Constrained Models

Regression with coefficient constraints: bounded and non-negative solutions.

---

BLS - Bounded Least Squares

Constrained regression with coefficient bounds.

Parameters

Parameter	Type	Required	Default	Description
y	DOUBLE	Yes	-	Target values
x	LIST(DOUBLE)	Yes	-	Predictors
options	MAP	Yes	-	lower_bounds, upper_bounds, fit_intercept

Example

-- Elasticity must be negative (law of demand)
SELECT anofox_stats_bls_fit_agg(
    log(quantity),
    [log(price), log(income)],
    MAP {
        'lower_bounds': '[-5.0, 0.0]',
        'upper_bounds': '[0.0, 3.0]'
    }
) as model
FROM demand_data;

When to use BLS:

• Economic constraints (negative elasticities)
• Physical constraints (positive effects)
• Domain knowledge requiring bounded coefficients

---

NNLS - Non-Negative Least Squares

Regression with non-negativity constraints on all coefficients.

Parameters

Parameter	Type	Required	Default	Description
y	DOUBLE	Yes	-	Target values
x	LIST(DOUBLE)	Yes	-	Predictors
options	MAP	No	-	fit_intercept (default: false), max_iterations, tolerance

Example

-- Channel contributions must be non-negative
SELECT
    (model).coefficients[1] as search_contribution,
    (model).coefficients[2] as social_contribution,
    (model).coefficients[3] as email_contribution
FROM (
    SELECT anofox_stats_nnls_fit_agg(
        conversions,
        [search_spend, social_spend, email_spend]
    ) as model
    FROM marketing_data
);

When to use NNLS:

• Marketing mix modeling (positive contributions only)
• Spectral decomposition
• Portfolio weights (no short selling)