【机器学习chp5代码示例】线性回归+岭回归+Lasso回归+LAR回归+弹性回归+胡伯Huber损失回归

已于 2025-02-22 14:16:58 修改
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分类专栏： 机器学习 文章标签： 机器学习 线性回归
于 2025-02-22 13:20:36 首次发布
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目录

一、线性回归

1、加载数据集

2、数据集各特征和标签可视化

3、最普通的线性回归，数据未标准化，无正则化

4、最普通的线性回归，有标准化，无正则化

二、岭回归

1、岭回归 Ridge

2、带有内置交叉验证的岭回归 RidgeCV

三、Lasso回归

1、Lasso回归

2、带有内置交叉验证的 Lasso回归

四、最小角度回归LAR

1、LAR回归

2、交叉验证的最小角度回归模型

五、弹性回归（L1+L2正则）

1、弹性回归

2、带交叉验证的弹性回归ElasticNetCV

六、Huber损失的回归—对异常值具有鲁棒性

本文使用的数据集是：加利福尼亚住房数据集（回归）

一、线性回归

1、加载数据集

# 加载加州住房数据集
from sklearn.datasets import fetch_california_housing

housing = fetch_california_housing()
X = housing.data
y = housing.target
print(type(X))
print(type(y))
print(X.shape)
print(y.shape)

输出结果为：

<class 'numpy.ndarray'>
<class 'numpy.ndarray'>
(20640, 8)
(20640,)

2、数据集各特征和标签可视化

#数据可视化
import seaborn as sns
from matplotlib import pyplot as plt

# 绘制特征X密度图
for col in range(0,8):
    sns.kdeplot(X[:,col], fill=True)
    # 设置图形标题和标签
    plt.title(f'X-{col} Distribution')
    plt.xlabel(col)
    plt.ylabel('Density')
    plt.show()


# 绘制标签y密度图
sns.kdeplot(y, fill=True)
# 设置图形标题和标签
plt.title('y Distribution')
plt.xlabel('y')
plt.ylabel('Density')

plt.show()

输出结果为：

3、最普通的线性回归，数据未标准化，无正则化

#最普通的最小二乘,数据未标准化，无正则化
import numpy as np
from sklearn.linear_model import LinearRegression

from sklearn.model_selection import train_test_split
X_train_1, X_test_1, y_train_1, y_test_1 = train_test_split(X, y, test_size=0.2, random_state=42)

#训练模型
reg_1 = LinearRegression().fit(X_train_1, y_train_1)
print("R^2系数为：",reg_1.score(X_train_1, y_train_1))
print("权重为：",reg_1.coef_)
print("截距为：",reg_1.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_1 = reg_1.predict(X_train_1)
# 计算均方误差
mse_in_train_1 = mean_squared_error(y_train_1, y_pred_in_train_1)
print(f"训练集上的均方误差为: {mse_in_train_1}")

#在测试集上计算均方误差
y_pred_in_test_1 = reg_1.predict(X_test_1)
# 计算均方误差
mse_in_test_1 = mean_squared_error(y_test_1, y_pred_in_test_1)
print(f"测试集上的均方误差为: {mse_in_test_1}")

输出结果为：

R^2系数为： 0.6125511913966952
权重为： [ 4.48674910e-01  9.72425752e-03 -1.23323343e-01  7.83144907e-01
 -2.02962058e-06 -3.52631849e-03 -4.19792487e-01 -4.33708065e-01]
截距为： -37.02327770606391
训练集上的均方误差为: 0.5179331255246699
测试集上的均方误差为: 0.5558915986952422

4、最普通的线性回归，有标准化，无正则化

#有标准化，无正则化

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])

import numpy as np
from sklearn.linear_model import LinearRegression

from sklearn.model_selection import train_test_split
X_train_2, X_test_2, y_train_2, y_test_2 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_2 = LinearRegression().fit(X_train_2, y_train_2)
print("R^2系数为：",reg_2.score(X_train_2, y_train_2))
print("权重为：",reg_2.coef_)
print("截距为：",reg_2.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_2 = reg_2.predict(X_train_2)
# 计算均方误差
mse_in_train_2 = mean_squared_error(y_train_2, y_pred_in_train_2)
print(f"训练集上的均方误差为: {mse_in_train_2}")

#在测试集上计算均方误差
y_pred_in_test_2 = reg_2.predict(X_test_2)
# 计算均方误差
mse_in_test_2 = mean_squared_error(y_test_2, y_pred_in_test_2)
print(f"测试集上的均方误差为: {mse_in_test_2}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125511913966952
权重为： [ 0.85238169  0.12238224 -0.30511591  0.37113188 -0.00229841 -0.03662363
 -0.89663505 -0.86892682]
截距为： 2.067862309508389
训练集上的均方误差为: 0.5179331255246699
测试集上的均方误差为: 0.555891598695244

