Regularized Models

Penalty-based regression for multicollinearity and high-dimensional data: Ridge and Elastic Net.

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Ridge - L2 Regularization

Handle multicollinearity by shrinking coefficients toward zero.

Variants

• Scalar Fit: anofox_stats_ridge_fit(y, x, alpha, [options]) -> STRUCT
• Aggregate Fit: anofox_stats_ridge_fit_agg(y, x, alpha, [options]) -> STRUCT
• Window Predict: anofox_stats_ridge_fit_predict(y, x, [options]) OVER (...) -> STRUCT
• Batch Predict: anofox_stats_ridge_predict_agg(y, x, [options]) -> LIST(STRUCT)

Parameters

Parameter	Type	Required	Default	Description
y	LIST(DOUBLE) / DOUBLE	Yes	-	Target values
x	LIST(LIST(DOUBLE)) / LIST(DOUBLE)	Yes	-	Predictor matrix
alpha	DOUBLE	Yes	-	Regularization strength (0.01-10.0)
options	MAP	No	-	Configuration options

Options MAP:

Option	Type	Default	Description
fit_intercept	BOOLEAN	true	Include intercept term
compute_inference	BOOLEAN	false	Compute inference statistics
confidence_level	DOUBLE	0.95	Confidence level

Example

SELECT
    region,
    (model).r_squared as fit,
    (model).coefficients[2] as price_elasticity
FROM (
    SELECT
        region,
        anofox_stats_ridge_fit_agg(
            sales,
            [price, promotion],
            0.5,
            MAP {'compute_inference': 'true'}
        ) as model
    FROM regional_sales
    GROUP BY region
);

When to use Ridge:

• VIF > 5 for any predictor
• More predictors than observations
• Coefficients unstable across samples

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Elastic Net - Combined L1+L2

Feature selection with regularization for high-dimensional data.

Variants

• Scalar Fit: anofox_stats_elasticnet_fit(y, x, alpha, l1_ratio, [options]) -> STRUCT
• Aggregate Fit: anofox_stats_elasticnet_fit_agg(y, x, alpha, l1_ratio, [options]) -> STRUCT
• Window Predict: anofox_stats_elasticnet_fit_predict(y, x, [options]) OVER (...) -> STRUCT
• Batch Predict: anofox_stats_elasticnet_predict_agg(y, x, [options]) -> LIST(STRUCT)

Parameters

Parameter	Type	Required	Default	Description
y	LIST(DOUBLE) / DOUBLE	Yes	-	Target values
x	LIST(LIST(DOUBLE)) / LIST(DOUBLE)	Yes	-	Predictor matrix
alpha	DOUBLE	Yes	-	Overall regularization (0.01-10.0)
l1_ratio	DOUBLE	Yes	-	L1/L2 balance (0=Ridge, 1=Lasso)
options	MAP	No	-	Configuration options

Options MAP:

Option	Type	Default	Description
fit_intercept	BOOLEAN	true	Include intercept term
max_iterations	INTEGER	1000	Convergence limit
tolerance	DOUBLE	1e-4	Convergence threshold

Example

SELECT anofox_stats_elasticnet_fit_agg(
    y,
    [x1, x2, x3, x4, x5],
    0.5,   -- alpha: regularization strength
    0.7    -- l1_ratio: 70% L1, 30% L2
) as model
FROM high_dim_data;

When to use Elastic Net:

• High-dimensional data (many predictors)
• Feature selection needed (sparse solutions)
• Correlated predictors with variable selection