广义可加模型 (GAM)¶
广义可加模型允许对广义线性模型中的平滑项进行惩罚估计。
参见 模块参考 以了解命令和参数。
示例¶
以下示例展示了高斯和泊松回归，其中分类变量被视为线性项，而两个解释变量的影响由惩罚 B 样条曲线捕获。数据来自汽车数据集 https://archive.ics.uci.edu/ml/datasets/automobile 我们可以在单元测试模块中加载一个包含选定列的 DataFrame。
In [1]: import statsmodels.api as sm
In [2]: from statsmodels.gam.api import GLMGam, BSplines
# import data
In [3]: from statsmodels.gam.tests.test_penalized import df_autos
# create spline basis for weight and hp
In [4]: x_spline = df_autos[['weight', 'hp']]
In [5]: bs = BSplines(x_spline, df=[12, 10], degree=[3, 3])
# penalization weight
In [6]: alpha = np.array([21833888.8, 6460.38479])
In [7]: gam_bs = GLMGam.from_formula('city_mpg ~ fuel + drive', data=df_autos,
...: smoother=bs, alpha=alpha)
...:
In [8]: res_bs = gam_bs.fit()
In [9]: print(res_bs.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable: city_mpg No. Observations: 203
Model: GLMGam Df Residuals: 189.13
Model Family: Gaussian Df Model: 12.87
Link Function: Identity Scale: 4.8825
Method: PIRLS Log-Likelihood: -441.81
Date: Thu, 03 Oct 2024 Deviance: 923.45
Time: 16:09:46 Pearson chi2: 923.
No. Iterations: 3 Pseudo R-squ. (CS): 0.9996
Covariance Type: nonrobust
================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------
Intercept 51.9923 1.997 26.034 0.000 48.078 55.906
fuel[T.gas] -5.8099 0.727 -7.989 0.000 -7.235 -4.385
drive[T.fwd] 1.3910 0.819 1.699 0.089 -0.213 2.995
drive[T.rwd] 1.0638 0.842 1.263 0.207 -0.587 2.715
weight_s0 -3.5556 0.959 -3.707 0.000 -5.436 -1.676
weight_s1 -9.0876 1.750 -5.193 0.000 -12.518 -5.658
weight_s2 -13.0303 1.827 -7.132 0.000 -16.611 -9.450
weight_s3 -14.2641 1.854 -7.695 0.000 -17.897 -10.631
weight_s4 -15.1805 1.892 -8.024 0.000 -18.889 -11.472
weight_s5 -15.9557 1.963 -8.128 0.000 -19.803 -12.108
weight_s6 -16.6297 2.038 -8.161 0.000 -20.624 -12.636
weight_s7 -16.9928 2.045 -8.308 0.000 -21.002 -12.984
weight_s8 -19.3480 2.367 -8.174 0.000 -23.987 -14.709
weight_s9 -20.7978 2.455 -8.472 0.000 -25.609 -15.986
weight_s10 -20.8062 2.443 -8.517 0.000 -25.594 -16.018
hp_s0 -1.4473 0.558 -2.592 0.010 -2.542 -0.353
hp_s1 -3.4228 1.012 -3.381 0.001 -5.407 -1.438
hp_s2 -5.9026 1.251 -4.717 0.000 -8.355 -3.450
hp_s3 -7.2389 1.352 -5.354 0.000 -9.889 -4.589
hp_s4 -9.1052 1.384 -6.581 0.000 -11.817 -6.393
hp_s5 -9.9865 1.525 -6.547 0.000 -12.976 -6.997
hp_s6 -13.3639 2.228 -5.998 0.000 -17.731 -8.997
hp_s7 -13.8902 3.194 -4.349 0.000 -20.150 -7.630
hp_s8 -11.9752 2.556 -4.685 0.000 -16.985 -6.965
================================================================================
# plot smooth components
In [10]: res_bs.plot_partial(0, cpr=True)
Out[10]: <Figure size 640x480 with 1 Axes>
In [11]: res_bs.plot_partial(1, cpr=True)
Out[11]: <Figure size 640x480 with 1 Axes>
In [12]: alpha = np.array([8283989284.5829611, 14628207.58927821])
In [13]: gam_bs = GLMGam.from_formula('city_mpg ~ fuel + drive', data=df_autos,
....: smoother=bs, alpha=alpha,
