python

超轻量级php框架startmvc

Python实现的简单线性回归算法实例分析

更新时间:2020-06-18 03:18:02 作者:startmvc
本文实例讲述了Python实现的简单线性回归算法。分享给大家供大家参考,具体如下:用python

本文实例讲述了Python实现的简单线性回归算法。分享给大家供大家参考,具体如下:

用python实现R的线性模型(lm)中一元线性回归的简单方法,使用R的women示例数据,R的运行结果:

> summary(fit) Call: lm(formula = weight ~ height, data = women) Residuals:     Min      1Q  Median      3Q     Max -1.7333 -1.1333 -0.3833  0.7417  3.1167 Coefficients:              Estimate Std. Error t value Pr(>|t|) (Intercept) -87.51667    5.93694  -14.74 1.71e-09 *** height        3.45000    0.09114   37.85 1.09e-14 *** --- Signif. codes:  0 ‘***' 0.001 ‘**' 0.01 ‘*' 0.05 ‘.' 0.1 ‘ ' 1 Residual standard error: 1.525 on 13 degrees of freedom Multiple R-squared:  0.991, Adjusted R-squared:  0.9903 F-statistic:  1433 on 1 and 13 DF,  p-value: 1.091e-14

python实现的功能包括:

  1. 计算pearson相关系数
  2. 使用最小二乘法计算回归系数
  3. 计算拟合优度判定系数R2R2
  4. 计算估计标准误差Se
  5. 计算显著性检验的F和P值

import numpy as np
import scipy.stats as ss
class Lm:
 """简单一元线性模型,计算回归系数、拟合优度的判定系数和
 估计标准误差,显著性水平"""
 def __init__(self, data_source, separator):
 self.beta = np.matrix(np.zeros(2))
 self.yhat = np.matrix(np.zeros(2))
 self.r2 = 0.0
 self.se = 0.0
 self.f = 0.0
 self.msr = 0.0
 self.mse = 0.0
 self.p = 0.0
 data_mat = np.genfromtxt(data_source, delimiter=separator)
 self.xarr = data_mat[:, :-1]
 self.yarr = data_mat[:, -1]
 self.ybar = np.mean(self.yarr)
 self.dfd = len(self.yarr) - 2 # 自由度n-2
 return
 # 计算协方差
 @staticmethod
 def cov_custom(x, y):
 result = sum((x - np.mean(x)) * (y - np.mean(y))) / (len(x) - 1)
 return result
 # 计算相关系数
 @staticmethod
 def corr_custom(x, y):
 return Lm.cov_custom(x, y) / (np.std(x, ddof=1) * np.std(y, ddof=1))
 # 计算回归系数
 def simple_regression(self):
 xmat = np.mat(self.xarr)
 ymat = np.mat(self.yarr).T
 xtx = xmat.T * xmat
 if np.linalg.det(xtx) == 0.0:
 print('Can not resolve the problem')
 return
 self.beta = np.linalg.solve(xtx, xmat.T * ymat) # xtx.I * (xmat.T * ymat)
 self.yhat = (xmat * self.beta).flatten().A[0]
 return
 # 计算拟合优度的判定系数R方,即相关系数corr的平方
 def r_square(self):
 y = np.mat(self.yarr)
 ybar = np.mean(y)
 self.r2 = np.sum((self.yhat - ybar) ** 2) / np.sum((y.A - ybar) ** 2)
 return
 # 计算估计标准误差
 def estimate_deviation(self):
 y = np.array(self.yarr)
 self.se = np.sqrt(np.sum((y - self.yhat) ** 2) / self.dfd)
 return
 # 显著性检验F
 def sig_test(self):
 ybar = np.mean(self.yarr)
 self.msr = np.sum((self.yhat - ybar) ** 2)
 self.mse = np.sum((self.yarr - self.yhat) ** 2) / self.dfd
 self.f = self.msr / self.mse
 self.p = ss.f.sf(self.f, 1, self.dfd)
 return
 def summary(self):
 self.simple_regression()
 corr_coe = Lm.corr_custom(self.xarr[:, -1], self.yarr)
 self.r_square()
 self.estimate_deviation()
 self.sig_test()
 print('The Pearson\'s correlation coefficient: %.3f' % corr_coe)
 print('The Regression Coefficient: %s' % self.beta.flatten().A[0])
 print('R square: %.3f' % self.r2)
 print('The standard error of estimate: %.3f' % self.se)
 print('F-statistic: %d on %s and %s DF, p-value: %.3e' % (self.f, 1, self.dfd, self.p))

python执行结果:

The Regression Coefficient: [-87.51666667   3.45      ] R square: 0.991 The standard error of estimate: 1.525 F-statistic:  1433 on 1 and 13 DF,  p-value: 1.091e-14

其中求回归系数时用矩阵转置求逆再用numpy内置的解线性方程组的方法是最快的:


a = np.mat(women.xarr); b = np.mat(women.yarr).T
timeit (a.I * b)
99.9 µs ± 941 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
timeit ata.I * (a.T*b)
64.9 µs ± 717 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
timeit np.linalg.solve(ata, a.T*b)
15.1 µs ± 126 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

Python 线性回归 算法