statsmodel 線性回歸 (ols) 的穩健性問題 - Python

Question

我正在使用 Stats 模型測試一些基本的類別回歸：我建立了一個確定性模型

Y = X + Z

其中 X 可以取 3 個值（a、b 或 c），而 Z 只有 2 個（d 或 e）。 在那個階段，模型純粹是確定性的，我為每個變量設置權重如下

a的權重=1

b的權重=2

c的權重=3

d的權重=1

e的權重=2

因此，如果 X=a，則 1(X=a) 為 1，否則為 0，模型很簡單：

Y = 1(X=a) + 2*1(X=b) + 3*1(X=c) + 1(Z=d) + 2*1(Z=e)

使用以下代碼，生成不同的變量並運行回歸

from statsmodels.formula.api import ols
nbData = 1000
rand1 = np.random.uniform(size=nbData)
rand2 = np.random.uniform(size=nbData)
a = 1 * (rand1 <= (1.0/3.0))
b = 1 * (((1.0/3.0)< rand1) & (rand1< (4/5.0)))
c = 1-b-a
d = 1 * (rand2 <= (3.0/5.0))
e = 1-d
weigths = [1,2,3,1,2]
y = a+2*b+3*c+4*d+5*e
df = pd.DataFrame({'y':y, 'a':a, 'b':b, 'c':c, 'd':d, 'e':e})

mod = ols(formula='y ~ a + b + c + d + e - 1', data=df)
res = mod.fit()
print(res.summary())

我最終得到了權利結果（必須考慮 coef 之間的區別，而不是 coef 本身）

                           OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 1.006e+30
Date:                Wed, 16 Sep 2015   Prob (F-statistic):               0.00
Time:                        03:05:40   Log-Likelihood:                 3156.8
No. Observations:                 100   AIC:                            -6306.
Df Residuals:                      96   BIC:                            -6295.
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
a              1.6000   7.47e-16   2.14e+15      0.000         1.600     1.600
b              2.6000   6.11e-16   4.25e+15      0.000         2.600     2.600
c              3.6000   9.61e-16   3.74e+15      0.000         3.600     3.600
d              3.4000   5.21e-16   6.52e+15      0.000         3.400     3.400
e              4.4000   6.85e-16   6.42e+15      0.000         4.400     4.400
==============================================================================
Omnibus:                       11.299   Durbin-Watson:                   0.833
Prob(Omnibus):                  0.004   Jarque-Bera (JB):                5.720
Skew:                          -0.381   Prob(JB):                       0.0573
Kurtosis:                       2.110   Cond. No.                     2.46e+15
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 1.67e-29. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

但是當我將數據點的數量增加到（比如）600 時，回歸產生了非常糟糕的結果。 我在 Excel 和 R 中嘗試過類似的回歸，無論數據點的數量如何，它們都產生一致的結果。 有誰知道對 statsmodel ols 解釋這種行為是否有一些限制，還是我遺漏了什么？

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.167
Model:                            OLS   Adj. R-squared:                  0.161
Method:                 Least Squares   F-statistic:                     29.83
Date:                Wed, 16 Sep 2015   Prob (F-statistic):           1.23e-22
Time:                        03:08:04   Log-Likelihood:                -701.02
No. Observations:                 600   AIC:                             1412.
Df Residuals:                     595   BIC:                             1434.
Df Model:                           4                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
a              5.8070   1.15e+13   5.05e-13      1.000     -2.26e+13  2.26e+13
b              6.4951   1.15e+13   5.65e-13      1.000     -2.26e+13  2.26e+13
c              6.9033   1.15e+13   6.01e-13      1.000     -2.26e+13  2.26e+13
d             -1.1927   1.15e+13  -1.04e-13      1.000     -2.26e+13  2.26e+13
e             -0.1685   1.15e+13  -1.47e-14      1.000     -2.26e+13  2.26e+13
==============================================================================
Omnibus:                       67.153   Durbin-Watson:                   0.328
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               70.964
Skew:                           0.791   Prob(JB):                     3.89e-16
Kurtosis:                       2.419   Cond. No.                     7.70e+14
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 9.25e-28. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

Answer 1

似乎正如 F 先生所提到的，主要問題是 statsmodel OLS 在這種情況下似乎不能像 Excel/R 一樣處理共線性 pb，但是如果不是為每個a, b, c, d and e定義一個變量a, b, c, d and e ，定義一個變量X和一個Z可以等於a, b or c和d or e ，然后回歸工作正常。 即更新代碼：

df['X'] = ['c']*len(df)
df.X[df.b!=0] = 'b'
df.X[df.a!=0] = 'a'
df['Z'] = ['e']*len(df)
df.Z[df.d!=0] = 'd'
mod = ols(formula='y ~ X + Z - 1', data=df)

導致預期的結果

                           OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 2.684e+27
Date:                Thu, 17 Sep 2015   Prob (F-statistic):               0.00
Time:                        06:22:43   Log-Likelihood:             2.5096e+06
No. Observations:              100000   AIC:                        -5.019e+06
Df Residuals:                   99996   BIC:                        -5.019e+06
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
X[a]           5.0000   1.85e-14    2.7e+14      0.000         5.000     5.000
X[b]           6.0000   1.62e-14   3.71e+14      0.000         6.000     6.000
X[c]           7.0000   2.31e-14   3.04e+14      0.000         7.000     7.000
Z[T.e]         1.0000   1.97e-14   5.08e+13      0.000         1.000     1.000
==============================================================================
Omnibus:                      145.367   Durbin-Watson:                   1.353
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             9729.487
Skew:                          -0.094   Prob(JB):                         0.00
Kurtosis:                       1.483   Cond. No.                         2.29
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.