....: family=sm.families.Poisson())
....:
In [14]: res_bs = gam_bs.fit()
In [15]: print(res_bs.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable: city_mpg No. Observations: 203
Model: GLMGam Df Residuals: 194.75
Model Family: Poisson Df Model: 7.25
Link Function: Log Scale: 1.0000
Method: PIRLS Log-Likelihood: -530.38
Date: Thu, 03 Oct 2024 Deviance: 37.569
Time: 16:09:46 Pearson chi2: 37.4
No. Iterations: 6 Pseudo R-squ. (CS): 0.7715
Covariance Type: nonrobust
================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------
Intercept 3.9960 0.130 30.844 0.000 3.742 4.250
fuel[T.gas] -0.2398 0.057 -4.222 0.000 -0.351 -0.128
drive[T.fwd] 0.0386 0.075 0.513 0.608 -0.109 0.186
drive[T.rwd] 0.0309 0.078 0.395 0.693 -0.122 0.184
weight_s0 -0.0811 0.030 -2.689 0.007 -0.140 -0.022
weight_s1 -0.1938 0.063 -3.067 0.002 -0.318 -0.070
weight_s2 -0.3160 0.082 -3.864 0.000 -0.476 -0.156
weight_s3 -0.3735 0.090 -4.160 0.000 -0.549 -0.198
weight_s4 -0.4187 0.096 -4.360 0.000 -0.607 -0.230
weight_s5 -0.4645 0.103 -4.495 0.000 -0.667 -0.262
weight_s6 -0.5092 0.112 -4.555 0.000 -0.728 -0.290
weight_s7 -0.5469 0.119 -4.598 0.000 -0.780 -0.314
weight_s8 -0.6211 0.137 -4.528 0.000 -0.890 -0.352
weight_s9 -0.6866 0.153 -4.486 0.000 -0.987 -0.387
weight_s10 -0.7370 0.174 -4.228 0.000 -1.079 -0.395
hp_s0 -0.0247 0.010 -2.378 0.017 -0.045 -0.004
hp_s1 -0.0557 0.022 -2.479 0.013 -0.100 -0.012
hp_s2 -0.1046 0.038 -2.719 0.007 -0.180 -0.029
hp_s3 -0.1438 0.050 -2.857 0.004 -0.242 -0.045
hp_s4 -0.1919 0.063 -3.047 0.002 -0.315 -0.068
hp_s5 -0.2567 0.079 -3.231 0.001 -0.412 -0.101
hp_s6 -0.4152 0.120 -3.455 0.001 -0.651 -0.180
hp_s7 -0.4889 0.152 -3.214 0.001 -0.787 -0.191
hp_s8 -0.5470 0.195 -2.810 0.005 -0.928 -0.166
================================================================================
# Optimal penalization weights alpha can be obtained through generalized
# cross-validation or k-fold cross-validation.
# The alpha above are from the unit tests against the R mgcv package.
In [16]: gam_bs.select_penweight()[0]
Out[16]: array([8.2839e+09, 1.4628e+07])
In [17]: gam_bs.select_penweight_kfold()[0]
Out[17]: (np.float64(10000000.0), np.float64(15848.931924611108))
参考文献¶
Hastie, Trevor, and Robert Tibshirani. 1986. Generalized Additive Models. Statistical Science 1 (3): 297-310.
Wood, Simon N. 2006. Generalized Additive Models: An Introduction with R. Texts in Statistical Science. Boca Raton, FL: Chapman & Hall/CRC.
Wood, Simon N. 2017. Generalized Additive Models: An Introduction with R. Second edition. Chapman & Hall/CRC Texts in Statistical Science. Boca Raton: CRC Press/Taylor & Francis Group.
模块参考¶
模型类¶
|
广义可加模型 (GAM) |
|
离散 Logit 的广义可加模型 |
结果类¶
|
广义可加模型 (GAM) 的结果类。 |
平滑基函数¶
目前已验证对两种样条基的支持
|
使用 B 样条的加性平滑分量 |
|
使用循环三次回归样条的加性平滑分量 |
statsmodels.gam.smooth_basis 包含其他样条和 (全局) 多项式平滑基，但尚未进行验证。
族和连接函数¶
GLMGam 中的分布族与 GLM 相同，相应的连接函数也相同。当前的单元测试只涵盖了高斯和泊松，GLMGam 可能会无法在 GLM 中支持所有可用的选项。
最后更新时间：2024 年 10 月 3 日