如何在 Python 擬合 GLM？

Generalized Linear Models in Python

Ita Cirovic Donev

Data Science Consultant

statsmodels

匯入 statsmodels
```
import statsmodels.api as sm
```
支援公式語法
```
import statsmodels.formula.api as smf
```

直接使用 glm()

from statsmodels.formula.api import glm

模型擬合流程

描述模型 $\rightarrow \texttt{\color{#007AFF}{glm()}}$
擬合模型 $\rightarrow \texttt{\color{#007AFF}{.fit()}}$
模型摘要 $\rightarrow \texttt{\color{#007AFF}{.summary()}}$
產生預測 $\rightarrow \texttt{\color{#007AFF}{.predict()}}$

描述模型

以 FORMULA 為基礎

from statsmodels.formula.api import glm

model = glm(formula, data, family)

以 ARRAY 為基礎

import statsmodels.api as sm

X = sm.add_constant(X)
model = sm.glm(y, X, family)

Formula 參數

$$\texttt{\color{#00A388}{response}} \sim \texttt{\color{#FF6138}{explanatory variable(s)}}$$ $$\texttt{\color{#00A388}{output}} \sim \texttt{\color{#FF6138}{input(s)}}$$

formula = 'y ~ x1 + x2'

$\texttt{\color{#FF6138}{C(x1)}}$ ：將 x1 視為類別變數
$\texttt{\color{#FF6138}{-1}}$ ：移除截距項
$\texttt{\color{#FF6138}{x1:x2}}$ ：x1 與 x2 的交互作用項
$\texttt{\color{#FF6138}{x1*x2}}$ ：含 x1、x2 及其交互作用
$\texttt{\color{#FF6138}{np.log(x1)}}$ ：對模型變數套用向量化函式

Family 參數

family = sm.families.____()

Family 函式：

$\texttt{\color{#007AFF}{Gaussian}(link = sm.families.links.\color{deeppink}{identity()})}$ $\rightarrow$ 預設 family
$\texttt{\color{#007AFF}{Binomial}(link = sm.families.links.\color{deeppink}{logit()})}$
- $\texttt{\color{deeppink}{probit()}}$、$\texttt{\color{deeppink}{cauchy()}}$、$\texttt{\color{deeppink}{log()}}$、$\texttt{\color{deeppink}{cloglog()}}$
$\texttt{\color{#007AFF}{Poisson}(link = sm.families.links.\color{deeppink}{log()})}$
- $\texttt{\color{deeppink}{identity()}}$、$\texttt{\color{deeppink}{sqrt()}}$

更多分配 family 請參見 statsmodels 網站。

模型摘要

print(model_GLM.summary())

                 Generalized Linear Model Regression Results                  
=============================================================================
Dep. Variable:                      y  No. Observations:                  173
Model:                            GLM  Df Residuals:                      171
Model Family:                Binomial  Df Model:                            1
Link Function:                  logit  Scale:                          1.0000
Method:                          IRLS  Log-Likelihood:                -97.226
Date:                Mon, 21 Jan 2019  Deviance:                       194.45
Time:                        11:30:01  Pearson chi2:                     165.
No. Iterations:                     4  Covariance Type:             nonrobust
=============================================================================
                 coef    std err         z      P>|z|      [0.025      0.975]
-----------------------------------------------------------------------------
Intercept    -12.3508      2.629    -4.698      0.000     -17.503      -7.199
width          0.4972      0.102     4.887      0.000       0.298       0.697
=============================================================================

迴歸係數

$\texttt{\color{#007AFF}{.params}}$ 輸出迴歸係數

model_GLM.params

Intercept   -12.350818
width         0.497231
dtype: float64

$\texttt{\color{#007AFF}{.conf\_int(alpha=0.05, cols=None)}}$ 輸出信賴區間

model_GLM.conf_int()

                   0         1
Intercept -17.503010 -7.198625
width       0.297833  0.696629

預測

在測試資料中指定所有模型變數
$\texttt{\color{#007AFF}{.predict(\color{#FF931B}{test\_data})}}$ 計算預測值

model_GLM.predict(test_data)

0    0.029309
1    0.470299
2    0.834983
3    0.972363
4    0.987941

一起來練習吧！

Generalized Linear Models in Python