Quantile Regression

Quantile regression for estimating conditional quantiles of the response distribution. Robust to outliers.

Functions

Function	Type	Description
quantile_fit	Scalar	Process complete arrays in a single call
quantile_fit_agg	Aggregate	Streaming row-by-row accumulation
quantile_fit_predict_agg	Aggregate	Fit and predict with GROUP BY support

anofox_stats_quantile_fit / quantile_fit

Quantile regression estimates conditional quantiles of the response variable distribution, rather than the conditional mean.

Signature:

anofox_stats_quantile_fit(
    y LIST(DOUBLE),
    x LIST(LIST(DOUBLE)),
    [options MAP]
) -> STRUCT

Options MAP:

Key	Type	Default	Description
tau	DOUBLE	0.5	Quantile to estimate (0 < tau < 1)
fit_intercept	BOOLEAN	true	Include intercept term
max_iterations	INTEGER	1000	Maximum iterations
tolerance	DOUBLE	1e-6	Convergence tolerance

Returns:

STRUCT(
    coefficients LIST(DOUBLE),  -- Regression coefficients
    intercept DOUBLE,           -- Intercept term (if fitted)
    tau DOUBLE,                 -- Quantile estimated
    n_observations BIGINT,      -- Number of observations
    n_features INTEGER          -- Number of features
)

Example:

-- Median regression (tau = 0.5) - robust to outliers
SELECT quantile_fit(
    [y1, y2, y3, y4, y5],
    [[x1, x2, x3, x4, x5]],
    {'tau': 0.5}
);

-- 90th percentile regression (upper bound estimation)
SELECT quantile_fit(
    prices,
    [size, location_score],
    {'tau': 0.9}
);

-- Compare different quantiles
SELECT
    0.25 as quantile, (quantile_fit(y, [x], {'tau': 0.25})).coefficients[1] as coef
UNION ALL
SELECT
    0.50 as quantile, (quantile_fit(y, [x], {'tau': 0.50})).coefficients[1] as coef
UNION ALL
SELECT
    0.75 as quantile, (quantile_fit(y, [x], {'tau': 0.75})).coefficients[1] as coef;

anofox_stats_quantile_fit_agg / quantile_fit_agg

Streaming quantile regression aggregate function.

-- Per-group median regression
SELECT
    region,
    (quantile_fit_agg(price, [sqft, bedrooms], {'tau': 0.5})).coefficients
FROM housing
GROUP BY region;

Common Tau Values

Tau	Description
0.10	10th percentile (lower tail)
0.25	First quartile
0.50	Median (robust central tendency)
0.75	Third quartile
0.90	90th percentile (upper tail)
0.95	95th percentile (risk analysis)

Use Cases

• Robust regression: Median regression is outlier-resistant
• Full response distribution: Understand effects across quantiles
• Risk analysis: VaR, conditional tail expectations
• Heteroscedastic data: Effects that vary across the distribution
• Asymmetric distributions: When mean doesn't represent typical values

See Also

• OLS - Mean regression
• ALM - Asymmetric Laplace for quantile regression
• Isotonic - Monotonic regression