Diagnostics

Model validation, assumption checking, and residual analysis.

Quick Reference

Function	Description	SQL Signature
VIF	Multicollinearity detection	anofox_stats_vif(x) -> LIST(DOUBLE)
VIF (agg)	VIF from row-wise data	anofox_stats_vif_agg(x) -> LIST(DOUBLE)
Residuals	Comprehensive residual analysis	anofox_stats_residuals_diagnostics_agg(y, y_hat, [x]) -> STRUCT
AIC	Akaike Information Criterion	anofox_stats_aic(rss, n, k) -> DOUBLE
BIC	Bayesian Information Criterion	anofox_stats_bic(rss, n, k) -> DOUBLE

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VIF (Variance Inflation Factor)

Detect multicollinearity among predictors.

Scalar Version

For matrix input.

Parameters

Parameter	Type	Required	Default	Description
x	LIST(LIST(DOUBLE))	Yes	-	Predictor matrix

Example

SELECT anofox_stats_vif(
    [[1.0, 2.0], [1.5, 2.5], [2.0, 3.0]]
) as vif_values;

Aggregate Version

For row-wise input.

Parameters

Parameter	Type	Required	Default	Description
x	LIST(DOUBLE)	Yes	-	Predictors per row

Output

Returns LIST(DOUBLE) - VIF for each predictor.

Interpretation

VIF Range	Multicollinearity
< 5	Acceptable
5-10	Moderate
> 10	Severe (consider removing/combining)

Example

WITH vif_result AS (
    SELECT anofox_stats_vif_agg([price, promotion, competitor_price]) as vif
    FROM sales_data
)
SELECT
    vif[1] as price_vif,
    vif[2] as promotion_vif,
    vif[3] as competitor_vif,
    CASE WHEN max(v) > 10 THEN 'SEVERE'
         WHEN max(v) > 5 THEN 'MODERATE'
         ELSE 'OK' END as status
FROM vif_result, (SELECT unnest(vif) as v FROM vif_result) t
GROUP BY vif[1], vif[2], vif[3];

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Residual Diagnostics

Comprehensive residual analysis including leverage and influence.

Scalar Version

Parameters

Parameter	Type	Required	Default	Description
y	LIST(DOUBLE)	Yes	-	Actual values
y_hat	LIST(DOUBLE)	Yes	-	Predicted values
x	LIST(LIST(DOUBLE))	No	-	Predictor matrix
options	MAP	No	-	Configuration options

Aggregate Version

Parameters

Parameter	Type	Required	Default	Description
y	DOUBLE	Yes	-	Actual value
y_hat	DOUBLE	Yes	-	Predicted value
x	LIST(DOUBLE)	No	-	Predictors

Output

Field	Type	Description
residuals	LIST(DOUBLE)	Raw residuals (y - y_hat)
standardized_residuals	LIST(DOUBLE)	Standardized residuals
studentized_residuals	LIST(DOUBLE)	Studentized residuals
leverage	LIST(DOUBLE)	Hat values (diagonal of H matrix)
cooks_distance	LIST(DOUBLE)	Cook's D (influence measure)
dffits	LIST(DOUBLE)	DFFITS (influence on fitted values)
outliers	LIST(BOOLEAN)	Flagged outliers

Interpretation

Metric	Threshold	Meaning
Leverage	> 2(k+1)/n	High leverage point
Cook's D	> 4/n	Influential observation
|Studentized residual|	> 3	Potential outlier

Example

WITH model AS (
    SELECT
        y,
        anofox_stats_predict([x1, x2], coefficients, intercept) as y_hat,
        [x1, x2] as x
    FROM sales_data, fitted_model
),
diag AS (
    SELECT anofox_stats_residuals_diagnostics_agg(y, y_hat, x) as d
    FROM model
)
SELECT
    row_number() OVER () as obs,
    d.residuals[obs] as residual,
    d.leverage[obs] as leverage,
    d.cooks_distance[obs] as cooks_d,
    d.outliers[obs] as is_outlier
FROM diag
WHERE d.cooks_distance[obs] > 4.0 / count(*) OVER ();

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Information Criteria

Model selection metrics.

