线性回归分析（python3）
1、python固定导入的包
# 工具：python3
#固定导入
import numpy as np #科学计算基础库，多维数组对象ndarray
import pandas as pd #数据处理库，DataFrame（二维数组）
import matplotlib as mpl #画图基础库
import matplotlib.pyplot as plt #最常用的绘图库
from scipy import stats #scipy库的stats模块
mpl.rcParams["font.family"]="SimHei" #使用支持的黑体中文字体
mpl.rcParams["axes.unicode_minus"]=False # 用来正常显示负号 "-"
plt.rcParams['font.sans-serif']=['SimHei'] # 用来正常显示中文标签
# % matplotlib inline #jupyter中用于直接嵌入图表，不用plt.show()
import warnings
warnings.filterwarnings("ignore") #用于排除警告
#用于显示使用库的版本
print("numpy_" + np.__version__)
print("pandas_" + pd.__version__)
print("matplotlib_"+ mpl.__version__)
numpy_1.17.4
pandas_0.23.4
matplotlib_2.2.3
2、常见的回归函数
3、线性回归
Y= aX + b + e ,e表示残差
线性回归：分析结论的前提：数据满足统计假设
- 正态性：预测变量固定时，因变量正态分布
- 独立性：因变量互相独立
- 线性：因变量与自变量线性（残差白噪声）
- 同方差性：残差方差均匀
(1)、scipy.optimize.curve_fit():回归函数拟合
from scipy.optimize import curve_fit #回归函数拟合包
def func(x,a,b):
y = a*x + b
return y
x= np.array([150,200,250,350,300,400,600])
ydata=np.array([6450,7450,8450,9450,11450,15450,18450])
popt,pcov=curve_fit(func,x,ydata)
print(curve_fit(func,x,ydata)) #拟合函数 y = 28.03191489*x + 2011.17021277
plt.title('分布图')
plt.xlabel('x')
plt.ylabel('y')
plt.plot(x,ydata,'bo',x,func(x,popt[0],popt[1]),'r-')
plt.legend(["ydata","y拟合"],loc="best", frameon=False, ncol=1)
plt.show()
(array([ 28.03191489, 2011.17021277]), array([[ 1.81614985e+01, -5.83762446e+03],
[-5.83762446e+03, 2.22478353e+06]]))
(2)、scipy.stats.linregress(): 线性拟合
from scipy import stats #线性回归包
x= np.array([150,200,250,350,300,400,600])
ydata=np.array([6450,7450,8450,9450,11450,15450,18450])
slope, intercept, r_value, p_value, std_err = stats.linregress(x,ydata)
#a , b r ， p ， 标准误
print(stats.linregress(x,ydata))
y = slope*x + intercept #拟合函数
LinregressResult(slope=28.03191489361702, intercept=2011.1702127659555,
rvalue=0.9467887456768281, pvalue=0.0012189021752212815, stderr=4.261630957224319)
(3)、statsmodels.formula.api.OLS():普通最小二乘模型拟合- - 常用
tips = pd.read_csv(r"E:\tips.txt",sep='\t',encoding='utf-8') #导入txt格式数据
display(tips.sample(3)) #随机抽取3行数据
import statsmodels.formula.api as sm #statsmodels.formula.api.ols 回归模型库
model = sm.OLS(tips["total_bill"], tips["tip"]) # OLS 是普通最小二乘，sm 中还有很多其它方法解决不同的问题
# total_bill：因变量, tip：自变量
result = model.fit() # 做拟合
print(result.summary()) #返回拟合总结信息
print(result.params) # 对应的各个变量的权重
resid = result.resid # 输出残差
print(type(resid)) # <class 'pandas.core.series.Series'>
print(stats.normaltest(result.resid.values)) #对残差进行正态分布检验，p<阀值，拒绝原假设，则是正态分布
figure = plt.figure(num="画布1",figsize=(8,4),dpi=80,facecolor="gray",edgecolor="blue",frameon=True)
axes1=figure.add_subplot(2,2,1)
plt.title("残差图")
axes1.plot(result.resid) #看看残差图
axes2=figure.add_subplot(2,2,2)
plt.title('分布/拟合图')
plt.xlabel('tip')
plt.ylabel('total_bill')
axes2.plot(tips["tip"],tips["total_bill"],'bo',tips["tip"],result.params[0]*tips["tip"],'r-')
plt.legend(["total_bill","total_bill拟合"],loc="best", frameon=False, ncol=2)
plt.show()
# '''
# 模型解释：
# result.summary()的输出解释：R-squared:0.892，Adj. R-squared:0.891 都比较大，说明模型是比较显著的，
# tip的权值检验p:0.000,非常小，拒绝原假设，说明系数的正确性也是非常显著的。
# stats.normaltest(result.resid.values) ：对残差进行正态分布检验，p<阀值,说明残差也是符合要求。
# 模型可用。
# 模型公式：total_bill = 6.2052 * tip
# '''
| total_bill | tip | sex | smoker | day | time | size | |
|---|---|---|---|---|---|---|---|
| 92 | 5.75 | 1.00 | Female | Yes | Fri | Dinner | 2 |
| 139 | 13.16 | 2.75 | Female | No | Thur | Lunch | 2 |
| 136 | 10.33 | 2.00 | Female | No | Thur | Lunch | 2 |
OLS Regression Results
==============================================================================
Dep. Variable: total_bill R-squared: 0.892
Model: OLS Adj. R-squared: 0.891
Method: Least Squares F-statistic: 2004.