二、岭回归

1、岭回归 Ridge

#岭回归


import numpy as np
from sklearn.linear_model import Ridge

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_3, X_test_3, y_train_3, y_test_3 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_3 = Ridge(alpha=1).fit(X_train_3, y_train_3)
print("R^2系数为：",reg_3.score(X_train_3, y_train_3))
print("权重为：",reg_3.coef_)
print("截距为：",reg_3.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_3 = reg_3.predict(X_train_3)
# 计算均方误差
mse_in_train_3 = mean_squared_error(y_train_3, y_pred_in_train_3)
print(f"训练集上的均方误差为: {mse_in_train_3}")

#在测试集上计算均方误差
y_pred_in_test_3 = reg_3.predict(X_test_3)
# 计算均方误差
mse_in_test_3 = mean_squared_error(y_test_3, y_pred_in_test_3)
print(f"测试集上的均方误差为: {mse_in_test_3}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.612551119993334
权重为： [ 0.85231009  0.12246004 -0.3048709   0.37081855 -0.00227294 -0.03662725
 -0.89588451 -0.86816501]
截距为： 2.067863077782602
训练集上的均方误差为: 0.5179332209751273
测试集上的均方误差为: 0.5558512007367514

2、带有内置交叉验证的岭回归 RidgeCV

RidgeCV 是基于 Ridge（岭回归）的一个扩展版本，其主要特点在于内置了交叉验证功能，用于自动选择最佳的正则化参数（α 值）。在 Ridge 回归中，通过添加 L2 正则化项来缓解多重共线性问题，并防止模型过拟合。而 RidgeCV 则在训练过程中，通过交叉验证来评估多个候选 α 值，从而确定一个在数据上表现最优的正则化强度。

#带内置交叉验证的岭回归


import numpy as np
from sklearn.linear_model import RidgeCV

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_4, X_test_4, y_train_4, y_test_4 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_4 = RidgeCV(alphas=[1e-3,1e-2,1e-1,1]).fit(X_train_4, y_train_4)
print("R^2系数为：",reg_4.score(X_train_4, y_train_4))
print("权重为：",reg_4.coef_)
print("截距为：",reg_4.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_4 = reg_4.predict(X_train_4)
# 计算均方误差
mse_in_train_4 = mean_squared_error(y_train_4, y_pred_in_train_4)
print(f"训练集上的均方误差为: {mse_in_train_4}")

#在测试集上计算均方误差
y_pred_in_test_4 = reg_4.predict(X_test_4)
# 计算均方误差
mse_in_test_4 = mean_squared_error(y_test_4, y_pred_in_test_4)
print(f"测试集上的均方误差为: {mse_in_test_4}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125511199933349
权重为： [ 0.85231009  0.12246004 -0.3048709   0.37081855 -0.00227294 -0.03662725
 -0.89588451 -0.86816501]
截距为： 2.0678630777825378
训练集上的均方误差为: 0.5179332209751261
测试集上的均方误差为: 0.5558512007384657

三、Lasso回归

1、Lasso回归

#Lasso回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_5, X_test_5, y_train_5, y_test_5 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_5 = linear_model.Lasso(alpha=0.01).fit(X_train_5, y_train_5)
print("R^2系数为：",reg_5.score(X_train_5, y_train_5))
print("权重为：",reg_5.coef_)
print("截距为：",reg_5.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_5 = reg_5.predict(X_train_5)
# 计算均方误差
mse_in_train_5 = mean_squared_error(y_train_5, y_pred_in_train_5)
print(f"训练集上的均方误差为: {mse_in_train_5}")

#在测试集上计算均方误差
y_pred_in_test_5 = reg_5.predict(X_test_5)
# 计算均方误差
mse_in_test_5 = mean_squared_error(y_test_5, y_pred_in_test_5)
print(f"测试集上的均方误差为: {mse_in_test_5}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6082259693662162
权重为： [ 0.79530412  0.12700165 -0.1593924   0.21628228 -0.         -0.02829497
 -0.79223    -0.75673715]
截距为： 2.067942536287589
训练集上的均方误差为: 0.5237149880961657
测试集上的均方误差为: 0.5479327795506

