python中matplotlib实现最小二乘法拟合的过程详解

import numpy as np 
import matplotlib.pyplot as plt

# 在直线 y = 3 + 5x 附近生成随机点
X = np.arange(0, 5, 0.1) 
Z = [3 + 5 * x for x in X] 
Y = [np.random.normal(z, 0.5) for z in Z]

plt.plot(X, Y, 'ro') 
plt.show() 

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() 

# 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) 

# 生成拟合曲线的绘制点
_X = np.arange(0, 5, 0.1) 
_Y = np.array([a0 + a1*x + a2*x**2 for x in _X])

plt.plot(X, Y, 'ro', _X, _Y, 'b', linewidth=2) 
plt.title("y = {} + {}x + {}$x^2$ ".format(a0, a1, a2)) 
plt.show()