普通最⼩⼆乘法回归-OLS（ordinaryleastsquare）
前⾔
这篇博客⽤来记录初学 普通最⼩⼆乘回归 遇到的相关知识点和解决问题的过程。
开发环境：Pycharm 2018.1.2
版本：Python 2.7.14 :: Anaconda, Inc.
回归 - 已有数据
数据集：
简 介：从 1990 年⾄今，美国加州所有街区⼈⼝普查的信息，关于 9 组变量，共 20640 个观测值。
Variables Bols tols
INTERCEPT (截距)11.4939275.7518
MEDIAN INCOME (收⼊中值)0.479045.7768
MEDIAN INCOME2 (收⼊中值2)-0.0166-9.4841
MEDIAN INCOME3 (收⼊中值3)-0.0002-1.9157
ln(MEDIAN AGE) (年龄中位数)0.157033.6123 ln(TOTAL ROOMS/ POPULATION) (总房屋数/⼈⼝)-0.8582-56.1280 ln(BEDROOMS/ POPULATION) (卧室/⼈⼝)0.804338.0685
ln(POPULATION/ HOUSEHOLDS) (⼈⼝/家庭)-0.4077-20.8762
ln(HOUSEHOLDS) (家庭)0.047713.0792⽤下⾯代码读⼊数据, 并弄清楚哪些是⾃变量哪个是因变量:
import pandas as pd
import numpy as np
data = pd.read_csv("cal_housing.csv")
name = lumns
X = data[name[:8]] # 第1-8列
y = data[name[8:9]] # 第9列
print("X name :", name[:8])
心神不宁焦虑怎么办print("y name :", name[8:9])
print(data.shape, X.shape, y.shape) # 返回⾏列数
---------------------------------------------
('X name :', Index([u'longitude', u'latitude', u'housingMedianAge', u'totalRooms',
hertzu'totalBedrooms', u'population', u'houholds', u'medianIncome'],
dtype='object'))
兔子的英文
('y name :', Index([u'medianHouValue'], dtype='object'))
((20640, 9), (20640, 8), (20640, 1))
把数据随机分成训练集和测试集
可⾃⼰决定随机种⼦(多少位数都可以)和测试集百分⽐(⼩于0.5即⼩于50%)
ed = 8888# 随机种⼦
proportion = 0.1# 测试集百分⽐
del_lection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=proportion, random_state=ed)
print(X_train.shape,X_test.shape, y_train.shape, y_test.shape)
---------------------------------------------------
((18576, 8), (2064, 8), (18576, 1), (2064, 1))
做回归, 并求出 R^2 和 N M S E
reg = LinearRegression() # 线性回归（Linear Regression）
res = reg.fit(X_train, y_train) # 对训练集X_train, y_train进⾏训练
y_hat = res.predict(X_test) # 使⽤训练得到的估计器对输⼊为X_test的集合进⾏预测，得到y_hat
e = y_test-y_hat # 计算残差
SSE_cv = np.mean(e**2) # 残差平⽅和
SSE_test = np.mean((an(y_test))**2) # 拍脑袋平⽅和
NMSE_cv = SSE_cv/SSE_test # 标准化均⽅误差 NMSE_cv
R2_cv = 1 - NMSE_cv # 可决系数R2_cv
print R2_cv
print NMSE_cv
-----------------------------------------------------------------
medianHouValue 0.657186
dtype: float64
英文版medianHouValue 0.342814
dtype: float64
回归 - 模拟数据
⾃⼰决定样本量(n), ⾃变量个数(p)和系数值(B), ⾃⼰决定正态误差的均值m和标准差s
ed = 8888# 随机种⼦
n = 100# 样本量
p = 7# ⾃变量个数
m = 0# 误差项均值
s = 5# 标准差
B = [2, 5, 16, 9, -3, -5, -2] # beta值
C = [2, 2]
np.random.ed(ed)
X = al(0, 1, (n, p))
y = X.dot(B)+al(m, s, n)
print(X.shape, y.shape)
----------------------------------------------------------
((100L, 7L), (100L,))
实施回归
import statsmodels.api as sm
# 增加截距项
mod = sm.OLS(y, X) # 普通最⼩⼆乘模型，ordinary least square model
res = mod.fit()
donna summer#输出R^2
print("R^2:",res2.rsquared,"\nNMSE:",1-res2.rsquared)
----------------------------------------------------------
R^2: 0.92564484308
NMSE: 0.0743551569196insisted
print (res2.summary())
---------------------------------
假期的英文OLS Regression Results2020年世界读书日主题
============================================================================== Dep. Variable: y R-squared: 0.926
Model: OLS Adj. R-squared: 0.920
Method: Least Squares F-statistic: 165.4
Date: Mon, 07 May 2018 Prob (F-statistic): 1.32e-49
Time: 09:54:25 Log-Likelihood: -304.71
No. Obrvations: 100 AIC: 623.4
Df Residuals: 93 BIC: 641.7
Df Model: 7
Covariance Type: nonrobust
============================================================================== coef std err t P>|t| [0.0250.975]
------------------------------------------------------------------------------
x1 2.34770.532 4.4150.000 1.292 3.404
x2 5.58290.51110.9290.000 4.568 6.597man in black
x3 15.76190.59626.4440.00014.57816.946
x4 8.95950.52117.1810.0007.9249.995
x5 -3.30480.530 -6.2330.000 -4.358 -2.252
x6 -4.99320.491 -10.1750.000 -5.968 -4.019
x7 -2.01260.536 -3.7540.000 -3.077 -0.948
============================================================================== Omnibus: 0.577 Durbin-Watson: 1.970
Prob(Omnibus): 0.749 Jarque-Bera (JB): 0.227
Skew: -0.078 Prob(JB): 0.893compaq
Kurtosis: 3.174 Cond. No. 1.51
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
附录
