H2O Python 因子列的 relevel 与 relevel_by_frequency

Question

Based on H2O's documentation it would seem as though relevel('most_frequency_category') and relevel_by_frequency() should accomplish the same thing. However the coefficient estimates are different depending on which method is used to set the reference level for a factor column.

Using an open source dataset from sklearn demonstrates how the GLM coefficients are misaligned when the base level is set using the two releveling methods. Why do the coefficient estimates vary when the base level is the same between the two models?

import pandas as pd
from sklearn.datasets import fetch_openml

import h2o
from h2o.estimators.glm import H2OGeneralizedLinearEstimator

h2o.init(max_mem_size=8)


def load_mtpl2(n_samples=100000):
    """
    Fetch the French Motor Third-Party Liability Claims dataset.
    https://scikit-learn.org/stable/auto_examples/linear_model/plot_tweedie_regression_insurance_claims.html
    
    Parameters
    ----------
    n_samples: int, default=100000
      number of samples to select (for faster run time). Full dataset has
      678013 samples.
    """
    # freMTPL2freq dataset from https://www.openml.org/d/41214
    df_freq = fetch_openml(data_id=41214, as_frame=True)["data"]
    df_freq["IDpol"] = df_freq["IDpol"].astype(int)
    df_freq.set_index("IDpol", inplace=True)

    # freMTPL2sev dataset from https://www.openml.org/d/41215
    df_sev = fetch_openml(data_id=41215, as_frame=True)["data"]

    # sum ClaimAmount over identical IDs
    df_sev = df_sev.groupby("IDpol").sum()

    df = df_freq.join(df_sev, how="left")
    df["ClaimAmount"].fillna(0, inplace=True)

    # unquote string fields
    for column_name in df.columns[df.dtypes.values == object]:
        df[column_name] = df[column_name].str.strip("'")
    return df.iloc[:n_samples]


df = load_mtpl2()
df.loc[(df["ClaimAmount"] == 0) & (df["ClaimNb"] >= 1), "ClaimNb"] = 0
df["Exposure"] = df["Exposure"].clip(upper=1)
df["ClaimAmount"] = df["ClaimAmount"].clip(upper=100000)
df["PurePremium"] = df["ClaimAmount"] / df["Exposure"]

X_freq = h2o.H2OFrame(df)
X_freq["VehBrand"] = X_freq["VehBrand"].asfactor()
X_freq["VehBrand"] = X_freq["VehBrand"].relevel_by_frequency()

X_relevel = h2o.H2OFrame(df)
X_relevel["VehBrand"] = X_relevel["VehBrand"].asfactor()
X_relevel["VehBrand"] = X_relevel["VehBrand"].relevel("B1") # most frequent category

response_col = "PurePremium"
weight_col = "Exposure"
predictors = "VehBrand"

glm_freq = H2OGeneralizedLinearEstimator(family="tweedie",
                                      solver='IRLSM',
                                      tweedie_variance_power=1.5,
                                      tweedie_link_power=0,
                                      lambda_=0,
                                      compute_p_values=True,
                                      remove_collinear_columns=True,
                                      seed=1)

glm_relevel = H2OGeneralizedLinearEstimator(family="tweedie",
                                      solver='IRLSM',
                                      tweedie_variance_power=1.5,
                                      tweedie_link_power=0,
                                      lambda_=0,
                                      compute_p_values=True,
                                      remove_collinear_columns=True,
                                      seed=1)

glm_freq.train(x=predictors, y=response_col, training_frame=X_freq, weights_column=weight_col)
glm_relevel.train(x=predictors, y=response_col, training_frame=X_relevel, weights_column=weight_col)

print('GLM with the reference level set using relevel_by_frequency()')
print(glm_freq._model_json['output']['coefficients_table'])
print('\n')
print('GLM with the reference level manually set using relevel()')
print(glm_relevel._model_json['output']['coefficients_table'])

Output

GLM with the reference level set using relevel_by_frequency()
Coefficients: glm coefficients
names         coefficients    std_error    z_value     p_value      standardized_coefficients
------------  --------------  -----------  ----------  -----------  ---------------------------
Intercept     5.40413         1.24082      4.35531     1.33012e-05  5.40413
VehBrand.B2   -0.398721       1.2599       -0.316472   0.751645     -0.398721
VehBrand.B12  -0.061573       1.46541      -0.0420176  0.966485     -0.061573
VehBrand.B3   -0.393908       1.30712      -0.301356   0.763144     -0.393908
VehBrand.B5   -0.282484       1.31929      -0.214118   0.830455     -0.282484
VehBrand.B6   -0.387747       1.25943      -0.307876   0.758177     -0.387747
VehBrand.B4   0.391771        1.45615      0.269047    0.787894     0.391771
VehBrand.B10  -0.0542706      1.35049      -0.040186   0.967945     -0.0542706
VehBrand.B13  -0.306381       1.4628       -0.209449   0.834098     -0.306381
VehBrand.B11  -0.435297       1.29155      -0.337035   0.736091     -0.435297
VehBrand.B14  -0.304243       1.34781      -0.225732   0.821411     -0.304243


GLM with the reference level manually set using relevel()
Coefficients: glm coefficients
names         coefficients    std_error    z_value     p_value     standardized_coefficients
------------  --------------  -----------  ----------  ----------  ---------------------------
Intercept     5.01639         0.215713     23.2549     2.635e-119  5.01639
VehBrand.B10  0.081366        0.804165     0.101181    0.919407    0.081366
VehBrand.B11  0.779518        0.792003     0.984237    0.325001    0.779518
VehBrand.B12  -0.0475497      0.41834      -0.113663   0.909505    -0.0475497
VehBrand.B13  0.326174        0.80891      0.403227    0.686782    0.326174
VehBrand.B14  0.387747        1.25943      0.307876    0.758177    0.387747
VehBrand.B2   -0.010974       0.306996     -0.0357465  0.971485    -0.010974
VehBrand.B3   -0.00616108     0.464188     -0.0132728  0.98941     -0.00616108
VehBrand.B4   0.333477        0.575082     0.579877    0.561999    0.333477
VehBrand.B5   0.105263        0.497431     0.211613    0.832409    0.105263
VehBrand.B6   0.0835042       0.568769     0.146816    0.883278    0.0835042

Answer 1

The two datasets are almost the same except at one place:

In the first dataset, number of rows for VehBrand with B1 = 72 In the second dataset, number of rows for VehBrand with B14 = 721.

If you look and compare the two datasets, you can map the equivalent names to the number of rows in the two dataset as follows:

Freq B2 == Relevel B2 with 26500 rows

Freq B12 == Relevel B13 with 1883 rows

Freq B3 == Relevel B3 with 8260 rows

Freq B5 == Relevel B5 with 6053 rows

Freq B6 == Relevel B1 with 27240 rows

Freq B4 == Relevel B11 with 1774 rows

Freq B10 == Relevel B4 with 3968 rows

Freq B13 == Relevel B10 with 2268 rows

Freq B11 == Relevel B12 with 16619 rows

Freq B14 == Relevel B6 with 4714 rows.

Since you are training the two GLM models with different datasets, you will get different coefficients and different prediction results.