2、带有内置交叉验证的 Lasso回归

#Lasso回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_6, X_test_6, y_train_6, y_test_6 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_6 = linear_model.LassoCV(cv=5, random_state=0).fit(X_train_6, y_train_6)
print("R^2系数为：",reg_6.score(X_train_6, y_train_6))
print("权重为：",reg_6.coef_)
print("截距为：",reg_6.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_6 = reg_6.predict(X_train_6)
# 计算均方误差
mse_in_train_6 = mean_squared_error(y_train_6, y_pred_in_train_6)
print(f"训练集上的均方误差为: {mse_in_train_6}")

#在测试集上计算均方误差
y_pred_in_test_6 = reg_6.predict(X_test_6)
# 计算均方误差
mse_in_test_6 = mean_squared_error(y_test_6, y_pred_in_test_6)
print(f"测试集上的均方误差为: {mse_in_test_6}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125069391599488
权重为： [ 0.84677361  0.12319172 -0.29057195  0.35575098 -0.00105138 -0.0358658
 -0.8857684  -0.8573757 ]
截距为： 2.067869881926557
训练集上的均方误差为: 0.5179922809505753
测试集上的均方误差为: 0.5544062174455687

四、最小角度回归LAR

1、LAR回归

#LAR回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_7, X_test_7, y_train_7, y_test_7 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_7 = linear_model.Lars().fit(X_train_7, y_train_7)
print("R^2系数为：",reg_7.score(X_train_7, y_train_7))
print("权重为：",reg_7.coef_)
print("截距为：",reg_7.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_7 = reg_7.predict(X_train_7)
# 计算均方误差
mse_in_train_7 = mean_squared_error(y_train_7, y_pred_in_train_7)
print(f"训练集上的均方误差为: {mse_in_train_7}")

#在测试集上计算均方误差
y_pred_in_test_7 = reg_7.predict(X_test_7)
# 计算均方误差
mse_in_test_7 = mean_squared_error(y_test_7, y_pred_in_test_7)
print(f"测试集上的均方误差为: {mse_in_test_7}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125511913966952
权重为： [ 0.85238169  0.12238224 -0.30511591  0.37113188 -0.00229841 -0.03662363
 -0.89663505 -0.86892682]
截距为： 2.067862309508389
训练集上的均方误差为: 0.5179331255246699
测试集上的均方误差为: 0.5558915986952446

2、交叉验证的最小角度回归模型

#LARCV回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_8, X_test_8, y_train_8, y_test_8 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_8 = linear_model.LarsCV().fit(X_train_8, y_train_8)
print("R^2系数为：",reg_8.score(X_train_8, y_train_8))
print("权重为：",reg_8.coef_)
print("截距为：",reg_8.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_8 = reg_8.predict(X_train_8)
# 计算均方误差
mse_in_train_8 = mean_squared_error(y_train_8, y_pred_in_train_8)
print(f"训练集上的均方误差为: {mse_in_train_8}")

#在测试集上计算均方误差
y_pred_in_test_8 = reg_8.predict(X_test_8)
# 计算均方误差
mse_in_test_8 = mean_squared_error(y_test_8, y_pred_in_test_8)
print(f"测试集上的均方误差为: {mse_in_test_8}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125284264168372
权重为： [ 0.84829912  0.12295169 -0.29457668  0.36001341 -0.00140771 -0.03607767
 -0.88897885 -0.86076655]
截距为： 2.0678677052360808
训练集上的均方误差为: 0.5179635572537357
测试集上的均方误差为: 0.5548051360099295

五、弹性回归（L1+L2正则）

1、弹性回归

#弹性回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_9, X_test_9, y_train_9, y_test_9 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_9 = linear_model.ElasticNet(alpha=0.1, l1_ratio=0.1).fit(X_train_9, y_train_9)
print("R^2系数为：",reg_9.score(X_train_9, y_train_9))
print("权重为：",reg_9.coef_)
print("截距为：",reg_9.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_9 = reg_9.predict(X_train_9)
# 计算均方误差
mse_in_train_9 = mean_squared_error(y_train_9, y_pred_in_train_9)
print(f"训练集上的均方误差为: {mse_in_train_9}")

#在测试集上计算均方误差
y_pred_in_test_9 = reg_9.predict(X_test_9)
# 计算均方误差
mse_in_test_9 = mean_squared_error(y_test_9, y_pred_in_test_9)
print(f"测试集上的均方误差为: {mse_in_test_9}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.5663150481219212
权重为： [ 0.72053917  0.15152293 -0.03202614  0.05370572  0.         -0.02739113
 -0.37827649 -0.33677859]
截距为： 2.0685307128041894
训练集上的均方误差为: 0.5797405944515618
测试集上的均方误差为: 0.591270122305316