Date: Sun, 16 Feb 2020 Prob (F-statistic): 2.26e-119
Time: 18:07:20 Log-Likelihood: -825.57
No. Observations: 244 AIC: 1653.
Df Residuals: 243 BIC: 1657.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
tip 6.2052 0.139 44.771 0.000 5.932 6.478
==============================================================================
Omnibus: 33.937 Durbin-Watson: 2.112
Prob(Omnibus): 0.000 Jarque-Bera (JB): 71.569
Skew: 0.688 Prob(JB): 2.88e-16
Kurtosis: 5.269 Cond. No. 1.00
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
tip 6.205158
dtype: float64
<class 'pandas.core.series.Series'>
NormaltestResult(statistic=33.937192138902724, pvalue=4.2720109911999064e-08)
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statsmodels.formula.api.ols():还可以采取公式的方式。
np.random.seed(123456789)
N = 100
x1 = np.random.randn(N) # 自变量
x2 = np.random.randn(N)
data = pd.DataFrame({"x1": x1, "x2": x2})
def y_true(x1, x2):
return 1 + 2 * x1 + 3 * x2 + 4 * x1 * x2
data["y_true"] = y_true(x1, x2) # 因变量
e = np.random.randn(N) # 标准正态分布
data["y"] = data["y_true"] + e # 加些噪音，加些扰动，噪音是符合标准正态分布的
data.head(3)
import statsmodels.formula.api as sm #statsmodels.formula.api.ols 回归模型库
model = sm.ols("y ~x1 + x2", data) # 这里没有写截距，截距是默认存在的
# model = sm.ols("y ~ -1 + x1 + x2", data) # 这里加上-1，表示不加截距
result = model.fit()
print(result.summary())
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.380
Model: OLS Adj. R-squared: 0.367
Method: Least Squares F-statistic: 29.76
Date: Sun, 16 Feb 2020 Prob (F-statistic): 8.36e-11
Time: 18:07:20 Log-Likelihood: -271.52
No. Observations: 100 AIC: 549.0
Df Residuals: 97 BIC: 556.9
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 0.9868 0.382 2.581 0.011 0.228 1.746
x1 1.0810 0.391 2.766 0.007 0.305 1.857
x2 3.0793 0.432 7.134 0.000 2.223 3.936
==============================================================================
Omnibus: 19.951 Durbin-Watson: 1.682
Prob(Omnibus): 0.000 Jarque-Bera (JB): 49.964
Skew: -0.660 Prob(JB): 1.41e-11
Kurtosis: 6.201 Cond. No. 1.32
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4、回归诊断流程
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对于多变量的时候，非常复杂，我们往往会去尝试很多模型，然后选择最优模型，常用剔除法（倒退法）
data = pd.read_csv(r"E:\regdata1.csv",sep=',',encoding='utf-8') #导入csv格式数据
display(tips.sample(3)) #随机抽取3行数据
import statsmodels.formula.api as sm #statsmodels.formula.api.ols 回归模型库
result=sm.ols(formula="y1~x1+x2+x3+x5+x6",data=data).fit()
print(result.summary()) #返回拟合总结信息
# x2 的 p:0.291 ,先去除看看
| total_bill | tip | sex | smoker | day | time | size | |
|---|---|---|---|---|---|---|---|
| 20 | 17.92 | 4.08 | Male | No | Sat | Dinner | 2 |
| 99 | 12.46 | 1.50 | Male | No | Fri | Dinner | 2 |
| 29 | 19.65 | 3.00 | Female | No | Sat | Dinner | 2 |
OLS Regression Results
==============================================================================
Dep. Variable: y1 R-squared: 0.894
Model: OLS Adj. R-squared: 0.876
Method: Least Squares F-statistic: 50.44
Date: Sun, 16 Feb 2020 Prob (F-statistic): 1.05e-13
Time: 18:07:21 Log-Likelihood: -108.08
No. Observations: 36 AIC: 228.2
Df Residuals: 30 BIC: 237.7
Df Model: 5
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 0.6359 5.862 0.108 0.914 -11.336 12.608
x1 1.4037 0.249 5.642 0.000 0.896 1.912
x2 -0.0676 0.063 -1.074 0.291 -0.196 0.061
x3 -0.2036 0.160 -1.271 0.214 -0.531 0.124
x5 -0.6946 0.255 -2.722 0.011 -1.216 -0.173
x6 4.6154 0.339 13.612 0.000 3.923 5.308
==============================================================================
Omnibus: 14.780 Durbin-Watson: 2.071
Prob(Omnibus): 0.001 Jarque-Bera (JB): 16.290
Skew: 1.306 Prob(JB): 0.000290
