GLM - Generalized Linear Models¶
Interpretable baseline models for distributional regression. Supports multiple distribution families with both statistical and neural network training approaches.
Class Definition¶
Overview¶
The GLM class provides PyTorch implementations of Generalized Linear Models with distributional outputs. Key features:
- Multiple Distribution Families - Gaussian, Gamma, Inverse Gaussian, Log-Normal
- Dual Training Modes - Statistical (statsmodels) or neural (PyTorch gradient descent)
- Interpretable Parameters - Access to coefficients, intercepts, and dispersion
- Seamless Integration - Perfect baseline for DRN refinement
- Automatic Parameter Estimation - Maximum likelihood via statsmodels
Supported Distributions¶
Gaussian Distribution¶
- Use Case: Continuous data with constant variance - Link Function: Identity (μ = Xβ)
- Parameters: Mean (μ), standard deviation (σ)
- Best For: Symmetric, unbounded data (temperatures, stock returns)
Gamma Distribution¶
- Use Case: Positive continuous data with right skew - Link Function: Log (log(μ) = Xβ)
- Parameters: Shape (α), scale (β)
- Best For: Insurance claims, waiting times, sales amounts
Inverse Gaussian Distribution¶
- Use Case: Positive data with extreme right tail - Link Function: Log (log(μ) = Xβ)
- Parameters: Mean (μ), dispersion (λ)
- Best For: First passage times, service durations
Log-Normal Distribution¶
- Use Case: Positive data from multiplicative processes - Link Function: Identity on log-scale - Parameters: Log-mean (μ), log-std (σ)- Best For: Income distributions, growth rates, file sizes
Quick Start¶
Basic Usage¶
from drn import GLM
import pandas as pd
# Load data
X = pd.DataFrame({'age': [25, 35, 45], 'income': [30000, 50000, 70000]})
y = pd.Series([1200, 1800, 2400])
# Fit GLM using statsmodels approach
glm = GLM('gamma')
glm.fit(X, y)
# Make predictions
predictions = glm.predict(X_test)
mean_pred = predictions.mean
quantiles = predictions.quantiles([10, 50, 90])
Alternative: Gradient Descent Training¶
# Neural network style training
glm = GLM('gaussian', learning_rate=0.001)
glm.fit(
X_train, y_train,
X_val, y_val,
grad_descent=True,
epochs=1000,
batch_size=128,
patience=50
)
Detailed Examples¶
Insurance Claims Modeling¶
import pandas as pd
import numpy as np
from drn import GLM
from drn.metrics import rmse, crps
# Load insurance data
claims_data = pd.read_csv('insurance_claims.csv')
X = claims_data[['age', 'vehicle_age', 'region', 'policy_type']]
y = claims_data['claim_amount']
# Split data
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.2, random_state=42
)
# Train Gamma GLM (ideal for insurance claims)
glm_gamma = GLM('gamma')
glm_gamma.fit(X_train, y_train)
# Evaluate predictions
pred_dist = glm_gamma.predict(X_test)
# Point prediction accuracy
rmse_score = rmse(y_test, pred_dist.mean)
print(f"RMSE: ${rmse_score:.2f}")
# Distributional accuracy
grid = np.linspace(0, y_test.max() * 1.2, 1000)
crps_score = crps(y_test, grid, pred_dist.cdf(grid)).mean()
print(f"CRPS: ${crps_score:.2f}")
# Risk measures
percentiles = [50, 75, 90, 95, 99]
risk_measures = glm_gamma.quantiles(X_test, percentiles)
print("Risk Measures:")
for i, p in enumerate(percentiles):
avg_risk = risk_measures[:, i].mean()
print(f"{p}th percentile: ${avg_risk:.2f}")
Parameter Access and Interpretation¶
Extracting Fitted Parameters¶
# After fitting
glm = GLM('gamma')
glm.fit(X_train, y_train)
# Access parameters
print("Model Parameters:")
print(f"Coefficients: {glm.linear.weight.data}")
print(f"Intercept: {glm.linear.bias.data}")
print(f"Dispersion: {glm.dispersion.data}")
# Interpret coefficients (for Gamma GLM with log link)
feature_names = X_train.columns
for i, name in enumerate(feature_names):
coef = glm.linear.weight[0, i].item()
effect = np.exp(coef) # Multiplicative effect
print(f"{name}: {coef:.4f} (multiplicative effect: {effect:.4f})")
Statsmodels equivalence¶
Under the hood, GLM uses statsmodels for statistical fitting.
You can make the equivalent GLM directly to access detailed statistical outputs.
import statsmodels.api as sm
from statsmodels.genmod.families import Gamma
import numpy as np
# For detailed statistical analysis, access underlying statsmodels results
X_np = X_train.values
y_np = y_train.values
# Fit using statsmodels directly for statistical inference
X_sm = sm.add_constant(X_np)
sm_model = sm.GLM(y_np, X_sm, family=Gamma(link=sm.families.links.Log()))
sm_results = sm_model.fit()
print(sm_results.summary())
print(f"AIC: {sm_results.aic:.2f}")
print(f"BIC: {sm_results.bic:.2f}")