Generalized Linear Models

AnoFox provides 2 generalized model types: Poisson GLM for count data (defects, arrivals, event counts) and the Augmented Linear Model (ALM) supporting 24 error distributions including Student-t, Cauchy, Laplace, Huber, Weibull, Gamma, and Log-Normal. Poisson GLM coefficients are on the log scale -- exp(coefficient) gives the rate ratio for a one-unit change in the predictor. ALM is the go-to choice when residuals violate the normality assumption, offering robust estimation that resists outlier contamination.

A Generalized Linear Model (GLM) extends ordinary linear regression to handle response variables that follow non-Gaussian distributions, such as counts, binary outcomes, or positive continuous values. It uses a link function to connect the linear predictor to the expected value of the response. The GLM framework unifies regression for exponential family distributions.

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Poisson GLM

The Poisson GLM is a generalized linear model that uses a log link function and Poisson distribution for modeling count data -- non-negative integers such as event counts, defects per unit, or arrivals per hour.

Parameters

Parameter	Type	Required	Default	Description
y	DOUBLE	Yes	-	Count target (non-negative integers)
x	LIST(DOUBLE)	Yes	-	Predictors
options	MAP	No	-	fit_intercept, max_iterations, tolerance

Example

SELECT
    production_line,
    (model).coefficients[2] as temperature_effect,
    exp((model).coefficients[2]) as rate_ratio,
    (model).p_values[2] as pvalue
FROM (
    SELECT
        production_line,
        anofox_stats_poisson_fit_agg(
            defect_count,
            [temperature, humidity, shift_hours]
        ) as model
    FROM quality_data
    GROUP BY production_line
);

Interpretation: Coefficients are on log scale. exp(coefficient) = rate ratio (multiplicative effect).

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ALM - Augmented Linear Model

The Augmented Linear Model (ALM) is a robust regression framework that replaces the Gaussian error assumption with any of 24 alternative distributions. By fitting the model under the correct error distribution (e.g., Student-t for heavy tails, Huber for outlier resistance, Weibull for survival data), ALM produces more efficient and reliable estimates than OLS when the normality assumption is violated.

Parameters

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

Supported Distributions

Distribution	Use Case
normal	Standard regression
student_t	Heavy tails, outliers
cauchy	Extreme outliers
laplace	LAD (median) regression
huber	Robust with breakdown
weibull	Survival, reliability
gamma	Positive, right-skewed
log_normal	Multiplicative errors

Example

SELECT anofox_stats_alm_fit_agg(
    revenue,
    [marketing_spend, competitor_activity],
    MAP {
        'distribution': 'student_t',
        'fit_intercept': 'true'
    }
) as model
FROM sales_with_outliers;

When to use ALM:

• Data has heavy tails or outliers
• Non-normal error distributions
• Robust estimation needed

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