AIC (Akaike Information Criterion)

Parameters

Parameter	Type	Required	Default	Description
rss	DOUBLE	Yes	-	Residual sum of squares
n	BIGINT	Yes	-	Number of observations
k	BIGINT	Yes	-	Number of parameters

Formula: AIC = n × log(RSS/n) + 2k

Example

SELECT anofox_stats_aic(rss, n, k) as aic;

BIC (Bayesian Information Criterion)

Parameters

Parameter	Type	Required	Default	Description
rss	DOUBLE	Yes	-	Residual sum of squares
n	BIGINT	Yes	-	Number of observations
k	BIGINT	Yes	-	Number of parameters

Formula: BIC = n × log(RSS/n) + k × log(n)

Example

SELECT anofox_stats_bic(rss, n, k) as bic;

Usage Guidelines

• Lower values = better model
• BIC penalizes complexity more than AIC
• Use AIC for prediction, BIC for explanation

Model Comparison Example

WITH models AS (
    SELECT
        'Model 1' as name,
        anofox_stats_ols_fit_agg(y, [x1]) as m
    FROM data
    UNION ALL
    SELECT
        'Model 2',
        anofox_stats_ols_fit_agg(y, [x1, x2])
    FROM data
    UNION ALL
    SELECT
        'Model 3',
        anofox_stats_ols_fit_agg(y, [x1, x2, x3])
    FROM data
)
SELECT
    name,
    m.r_squared,
    m.aic,
    m.bic,
    ROW_NUMBER() OVER (ORDER BY m.bic) as bic_rank
FROM models
ORDER BY m.bic;

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Prediction

Generate predictions from fitted coefficients.

Parameters

Parameter	Type	Required	Default	Description
x	LIST(LIST(DOUBLE))	Yes	-	New predictor values
coefficients	LIST(DOUBLE)	Yes	-	Fitted coefficients
intercept	DOUBLE	Yes	-	Intercept term

Returns: LIST(DOUBLE) - predicted values

Example

WITH fitted AS (
    SELECT
        (anofox_stats_ols_fit_agg(y, [x1, x2])).coefficients as coef,
        (anofox_stats_ols_fit_agg(y, [x1, x2])).intercept as intercept
    FROM training_data
)
SELECT
    anofox_stats_predict([[10, 5], [15, 8], [20, 10]], coef, intercept) as forecasts
FROM fitted;

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Normality Tests

For testing residual normality, see Parametric Tests:

• anofox_stats_shapiro_wilk_agg - Shapiro-Wilk test
• anofox_stats_jarque_bera_agg - Jarque-Bera test
• anofox_stats_dagostino_k2_agg - D'Agostino K² test

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Complete Diagnostic Workflow

Full model validation workflow:

-- 1. Fit model
WITH fitted AS (
    SELECT anofox_stats_ols_fit_agg(
        revenue,
        [marketing_spend, seasonality, competitor_activity],
        true, true, 0.95
    ) as model
    FROM sales_data
),

-- 2. Check multicollinearity
vif_check AS (
    SELECT anofox_stats_vif_agg([marketing_spend, seasonality, competitor_activity]) as vif
    FROM sales_data
),

-- 3. Get predictions
predictions AS (
    SELECT
        s.revenue as actual,
        anofox_stats_predict(
            [[s.marketing_spend, s.seasonality, s.competitor_activity]],
            f.model.coefficients,
            f.model.intercept
        )[1] as predicted
    FROM sales_data s, fitted f
),

-- 4. Test residual normality
normality AS (
    SELECT anofox_stats_jarque_bera_agg(actual - predicted) as jb
    FROM predictions
)

SELECT
    'R-squared' as metric, round(model.r_squared, 3)::VARCHAR as value FROM fitted
UNION ALL
SELECT 'Max VIF', round(max(v), 2)::VARCHAR FROM vif_check, unnest(vif) as v
UNION ALL
SELECT 'Normality p-value', round(jb.p_value, 4)::VARCHAR FROM normality
UNION ALL
SELECT 'Model Valid',
    CASE WHEN model.r_squared > 0.5
         AND (SELECT max(v) FROM vif_check, unnest(vif) as v) < 10
         AND jb.p_value > 0.05
    THEN 'YES' ELSE 'NO' END
FROM fitted, normality;

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