python 线性回归分析模型检验标准--拟合优度详解

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from pandas import DataFrame,Series
from sklearn.cross_validation import train_test_split
from sklearn.linear_model import LinearRegression
 
#创建数据集
examDict = {'学习时间':[0.50,0.75,1.00,1.25,1.50,1.75,1.75,
           2.00,2.25,2.50,2.75,3.00,3.25,3.50,4.00,4.25,4.50,4.75,5.00,5.50],
       '分数':[10,22,13,43,20,22,33,50,62,
          48,55,75,62,73,81,76,64,82,90,93]}
 
#转换为DataFrame的数据格式
examDf = DataFrame(examDict)
#examDf
#绘制散点图
plt.scatter(examDf.分数,examDf.学习时间,color = 'b',label = "Exam Data")
 
#添加图的标签（x轴，y轴）
plt.xlabel("Hours")
plt.ylabel("Score")
#显示图像
plt.show()
 
#将原数据集拆分训练集和测试集
exam_X = examDf.学习时间
exam_Y = examDf.分数
X_train,X_test,Y_train,Y_test = train_test_split(exam_X,exam_Y,train_size=0.8)
#X_train为训练数据标签,X_test为测试数据标签,exam_X为样本特征,exam_y为样本标签，train_size 训练数据占比
 
print("原始数据特征:",exam_X.shape,
   ",训练数据特征:",X_train.shape,
   ",测试数据特征:",X_test.shape)
 
print("原始数据标签:",exam_Y.shape,
   ",训练数据标签:",Y_train.shape,
   ",测试数据标签:",Y_test.shape)
 
model = LinearRegression()
 
#对于模型错误我们需要把我们的训练集进行reshape操作来达到函数所需要的要求
# model.fit(X_train,Y_train)
 
#reshape如果行数=-1的话可以使我们的数组所改的列数自动按照数组的大小形成新的数组
#因为model需要二维的数组来进行拟合但是这里只有一个特征所以需要reshape来转换为二维数组
X_train = X_train.values.reshape(-1,1)
X_test = X_test.values.reshape(-1,1)
 
model.fit(X_train,Y_train)
a = model.intercept_#截距
b = model.coef_#回归系数
print("最佳拟合线:截距",a,",回归系数：",b)
 

# 用训练集进行拟合优度，验证回归方程是否合理
def get_lr_stats(x, y, model):
  message0 = '一元线性回归方程为: '+'\ty' + '=' + str(model.intercept_)+' + ' +str(model.coef_[0]) + '*x'
  from scipy import stats
  n = len(x)
  y_prd = model.predict(x)
  Regression = sum((y_prd - np.mean(y))**2) # 回归平方和
  Residual  = sum((y - y_prd)**2)     # 残差平方和
  total = sum((y-np.mean(y))**2) #总体平方和
  R_square  = 1-Residual / total # 相关性系数R^2
  message1 = ('相关系数(R^2)： ' + str(R_square) + '；' + '\n'+ '总体平方和(TSS)： ' + str(total) + '；' + '\n')
  message2 = ('回归平方和(RSS)： ' + str(Regression) + '；' + '\n残差平方和(ESS)： ' + str(Residual) + '；' + '\n')
  return print(message0 +'\n' +message1 + message2 )
 
get_lr_stats(X_train,Y_train,model)

#训练数据的预测值
y_train_pred = model.predict(X_train)
#绘制最佳拟合线：标签用的是训练数据集中的极值预测值
X_train_pred = [min(X_train),max(X_train)]
y_train_pred = [a+b*min(X_train),a+b*max(X_train)]
plt.plot(X_train_pred, y_train_pred, color='green', linewidth=3, label="best line")
 
#测试数据散点图
plt.scatter(X_test, Y_test, color='red', label="test data")
plt.scatter(X_train, Y_train, color="blue", label="train data")
 
#添加图标标签
plt.legend(loc=2)
plt.xlabel("Hours")
plt.ylabel("Score")
#显示图像
plt.savefig("lines.jpg")
plt.show()
#计算拟合优度
score = model.score(X_test,Y_test)
print(score)