Least square (最小二乘法)

 

线性模型

 

可以重写为向量形式

其中y 为常量，

通常来说，输出y是一个k维向量，则β是一个(p + 1) * k维的矩阵

最小二乘法

选择系数矩阵β使得在数据集上，预测值与真实值的距离平方和最小。

RSS是二次函数，它的最小值总是存在的。

 

 

Sklearn示例代码

 

 

import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets, linear_model
from sklearn.metrics import mean_squared_error, r2_score


#Load the diabetes dataset
diabetes = datasets.load_diabetes()

diabetes_X = diabetes.data[:, np.newaxis, 2]

# Split the data into training/testing sets
diabetes_X_train = diabetes_X[:-20]
diabetes_X_test = diabetes_X[-20:]

# Split the target into training/testing sets
diabetes_y_train = diabetes.target[:-20]
diabetes_y_test = diabetes.target[-20:]

# Create a learning regression object
regr = linear_model.LinearRegression()

# Train the model using the training sets
regr.fit(diabetes_X_train, diabetes_y_train)

# Make predictions using the testing set
diabetes_y_pred = regr.predict(diabetes_X_test)

# The coefficients
print('Coefficients: \n', regr.coef_)

# The mean squared error
mse = mean_squared_error(diabetes_y_test, diabetes_y_pred)
print("Mean squared eror: %.2f" % mse)

# Explained variance score: 1 is perfect prediction
print('Variable score: %.2f' % r2_score(diabetes_y_test, diabetes_y_pred))

# Plot outputs
plt.scatter(diabetes_X_test, diabetes_y_test, color='black')
plt.plot(diabetes_X_test, diabetes_y_pred, color='blue', linewidth=3)

plt.xticks(())
plt.yticks(())

plt.show()

print(mse)

 

缺点

1. 最小二乘法基于线性模型，线性模型的表达能力不足，不能够拟合真实世界中的非线性关系；
2. 如果样本X的维度很高，即(XTX)-1是一个PXP维的矩阵，而且P很大，求其逆矩阵将很复杂。