l2正则化python_回归分析_L2正则化（岭回归）【python实现】

样本

[x_i=(x_{i1};x_{i2}; ...; x_{im}) , {函数值 y_i}

]

每个样本有m个变量

回归面

[f(x_i) = x_i^T omega +b

]

(omega = (omega_1; omega_2; ...; omega_m))

[hat x_i=(x_{i1};x_{i2}; ...; x_{im}; 1)

]

[hat omega = (omega_1; omega_2; ...; omega_m; b)

]

则

[f(x_i) = hat x_i^T hat omega

]

假设有n个样本令

[X = [hat x_1^T; hat x_2^T; ..., hat x_n^T]

]

则均方误差为

[frac 1n (X hat omega - Y)^T (X hat omega - Y)

]

其中(Y = (y_1; y_2; ...; y_n))

损失函数

[J(hat omega^*) = (X hat omega^* - Y)^T (Xhat omega^* - Y)

]

令

[left. frac{{rm d} J(hat omega^*)}{{rm d} hat omega^*} = 0right.

]

[hat omega^* = (X^TX)^{-1}X^TY

]

当样本变量较多，样本数量不足时(m>n), (hat omega^*)解不唯一

L2正则化

引入对于(hat omega^*)的L2正则化项

[hat J(hat omega^*) = (X hat omega^* - Y)^T (Xhat omega^* - Y) + frac{lambda}{2} ||hat omega^*||_2^2

]

可以发现，正则化项为(hat omega^*)二范数平方，乘以(frac{lambda}{2})

(lambda)用于控制正则化项和均方误差的权重

[left. frac{{rm d} hat J(hat omega^*)}{{rm d} hat omega^*} = 0right.

]

[hat omega^* = (X^TX + frac{lambda}{2}I)^{-1}X^TY

]

python程序

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

#随机生成两个变量的N个样本

N = 50

feature1 = np.random.rand(N)*10

feature2 = np.random.rand(N)*10

splt = np.ones((1, N))

#

temp_X1 = np.row_stack((feature1, feature2))

temp_X = np.vstack((temp_X1, splt))

X_t = np.mat(temp_X)

X = X_t.T

temp_Y = np.random.rand(N)*10

Y_t = np.mat(temp_Y)

Y = Y_t.T

#画样本散点图

fig = plt.figure()

ax1 = Axes3D(fig)

ax1.scatter(feature1, feature2, temp_Y)

#求Omega

Lbd = 0.01 #确定lambda

I_11 = np.eye(2)

I_12 = np.zeros((2, 1))

I_2 = np.zeros((1, 3))

I_t1 = np.hstack((I_11, I_12))

I_t = np.vstack((I_t1, I_2))

I = np.mat(I_t)

Omega = (X.T*X + Lbd/2*I).I*X.T*Y

#画分回归面

xx = np.linspace(0,10, num=50)

yy = np.linspace(0,10, num=50)

xx_1, yy_1 = np.meshgrid(xx, yy)

Omega_h = np.array(Omega.T)

zz_1 = Omega_h[0, 0]*xx_1 + Omega_h[0, 1]*yy_1 + Omega_h[0, 2]

ax1.plot_surface(xx_1, yy_1, zz_1, alpha= 0.6, color= "r")

plt.show()

效果

参考