重回帰分析(Multiple Regression Analysis)

In [1]:

    

%matplotlib inline
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings('ignore')


    

In [2]:

    

# データ読み込み
data = pd.read_csv('data/tab26.csv')
data.head()


    

    
Out[2]:






  

    

      

      
y
      
x1
      
x2
      
x3
      
x4
    
  
  

    

      
0
      
2.9
      
2
      
0
      
0
      
1
    
    

      
1
      
10.5
      
8
      
1
      
0
      
3
    
    

      
2
      
7.5
      
6
      
1
      
1
      
2
    
    

      
3
      
6.5
      
4
      
1
      
0
      
1
    
    

      
4
      
6.0
      
3
      
1
      
1
      
1
    
  

In [3]:

    

# 説明変数設定
X = data[['x1', 'x2', 'x3', 'x4']]
X = sm.add_constant(X)
# 非説明変数設定
Y = data['y']


    

In [4]:

    

# OLSの実行
model1 = sm.OLS(Y,X)
results1 = model1.fit()
results1.summary()


    

    
Out[4]:





OLS Regression Results

  
Dep. Variable:
            
y
        
  R-squared:         
 
   0.897


  
Model:
                   
OLS
       
  Adj. R-squared:    
 
   0.814


  
Method:
             
Least Squares
  
  F-statistic:       
 
   10.86


  
Date:
             
Wed, 23 Dec 2015
 
  Prob (F-statistic):
  
0.0111
 


  
Time:
                 
00:28:14
     
  Log-Likelihood:    
 
 -9.7935


  
No. Observations:
      
    10
      
  AIC:               
 
   29.59


  
Df Residuals:
          
     5
      
  BIC:               
 
   31.10


  
Df Model:
              
     4
      
                     
     
 
   


  
Covariance Type:
      
nonrobust
    
                     
     
 
   




    
       
coef
     
std err
      
t
      
P>|t|
 
[95.0% Conf. Int.]
 


  
const
 
    2.2947
 
    0.904
 
    2.539
 
 0.052
 
   -0.029     4.618


  
x1
    
    0.5771
 
    0.395
 
    1.463
 
 0.203
 
   -0.437     1.591


  
x2
    
    1.4843
 
    0.723
 
    2.053
 
 0.095
 
   -0.374     3.343


  
x3
    
   -0.8356
 
    0.707
 
   -1.182
 
 0.290
 
   -2.653     0.981


  
x4
    
    0.5600
 
    1.151
 
    0.487
 
 0.647
 
   -2.398     3.518




  
Omnibus:
       
 0.141
 
  Durbin-Watson:     
 
   2.294


  
Prob(Omnibus):
 
 0.932
 
  Jarque-Bera (JB):  
 
   0.156


  
Skew:
          
 0.162
 
  Prob(JB):          
 
   0.925


  
Kurtosis:
      
 2.483
 
  Cond. No.          
 
    28.8

In [5]:

    

# 説明変数設定
X = data[['x1', 'x2', 'x3']]
X = sm.add_constant(X)
# OLSの実行
model2 = sm.OLS(Y,X)
results2 = model2.fit()
results2.summary()


    

    
Out[5]:





OLS Regression Results

  
Dep. Variable:
            
y
        
  R-squared:         
 
   0.892


  
Model:
                   
OLS
       
  Adj. R-squared:    
 
   0.838


  
Method:
             
Least Squares
  
  F-statistic:       
 
   16.51


  
Date:
             
Wed, 23 Dec 2015
 
  Prob (F-statistic):
  
0.00265


  
Time:
                 
00:28:14
     
  Log-Likelihood:    
 
 -10.025


  
No. Observations:
      
    10
      
  AIC:               
 
   28.05


  
Df Residuals:
          
     6
      
  BIC:               
 
   29.26


  
Df Model:
              
     3
      
                     
     
 
   


  
Covariance Type:
      
nonrobust
    
                     
     
 
   




    
       
coef
     
std err
      
t
      
P>|t|
 
[95.0% Conf. Int.]
 


  
const
 
    2.5003
 
    0.746
 
    3.350
 
 0.015
 
    0.674     4.327


  
x1
    
    0.7599
 
    0.112
 
    6.765
 
 0.001
 
    0.485     1.035


  
x2
    
    1.3900
 
    0.651
 
    2.136
 
 0.077
 
   -0.202     2.982


  
x3
    
   -0.8633
 
    0.658
 
   -1.311
 
 0.238
 
   -2.474     0.747




  
Omnibus:
       
 0.107
 
  Durbin-Watson:     
 
   2.260


  
Prob(Omnibus):
 
 0.948
 
  Jarque-Bera (JB):  
 
   0.300


  
Skew:
          
 0.158
 
  Prob(JB):          
 
   0.861


  
Kurtosis:
      
 2.213
 
  Cond. No.          
 
