WLS (Weighted Least Squares)

March 3, 2026 · View on GitHub

Weighted Least Squares regression for heteroscedastic data. Supports SVD, QR, and Cholesky decomposition with optional HC standard errors.

Functions

Function	Type	Description
`wls_fit`	Scalar	Process complete arrays in a single call
`wls_fit_agg`	Aggregate	Streaming row-by-row accumulation
`wls_fit_predict`	Window	Fit and predict in a single pass
`wls_fit_predict_agg`	Aggregate	Fit and predict with GROUP BY support
`wls_fit_predict_by`	Table Macro	Per-group regression with long-format output

anofox_stats_wls_fit

Signature:

anofox_stats_wls_fit(
    y LIST(DOUBLE),
    x LIST(LIST(DOUBLE)),
    weights LIST(DOUBLE),
    [fit_intercept BOOLEAN DEFAULT true],
    [compute_inference BOOLEAN DEFAULT false],
    [confidence_level DOUBLE DEFAULT 0.95]
) -> STRUCT

Parameters:

Parameter	Type	Description
y	LIST(DOUBLE)	Response variable values
x	LIST(LIST(DOUBLE))	Feature arrays
weights	LIST(DOUBLE)	Observation weights (same length as y)
fit_intercept	BOOLEAN	Include intercept term (default: true)
compute_inference	BOOLEAN	Compute t-tests, p-values, CIs (default: false)
confidence_level	DOUBLE	CI confidence level (default: 0.95)

Returns: FitResult STRUCT

Example:

SELECT anofox_stats_wls_fit(
    [3.0, 5.0, 7.0, 9.0, 11.0],
    [[1.0, 2.0, 3.0, 4.0, 5.0]],
    [1.0, 2.0, 3.0, 2.0, 1.0]  -- higher weight for middle observations
);

anofox_stats_wls_fit_agg

Streaming WLS aggregate function.

SELECT anofox_stats_wls_fit_agg(y, [x], weight) FROM data;

MAP Options

All WLS functions accept an optional MAP parameter for advanced configuration:

Key	Type	Default	Description
`fit_intercept`	BOOLEAN	`true`	Include intercept term
`compute_inference`	BOOLEAN	`false`	Compute t-tests, p-values, CIs
`confidence_level`	DOUBLE	`0.95`	CI confidence level
`solver`	VARCHAR	`'svd'`	Decomposition method: `'qr'`, `'svd'`, `'cholesky'`
`hc_type`	VARCHAR	`'none'`	Heteroscedasticity-consistent SEs: `'none'`, `'hc0'`, `'hc1'`, `'hc2'`, `'hc3'`

Example with MAP options:

-- WLS with Cholesky decomposition
SELECT wls_fit_agg(
    y, [x], weight,
    {'solver': 'cholesky', 'compute_inference': true}
) FROM data;

Choosing Weights

Inverse variance: weight = 1 / variance when variance is known
Sample size: weight = n when observations are group means
Reliability: Higher weights for more reliable observations

Use Cases

Heteroscedastic data (non-constant variance)
Aggregated data (weighted by sample size)
When observation reliability varies
Survey data with sampling weights