如何在 Python 中将总和为零约束添加到 GLM？

Question

我使用 statsmodel glm 函数在 Python 中设置了一个模型，但现在我想在模型中添加一个总和为零的约束。

模型定义如下：

import statsmodels.formula.api as smf
model = smf.glm(formula="A ~ B + C + D", data=data, family=sm.families.Poisson()).fit()

在 R 中，要添加约束，我只需执行以下操作：

model <- glm(A ~ B + C + D –1, family=poisson(), data=data, contrasts=list(C="contr.sum", D="contr.sum"))

这将总和为零约束添加到 C 和 D，但我不确定如何在 Python 中实现相同的目标。

我已经看到有一个fit_constraint() 方法可用，但我不太确定如何使用它，或者它是否适合用来实现我的要求。

http://statsmodels.sourceforge.net/devel/generated/statsmodels.genmod.generalized_linear_model.GLM.fit_constrained.html#statsmodels.genmod.generalized_linear_model.GLM.fit_constrained

任何人都可以提供任何建议来应用此约束吗？

Answer 1

这里是一个例子来说明fit_constrained，使用高斯族，因为我没有很快找到一个带有分类变量的泊松例子

import pandas
import statsmodels.api as sm
from statsmodels.formula.api import glm

url = 'http://www.ats.ucla.edu/stat/data/hsb2.csv'
hsb2 = pandas.read_table(url, delimiter=",")

mod = glm("write ~ C(race) - 1", data=hsb2)
res = mod.fit()
print(res.summary())

所有系数相加为零的约束

res_c = mod.fit_constrained('C(race)[1] + C(race)[2] + C(race)[3] + C(race)[4] = 0')
print(res_c.summary())

                 Generalized Linear Model Regression Results                  
==============================================================================
Dep. Variable:                  write   No. Observations:                  200
Model:                            GLM   Df Residuals:                      197
Model Family:                Gaussian   Df Model:                            2
Link Function:               identity   Scale:                   1232.08314649
Method:                          IRLS   Log-Likelihood:                -993.41
Date:                Wed, 25 Mar 2015   Deviance:                   2.4149e+05
Time:                        16:42:37   Pearson chi2:                 2.41e+05
No. Iterations:                     1                                         
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
C(race)[1]     1.0002    221.565      0.005      0.996      -433.260   435.260
C(race)[2]   -41.1814    267.253     -0.154      0.878      -564.988   482.626
C(race)[3]    -6.3498    235.771     -0.027      0.979      -468.453   455.754
C(race)[4]    46.5311    100.184      0.464      0.642      -149.827   242.889
==============================================================================

Model has been estimated subject to linear equality constraints.

约束以逗号分隔，默认为零：

res_c2 = mod.fit_constrained('C(race)[1] + C(race)[2], C(race)[3] + C(race)[4]')
print(res_c2.summary())

最后一张照片

                 Generalized Linear Model Regression Results                  
==============================================================================
Dep. Variable:                  write   No. Observations:                  200
Model:                            GLM   Df Residuals:                      198
Model Family:                Gaussian   Df Model:                            1
Link Function:               identity   Scale:                   1438.99574167
Method:                          IRLS   Log-Likelihood:                -1008.9
Date:                Wed, 25 Mar 2015   Deviance:                   2.8204e+05
Time:                        16:42:37   Pearson chi2:                 2.82e+05
No. Iterations:                     1                                         
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
C(race)[1]    13.6286    242.003      0.056      0.955      -460.689   487.946
C(race)[2]   -13.6286    242.003     -0.056      0.955      -487.946   460.689
C(race)[3]   -41.6606    111.458     -0.374      0.709      -260.115   176.794
C(race)[4]    41.6606    111.458      0.374      0.709      -176.794   260.115
==============================================================================

Model has been estimated subject to linear equality constraints.

我不确定 patsy 公式是如何工作的，因此如果有多个分类解释变量，则不会删除任何级别。