Linear Models

Classical least squares regression variants: OLS, WLS, and RLS.

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OLS - Ordinary Least Squares

The statistical workhorse for linear regression with full inference support.

Variants

• Scalar Fit: anofox_stats_ols_fit(y, x, [options]) -> STRUCT
• Aggregate Fit: anofox_stats_ols_fit_agg(y, x, [options]) -> STRUCT
• Window Predict: anofox_stats_ols_fit_predict(y, x, [options]) OVER (...) -> STRUCT
• Batch Predict: anofox_stats_ols_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
options	MAP	No	-	Configuration options

Options MAP:

Option	Type	Default	Description
fit_intercept	BOOLEAN	true	Include intercept term
compute_inference	BOOLEAN	false	Compute t-stats, p-values, CIs
confidence_level	DOUBLE	0.95	Confidence level for intervals

Example

SELECT
    (model).coefficients[2] as marketing_effect,
    (model).p_values[2] as p_value,
    (model).r_squared as fit
FROM (
    SELECT anofox_stats_ols_fit_agg(
        revenue,
        [marketing_spend, seasonality_index],
        MAP {
            'fit_intercept': 'true',
            'compute_inference': 'true',
            'confidence_level': '0.95'
        }
    ) as model
    FROM sales_data
);

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WLS - Weighted Least Squares

For heteroscedastic data where variance differs across observations.

Variants

• Scalar Fit: anofox_stats_wls_fit(y, x, weights, [options]) -> STRUCT
• Aggregate Fit: anofox_stats_wls_fit_agg(y, x, weight, [options]) -> STRUCT
• Window Predict: anofox_stats_wls_fit_predict(y, x, weight, [options]) OVER (...) -> STRUCT
• Batch Predict: anofox_stats_wls_predict_agg(y, x, weight, [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
weights / weight	LIST(DOUBLE) / DOUBLE	Yes	-	Observation weights
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 anofox_stats_wls_fit_agg(
    y,
    [x1, x2],
    sample_size,
    MAP {'compute_inference': 'true'}
) as model
FROM aggregated_data;

When to use WLS:

• Residual variance changes with predictor values
• Aggregated data with different sample sizes
• Known measurement precision differences

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RLS - Recursive Least Squares

For streaming/online learning with concept drift.

Variants

• Scalar Fit: anofox_stats_rls_fit(y, x, [options]) -> STRUCT
• Aggregate Fit: anofox_stats_rls_fit_agg(y, x, [options]) -> STRUCT
• Window Predict: anofox_stats_rls_fit_predict(y, x, [options]) OVER (...) -> STRUCT
• Batch Predict: anofox_stats_rls_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
options	MAP	No	-	Configuration options

Options MAP:

Option	Type	Default	Description
forgetting_factor	DOUBLE	1.0	Weight decay (0.9-1.0)
fit_intercept	BOOLEAN	true	Include intercept term
initial_p_diagonal	DOUBLE	100.0	Initial covariance diagonal

Example

SELECT
    timestamp,
    (adaptive_model).coefficients[2] as current_beta
FROM (
    SELECT
        timestamp,
        anofox_stats_rls_fit_predict(
            y,
            [x],
            MAP {'forgetting_factor': '0.95'}
        ) OVER (
            ORDER BY timestamp
            ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW
        ) as adaptive_model
    FROM sensor_data
);

When to use RLS:

• Real-time model updates
• Non-stationary relationships
• Concept drift detection