# gizlang:stats descriptive statistics over f64 samples: moments, order statistics and the sort they share; streaming moments, variance and least squares (the SQL aggregates and their merges). ```gizlang import * as Stats from "gizlang:stats"; ``` ## Types ### `VarianceState` ```gizlang type VarianceState = { mean: f64; m2: f64; } ``` A running mean and the sum of squared differences (Welford's method). - `mean`: `f64` - `m2`: `f64` ### `OlsFit` ```gizlang type OlsFit = { coefficients: Vec; units: Vec; } ``` The fit of a least-squares regression: the coefficients and the unit of each. - `coefficients`: `Vec` - `units`: `Vec` ### `OlsInference` ```gizlang type OlsInference = { rss: f64; degreesOfFreedom: i64; sigma: Option; covariance: Vec>; standardErrors: Vec; } ``` The inference of a least-squares fit: residual sum of squares, degrees of freedom, sigma, covariance and standard errors. - `rss`: `f64` - `degreesOfFreedom`: `i64` - `sigma`: `_gizlang_sdk_core.Option` - `covariance`: `Vec>` - `standardErrors`: `Vec` ## Functions ### `mean` ```gizlang function mean(xs: Vec): f64 ``` The mean of `xs`. Effects: fail. ### `variance` ```gizlang function variance(xs: Vec, ddof: i64): f64 ``` The variance with `ddof` degrees of freedom removed (0: population, 1: sample). Effects: fail. ### `argsort` ```gizlang function argsort(keys: Vec): Vec ``` The indices of `keys` in ascending order of their keys; equal keys keep their order (a merge sort). Effects: fail. ### `sortIndices` ```gizlang function sortIndices(keys: Vec, idx: mut Vec): void ``` Sorts `idx` in place by ascending `keys[idx[i]]`, stable. Effects: fail. ### `sortedValues` ```gizlang function sortedValues(xs: Vec): Vec ``` The values in ascending order: the merge sort of sortIndices on the values themselves (no index to follow), so equal and unordered (NaN) values land where argsort puts them. Effects: fail. ### `quantileSorted` ```gizlang function quantileSorted(sorted: Vec, q: f64): f64 ``` The q-quantile (0 ≤ q ≤ 1) of sorted values by linear interpolation between the closest ranks (NumPy's default). Effects: fail. ### `median` ```gizlang function median(xs: Vec): f64 ``` The median: the middle value, or the mean of the two middle values. Effects: fail. ### `weightedMedian` ```gizlang function weightedMedian(xs: Vec, weights: Vec): f64 ``` The smallest value whose cumulative weight reaches half of the total weight (scikit-learn's weighted median). Effects: fail. ### `nearestHistogramRank` ```gizlang function nearestHistogramRank(probability: f64, count: i64): i64 ``` Exact ceil(probability * count), with zero selecting the first observation. Preconditions: probability is finite in [0, 1], count is positive signed i64. Splitting the 128-bit product preserves ranks above f64's exact integer range. Effects: fail. ### `histogramBucket` ```gizlang function histogramBucket(counts: Vec, probability: f64, count: i64): Option ``` Locate a nearest-rank histogram bucket; populations are validated by the caller. Effects: fail. ### `validateMomentMatrix` ```gizlang function validateMomentMatrix(co: Vec, dimensions: i64, count: i64): bool throws string ``` Validate retained means and centered products without storage or SQL state. Effects: fail. ### `transitionMomentMatrix` ```gizlang function transitionMomentMatrix(co: mut Vec, count: i64, values: Vec, weight: i64): i64 throws string ``` Signed repeated-observation update. A refused downdate leaves the prior matrix intact. Effects: fail. ### `interpolate` ```gizlang function interpolate(left: f64, right: f64, fraction: f64): f64 ``` Linear interpolation avoids overflowing the difference of opposite extremes. ### `stepVariance` ```gizlang function stepVariance(mean: f64, m2: f64, value: f64, count: i64): VarianceState ``` Welford update; count includes this observation. ### `mergeVariance` ```gizlang function mergeVariance(mean: f64, m2: f64, count: i64, laterMean: f64, laterM2: f64, laterCount: i64): VarianceState ``` Chan merge retaining the scalar fold's exact arithmetic order. Effects: fail. ### `stepMomentMatrix` ```gizlang function stepMomentMatrix(co: mut Vec, a: Vec, count: i64): void ``` Means followed by the row-major centered-product matrix. count includes row. Effects: fail. ### `mergeMomentMatrix` ```gizlang function mergeMomentMatrix(co: mut Vec, count: i64, later: Vec, laterCount: i64, scaled: bool): void ``` Chan combination. scaled=true normalizes populations before products; false preserves ordered SQL folds' inherited floating operation order. Effects: fail. ### `boundedCorrelation` ```gizlang function boundedCorrelation(cross: f64, left: f64, right: f64): f64 ``` Centered products on an exact line can differ by rounding ulps. ### `stepOlsFactor` ```gizlang function stepOlsFactor(co: mut Vec, a: Vec, count: i64): bool ``` Complete numerical row; count includes it. True records nonfinite input. State holds d means followed by the d-by-d triangular design/RHS factor. Effects: fail. ### `mergeOlsFactor` ```gizlang function mergeOlsFactor(co: mut Vec, count: i64, later: Vec, laterCount: i64): void ``` Stack factors and the centered-mean correction without raw row replay. Effects: fail. ### `qrSolve` ```gizlang function qrSolve(co: Vec, d: i64, units: Vec, rhs: Vec): Vec throws string ``` Solve in exact power-of-two column units with compensated term order. Wider temporaries use exponent-aware compensation instead of a global RHS scale. Effects: fail. ### `olsInference` ```gizlang function olsInference(co: Vec, count: i64, fit: OlsFit): OlsInference throws string ``` Residual uncertainty from the same centered square-root fit, independent of result formatting. Effects: fail. ### `fitOls` ```gizlang function fitOls(co: Vec, count: i64, overflow: bool): Option throws string ``` Intercept-first least squares. None means too few rows or deficient rank. Effects: fail. ## Constants ### `OLS_RANK_SQUARED_TOLERANCE` ```gizlang const OLS_RANK_SQUARED_TOLERANCE = 0.000000000001 ``` The squared tolerance under which a column of a least-squares fit is counted as dependent. ### `MOMENT_ROUNDOFF_ULPS` ```gizlang const MOMENT_ROUNDOFF_ULPS = 64.0 ``` The roundoff, in units in the last place, that a moment calculation allows for.