机器学习3：岭回归 ~LASSO回归 ~弹性网

岭回归

原理

算法实现

手工实现

import numpy as np
from numpy import genfromtxt
import matplotlib.pyplot as plt  

# 读入数据 
data = genfromtxt(r"longley.csv",delimiter=',')
print(data)
           

# 切分数据
x_data = data[1:,2:]
y_data = data[1:,1,np.newaxis]
print(x_data)
print(y_data)

print(np.mat(x_data).shape)
print(np.mat(y_data).shape)
# 给样本添加偏置项
X_data = np.concatenate((np.ones((16,1)),x_data),axis=1)
print(X_data.shape)
           

(16, 6)

(16, 1)

(16, 7)

# 岭回归标准方程法求解回归参数
def weights(xArr, yArr,lam=0.2):
    xMat = np.mat(xArr)
    yMat = np.mat(yArr)
    xTx = xMat.T*xMat # 矩阵乘法
    rxTx = xTx + np.eye(xMat.shape[1])*lam
    # 计算矩阵的值,如果值为0，说明该矩阵没有逆矩阵
    if np.linalg.det(rxTx) == 0.0:
        print("This matrix cannot do inverse")
        return
    # xTx.I为xTx的逆矩阵
    ws = rxTx.I*xMat.T*yMat
    return ws

ws = weights(X_data,y_data)
print(ws)
           

[[ 7.38107538e-04]

[ 2.07703836e-01]

[ 2.10076376e-02]

[ 5.05385441e-03]

[-1.59173066e+00]

[ 1.10442920e-01]

[-2.42280461e-01]]

# 计算预测值
print(np.mat(X_data)*np.mat(ws))
           

matrix([[ 83.55075226],

[ 86.92588689],

[ 88.09720228],

[ 90.95677622],

[ 96.06951002],

[ 97.81955375],

[ 98.36444357],

[ 99.99814266],

[103.26832266],

[105.03165135],

[107.45224671],

[109.52190685],

[112.91863666],

[113.98357055],

[115.29845063],

[117.64279933]])

sklearn实现

import numpy as np
from numpy import genfromtxt
from sklearn import linear_model
import matplotlib.pyplot as plt

# 读入数据 
data = genfromtxt(r"longley.csv",delimiter=',')
print(data)
           

# 切分数据
x_data = data[1:,2:]
y_data = data[1:,1]
print(x_data)
print(y_data)
           

# 创建模型
# 默认生成50个值,为岭回归系数
alphas_to_test = np.linspace(0.001, 1 ,100)
# 创建模型，保存误差值  Ridge岭回归 CV表示交叉验证
model = linear_model.RidgeCV(alphas=alphas_to_test, store_cv_values=True)
model.fit(x_data, y_data)

# 岭系数,选取50个最佳岭系数
print(model.alpha_)
# loss值  (交叉验证法误差)
print(model.cv_values_.shape)
print(model.cv_values_)
           

# 画图
# 岭系数跟loss值的关系,对16个样本求平均值
plt.plot(alphas_to_test, model.cv_values_.mean(axis=0))
# 选取的岭系数值的位置
plt.plot(model.alpha_, min(model.cv_values_.mean(axis=0)),'ro')
plt.show()
           

[88.11196241]

LASSO回归

原理

从图中可以看出，LASSO回归可以让有些特征值等于0，而岭回归估计系数等于0的机会微乎其微

蓝色部分是限制正则项<=t，红色部分是代价函数等高线

算法实现

sklearn实现

数据和上述一样

import numpy as np
from numpy import genfromtxt
from sklearn import linear_model

#读入数据
data = genfromtxt(r"longley.csv",delimiter=',')
print(data)

# 切分数据
x_data = data[1:,2:]
y_data = data[1:,1]
print(x_data)
print(y_data)
           

#创建模型
model=linear_model.LassoCV()
model.fit(x_data,y_data)

#LASSO系数
print(model.alpha_)
#相关系数
print(model.coef_)
           

20.03464209711722

[0.10206856 0.00409161 0.00354815 0. 0. 0. ]

[115.6461414]

弹性网

原理

算法实现

sklearn实现

数据和上述相同

import numpy as np
from numpy import genfromtxt
from sklearn import linear_model

# 读入数据 
data = genfromtxt(r"longley.csv",delimiter=',')
print(data)

# 切分数据
x_data = data[1:,2:]
y_data = data[1:,1]
print(x_data)
print(y_data)

# 创建模型
model = linear_model.ElasticNetCV()
model.fit(x_data, y_data)

# 弹性网系数
print(model.alpha_)
# 相关系数
print(model.coef_)
           

42.96498005089394

[0.1016487 0.00416716 0.00349843 0. 0. 0. ]

[115.6037171]