    19.5

In [6]:

    

# 説明変数設定
X = data[['x1', 'x2']]
X = sm.add_constant(X)
# OLSの実行
model3 = sm.OLS(Y,X)
results3 = model3.fit()
results3.summary()


    

    
Out[6]:





OLS Regression Results

  
Dep. Variable:
            
y
        
  R-squared:         
 
   0.861


  
Model:
                   
OLS
       
  Adj. R-squared:    
 
   0.821


  
Method:
             
Least Squares
  
  F-statistic:       
 
   21.67


  
Date:
             
Wed, 23 Dec 2015
 
  Prob (F-statistic):
  
0.00100


  
Time:
                 
00:28:14
     
  Log-Likelihood:    
 
 -11.285


  
No. Observations:
      
    10
      
  AIC:               
 
   28.57


  
Df Residuals:
          
     7
      
  BIC:               
 
   29.48


  
Df Model:
              
     2
      
                     
     
 
   


  
Covariance Type:
      
nonrobust
    
                     
     
 
   




    
       
coef
     
std err
      
t
      
P>|t|
 
[95.0% Conf. Int.]
 


  
const
 
    2.6155
 
    0.778
 
    3.360
 
 0.012
 
    0.775     4.456


  
x1
    
    0.7369
 
    0.117
 
    6.324
 
 0.000
 
    0.461     1.012


  
x2
    
    1.0233
 
    0.617
 
    1.658
 
 0.141
 
   -0.436     2.483




  
Omnibus:
       
 0.240
 
  Durbin-Watson:     
 
   2.132


  
Prob(Omnibus):
 
 0.887
 
  Jarque-Bera (JB):  
 
   0.396


  
Skew:
          
-0.021
 
  Prob(JB):          
 
   0.820


  
Kurtosis:
      
 2.026
 
  Cond. No.          
 
    18.0

In [7]:

    

# 説明変数設定
X = data[['x1']]
X = sm.add_constant(X)
# OLSの実行
model4 = sm.OLS(Y,X)
results4 = model4.fit()
results4.summary()


    

    
Out[7]:





OLS Regression Results

  
Dep. Variable:
            
y
        
  R-squared:         
 
   0.806


  
Model:
                   
OLS
       
  Adj. R-squared:    
 
   0.782


  
Method:
             
Least Squares
  
  F-statistic:       
 
   33.31


  
Date:
             
Wed, 23 Dec 2015
 
  Prob (F-statistic):
 
0.000419


  
Time:
                 
00:28:14
     
  Log-Likelihood:    
 
 -12.942


  
No. Observations:
      
    10
      
  AIC:               
 
   29.88


  
Df Residuals:
          
     8
      
  BIC:               
 
   30.49


  
Df Model:
              
     1
      
                     
     
 
   


  
Covariance Type:
      
nonrobust
    
                     
     
 
   




    
       
coef
     
std err
      
t
      
P>|t|
 
[95.0% Conf. Int.]
 


  
const
 
    3.3053
 
    0.726
 
    4.551
 
 0.002
 
    1.630     4.980


  
x1
    
    0.7421
 
    0.129
 
    5.771
 
 0.000
 
    0.446     1.039




  
Omnibus:
       
 1.684
 
  Durbin-Watson:     
 
   2.321


  
Prob(Omnibus):
 
 0.431
 
  Jarque-Bera (JB):  
 
   0.731


  
Skew:
          
-0.652
 
  Prob(JB):          
 
   0.694


  
Kurtosis:
      
 2.766
 
  Cond. No.          
 
    13.5

In [8]:

    

# モデル選択
criteria = pd.DataFrame(index=['results1', 'results2', 'results3', 'results4'])
criteria["AIC"] = [results1.aic, results2.aic, results3.aic, results4.aic]
criteria["BIC"] = [results1.bic, results2.bic, results3.bic, results4.bic]
criteria


    

    
Out[8]:






  

    

      

      
AIC
      
BIC
    
  
  

    

      
results1
      
29.587070
      
31.099996
    
    

      
results2
      
28.049772
      
29.260113
    
    

      
results3
      
28.570178
      
29.477933
    
    

      
results4
      
29.883355
      
30.488525