2、带交叉验证的弹性回归ElasticNetCV

#带交叉验证的弹性回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_10, X_test_10, y_train_10, y_test_10 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_10 = linear_model.ElasticNetCV(cv=5, random_state=0).fit(X_train_10, y_train_10)
print("R^2系数为：",reg_10.score(X_train_10, y_train_10))
print("权重为：",reg_10.coef_)
print("截距为：",reg_10.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_10 = reg_10.predict(X_train_10)
# 计算均方误差
mse_in_train_10 = mean_squared_error(y_train_10, y_pred_in_train_10)
print(f"训练集上的均方误差为: {mse_in_train_10}")

#在测试集上计算均方误差
y_pred_in_test_10 = reg_10.predict(X_test_10)
# 计算均方误差
mse_in_test_10 = mean_squared_error(y_test_10, y_pred_in_test_10)
print(f"测试集上的均方误差为: {mse_in_test_10}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6124786022650697
权重为： [ 0.84699182  0.12403155 -0.29038901  0.35484741 -0.00097329 -0.03606413
 -0.87819697 -0.84980088]
截距为： 2.0678783701103045
训练集上的均方误差为: 0.5180301610942394
测试集上的均方误差为: 0.554245232686696

六、Huber损失的回归—对异常值具有鲁棒性

参数epsilon 的作用与含义

  • 定义阈值：
    epsilon 定义了在损失函数中何时从二次（L2）损失转为线性（L1）损失的临界点。

    • 当预测误差 |y - \hat{y}| 小于等于 epsilon 时，使用 L2 损失，模型对这些较小误差进行精确拟合；
    • 当误差超过 epsilon 时，损失函数切换为 L1 损失，从而降低异常值对模型的影响。

  • 鲁棒性调节：

    • 较小的 epsilon：将阈值设定得较低，会使得更多的误差被认为是“异常”并使用 L1 损失，这增强了模型对异常值的鲁棒性，但可能牺牲一部分在正常数据上的拟合效率。
    • 较大的 epsilon：阈值较高时，大部分误差依然在 L2 损失范围内，模型会更像传统的最小二乘回归，可能在异常值存在时表现不够鲁棒。

  • 默认值与适用场景：
    sklearn 默认的 epsilon 值为 1.35。这个默认值在许多情况下能取得一个较好的平衡，假设残差大致服从正态分布。但如果数据的噪声分布或者异常值特性与正态分布有较大偏离，则可能需要通过交叉验证或其他方式调整 epsilon 以获得更好的模型表现。

#带交叉验证的弹性回归


import numpy as np
from sklearn import linear_model

#对X进行标准化
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_standardized = scaler.fit_transform(X)
print(X_standardized[[0,1,2],:])


from sklearn.model_selection import train_test_split
X_train_11, X_test_11, y_train_11, y_test_11 = train_test_split(X_standardized, y, test_size=0.2, random_state=42)

#训练模型
reg_11 = linear_model.HuberRegressor(epsilon=5).fit(X_train_11, y_train_11)
print("R^2系数为：",reg_11.score(X_train_11, y_train_11))
print("权重为：",reg_11.coef_)
print("截距为：",reg_11.intercept_)

#计算均方误差
from sklearn.metrics import mean_squared_error
#在训练集上计算均方误差
y_pred_in_train_11 = reg_11.predict(X_train_11)
# 计算均方误差
mse_in_train_11 = mean_squared_error(y_train_11, y_pred_in_train_11)
print(f"训练集上的均方误差为: {mse_in_train_11}")

#在测试集上计算均方误差
y_pred_in_test_11 = reg_11.predict(X_test_11)
# 计算均方误差
mse_in_test_11 = mean_squared_error(y_test_11, y_pred_in_test_11)
print(f"测试集上的均方误差为: {mse_in_test_11}")

输出结果为：

[[ 2.34476576  0.98214266  0.62855945 -0.15375759 -0.9744286  -0.04959654
   1.05254828 -1.32783522]
 [ 2.33223796 -0.60701891  0.32704136 -0.26333577  0.86143887 -0.09251223
   1.04318455 -1.32284391]
 [ 1.7826994   1.85618152  1.15562047 -0.04901636 -0.82077735 -0.02584253
   1.03850269 -1.33282653]]
R^2系数为： 0.6125474458474925
权重为： [ 0.8554186   0.12299684 -0.30901178  0.37582821 -0.00230365 -0.03672557
 -0.89469167 -0.86738322]
截距为： 2.0680122661277878
训练集上的均方误差为: 0.5179381324932351
测试集上的均方误差为: 0.5565907434185678

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