Kurtosis: 5.009 Cond. No. 175.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
result=sm.ols(formula="y1~x1+x3+x5+x6",data=data).fit()
print(result.summary()) #返回拟合总结信息
# x3 的 p:0.222,去除看看
OLS Regression Results
==============================================================================
Dep. Variable: y1 R-squared: 0.890
Model: OLS Adj. R-squared: 0.875
Method: Least Squares F-statistic: 62.45
Date: Sun, 16 Feb 2020 Prob (F-statistic): 2.17e-14
Time: 18:07:21 Log-Likelihood: -108.76
No. Observations: 36 AIC: 227.5
Df Residuals: 31 BIC: 235.4
Df Model: 4
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -0.5756 5.767 -0.100 0.921 -12.337 11.186
x1 1.4259 0.249 5.737 0.000 0.919 1.933
x3 -0.2004 0.161 -1.248 0.222 -0.528 0.127
x5 -0.6967 0.256 -2.723 0.011 -1.218 -0.175
x6 4.6147 0.340 13.577 0.000 3.921 5.308
==============================================================================
Omnibus: 12.513 Durbin-Watson: 2.147
Prob(Omnibus): 0.002 Jarque-Bera (JB): 12.554
Skew: 1.179 Prob(JB): 0.00188
Kurtosis: 4.675 Cond. No. 137.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
result=sm.ols(formula="y1~x1+x5+x6",data=data).fit()
print(result.summary()) #返回拟合总结信息
# Intercept 的 p:0.990,去除看看
OLS Regression Results
==============================================================================
Dep. Variable: y1 R-squared: 0.884
Model: OLS Adj. R-squared: 0.873
Method: Least Squares F-statistic: 81.34
Date: Sun, 16 Feb 2020 Prob (F-statistic): 4.65e-15
Time: 18:07:21 Log-Likelihood: -109.64
No. Observations: 36 AIC: 227.3
Df Residuals: 32 BIC: 233.6
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -0.0701 5.802 -0.012 0.990 -11.889 11.749
x1 1.4427 0.250 5.763 0.000 0.933 1.953
x5 -0.8027 0.243 -3.298 0.002 -1.298 -0.307
x6 4.5462 0.338 13.437 0.000 3.857 5.235
==============================================================================
Omnibus: 16.154 Durbin-Watson: 2.089
Prob(Omnibus): 0.000 Jarque-Bera (JB): 19.372
Skew: 1.343 Prob(JB): 6.21e-05
Kurtosis: 5.389 Cond. No. 131.
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
result=sm.ols(formula="y1~-1+x1+x5+x6",data=data).fit() #加-1表示不要截距
print(result.summary()) #返回拟合总结信息
print(result.params) # 对应的各个变量的权重
resid = result.resid # 输出残差
print(type(resid)) # <class 'pandas.core.series.Series'>
print(stats.normaltest(result.resid.values)) #对残差进行正态分布检验，p<阀值，拒绝原假设，则是正态分布
figure = plt.figure(num="画布1",figsize=(8,4),dpi=80,facecolor="gray",edgecolor="blue",frameon=True)
axes1=figure.add_subplot(2,2,1)
plt.title("残差图")
axes1.plot(result.resid) #看看残差图
plt.show()
OLS Regression Results
==============================================================================
Dep. Variable: y1 R-squared: 0.995
Model: OLS Adj. R-squared: 0.994
Method: Least Squares F-statistic: 2064.
Date: Sun, 16 Feb 2020 Prob (F-statistic): 1.33e-37
Time: 18:07:21 Log-Likelihood: -109.64
No. Observations: 36 AIC: 225.3
Df Residuals: 33 BIC: 230.0
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
x1 1.4411 0.211 6.845 0.000 1.013 1.869
x5 -0.8040 0.215 -3.737 0.001 -1.242 -0.366
x6 4.5432 0.233 19.539 0.000 4.070 5.016
==============================================================================
Omnibus: 16.185 Durbin-Watson: 2.089
Prob(Omnibus): 0.000 Jarque-Bera (JB): 19.437
Skew: 1.344 Prob(JB): 6.02e-05
Kurtosis: 5.394 Cond. No. 6.78
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
x1 1.441105
x5 -0.803997
x6 4.543234
dtype: float64
<class 'pandas.core.series.Series'>
NormaltestResult(statistic=16.185229150896554, pvalue=0.0003057892020932761)
5、避免过度拟合：
- 原则简单模型和复杂模型都通过检验时选择简单模型
- R^2增大的同时，调整后的R^2_p也应该同样增大
