定函数的形状
def func(p,x):
 m,n,o = p
 return m + n*x + o*x**2
# 偏差函数：x,y都是列表:这⾥的x,y跟上⾯的Xi,Yi中是⼀⼀对应的
def error(p,x,y):
 return func(p,x)-y
p0=[0,5,10]
Para=leastsq(error,p0,args=(X,Y))
m,n,o=Para[0]
_X = np.arange(0, 5, 0.1) 
_Y = np.array([m + n*x + o*x**2 for x in _X])
plt.plot(X, Y, 'ro', _X, _Y, 'b', linewidth=2) 
plt.title("y = {} + {}x + {}$x^2$ ".format(m, n, o)) 
plt.show()

# ⽣成系数矩阵A
def gen_coefficient_matrix(X, Y): 
 N = len(X)
 m = 3
 A = []
# 计算每⼀个⽅程的系数
 for i in range(m):
 a = []
 # 计算当前⽅程中的每⼀个系数
 for j in range(m):
 a.append(sum(X ** (i+j)))
 A.append(a)
 return A
 
# 计算⽅程组的右端向量b
def gen_right_vector(X, Y): 
 N = len(X)
 m = 3
 b = []
 for i in range(m):
 b.append(sum(X**i * Y))
 return b A = gen_coefficient_matrix(X, Y) 
b = gen_right_vector(X, Y)
 
a0, a1, a2 = np.linalg.solve(A, b)
print(a0,a1,a2)
2.760380334517119 2.771490852340913 3.99403510389479
# ⽣成拟合曲线的绘制点
_X2 = np.arange(0, 5, 0.1) 
_Y2 = np.array([a0 + a1*x + a2*x**2 for x in _X2])
 
plt.plot(X, Y, 'ro', _X2, _Y2, 'b', linewidth=2) 
plt.title("y = {} + {}x + {}$x^2$ ".format(a0, a1, a2)) 
plt.show()

六、总结
向量和矩阵的概念和计算
向量和矩阵的区别与联系
损失函数最⼩化的意义