线性回归——最小二乘法拟合

记录下用最小二乘法拟合线性模型的代码实现：

1、应用正规方程（Normal Equation)求解最小二乘法举例：

最简单的y关于x的线性方程\(y=\beta_0+\beta_1x\)

预测值和观察值：

 

 写成矩阵形式：

\(X\beta=y\)，其中\(X=\begin{bmatrix} 1& x_1\\ 1& x_2\\ ...&...\\ 1& x_n\\ \end{bmatrix}\) ， \(\beta=\begin{bmatrix} \beta_0\\ \beta_1\\ \end{bmatrix}\) ， \(y=\begin{bmatrix} y_1\\ y_2\\ ...\\ y_n\\ \end{bmatrix}\)

最小二乘法的解\(\beta\)可以通过解正规方程获得： \(X^TX\beta=X^Ty\)

python代码实现如下：

import numpy as np
from numpy.linalg import inv


x=[0.015348106072082663,0.021715879765738008,0.0316253067336889,
   0.03212431406271639,0.03828189026158509,0.03965578961513808,
   0.05502733389871053,0.06116957740576664,0.06170785924013203,
   0.07206404835977503]
y=[22, 0, 22, 11, 9, 31, 20, 31, 2, 20]


def plotLinearRegression1():
    X = np.array(x).reshape(len(x),1)
    reg=linear_model.LinearRegression(fit_intercept=True,normalize=False)
    reg.fit(X,y)
    k=reg.coef_#获取斜率w1,w2,w3,...,wn
    b=reg.intercept_#获取截距w0
    #x0=np.arange(0,10,0.2)
    x0 = np.array(x).reshape(len(x),)
    y0=k*x0+b
    plt.scatter(x,y)
    plt.plot(x0, y0)
    print(\'k\',k);
    print(\'b\',b)
  
      
\'\'\'
用least squares
\'\'\'
def plotLinearRegression2():
    X = np.vstack([np.ones((1,len(x))),x]).T
    print(\'X:\n\',X)
    Xt = X.T
    Z = Xt.dot(X)
    invZ = inv(Z)
    print(\'invZ:\n\',invZ)
    
    W = (invZ.dot(Xt)).dot(y)
    print(\'W:\n\',W)
    
    #plot
    plt.scatter(x,y)
    x0 = np.array(x).reshape(len(x),)
   # x0=np.arange(0,10,0.2)
    plt.plot(x0, W[1]*x0+W[0])
    
    
def plotLinearRegression3():
    A = np.vstack([x, np.ones(len(x))]).T
    print(\'A:\n\',A)
    m, c = np.linalg.lstsq(A, y, rcond=None)[0]
    print(\'m: \',m,\'  c:\',c)
   
    x0 = np.array(x).reshape(len(x),)
    plt.plot(x, y, \'o\', label=\'Original data\', markersize=10)
    plt.plot(x0, m*x0 + c, \'r\', label=\'Fitted line\')
    plt.legend()
    plt.show()  
    
        
if __name__ == \'__main__\':
    plotLinearRegression2()