robust-rs

Robust statistics for Rust, built around one abstraction: the M-estimator. Almost every robust method is a loss ρ with a score ψ = ρ' and an IRLS weight ψ(r)/r. Implement the RhoFunction trait and you get location, scale and regression estimation, plus the sampling theory (influence function, efficiency, covariance) reported off each fit.

Features

• Losses behind one RhoFunction trait: Huber, Tukey biweight, Cauchy, Welsch, Andrews, Hampel, L1 and least squares.
• Robust scale: MAD, Huber Proposal 2, the S-scale, Qn and Sn, Fisher-consistent for σ at the Gaussian.
• Robust regression: M, S, MM (the lmrob equivalent), LTS (FAST-LTS) and Theil–Sen. Randomized fits are reproducible by default via seeded, thread-count-invariant RNG sub-streams.
• Multivariate location/scatter: FAST-MCD, OGK and monotone M-scatter, sharing a robust-Mahalanobis / χ² outlier map, plus Tyler's distribution-free shape estimator.
• Sampling theory off each fit: influence function, asymptotic variance, Gaussian efficiency and coefficient covariance, besides point estimates.
• Weighted least squares via rank-revealing QR (never the normal equations); Gauss–Hermite quadrature (Golub–Welsch) with E[ψ'] via Stein's identity, so kinked scores and L1 stay exact.
• robust-rs-core is dependency-light and wasm32-friendly (no linear algebra).

Choosing an estimator

If you know the shape of your problem but not the name of the method:

Rule of thumb: for regression reach for MMEstimator::default() unless you know the contamination is purely vertical (then MEstimator is cheaper); for covariance and multivariate outlier flagging reach for Mcd::new().

Installation

[dependencies]
robust-rs = "0.1"

Quick start

A robust location estimate is unmoved by a gross outlier that wrecks the mean:

use robust_rs::prelude::*;

let data = [2.1, 2.3, 1.9, 2.0, 2.2, 47.0];   // one gross outlier
let fit = m_location(&data, &Huber::default(), &Mad::default(), &Control::default())?;
println!("{:.2}", fit.estimate);              // 2.16   (the mean is 9.58)

Usage

Robust regression

An outlier in the response drags ordinary least squares off the trend; a Huber M-estimator resists it and down-weights the offending observation toward zero.

use ndarray::array;
use robust_rs::prelude::*;

// y ≈ 1 + 2x, with one gross outlier at x = 5.
let x = array![
    [1.0, 1.0], [1.0, 2.0], [1.0, 3.0], [1.0, 4.0], [1.0, 5.0],
    [1.0, 6.0], [1.0, 7.0], [1.0, 8.0], [1.0, 9.0], [1.0, 10.0],
];
let y = array![3.2, 4.8, 7.3, 9.1, 40.0, 12.7, 15.2, 17.1, 18.8, 21.3];

let fit = MEstimator::new(Huber::default(), Mad::default()).fit(&x, &y)?;
let beta = fit.coefficients();

println!("intercept = {:.2}, slope = {:.2}", beta[0], beta[1]); // 1.17, 1.99  (OLS: 4.92, 1.82)
println!("outlier weight = {:.3}", fit.weights[4]);             // 0.026

Sampling theory

A fitted estimator reports the sampling theory derived from its loss:

let fit = MEstimator::new(Huber::default(), Mad::default()).fit(&x, &y)?;

fit.gaussian_efficiency();   // 0.950, asymptotic efficiency at the Gaussian
fit.asymptotic_variance();   // V(ψ) = E[ψ²] / (E[ψ'])²
fit.coef_covariance(&x);     // s²·V·(XᵀX)⁻¹  (coefficient covariance)
fit.influence_function();    // x ↦ ψ(x) / E[ψ'], a bounded closure

Examples

cargo run --example demo           # OLS dragged by vertical outliers; Huber recovers the line
cargo run --example stars_mm       # the classic starsCYG high-leverage example
cargo run --example mcd_outliers   # multivariate on Brownlee's stackloss data

Workspace

• robust-rs-core: losses, robust scale and the influence/variance/efficiency theory. No linear-algebra dependency; usable on its own and on wasm32.
• robust-rs: the location, regression and multivariate estimators, built on the core.

Status

Working and tested; see CHANGELOG.md for per-stage detail.

• v0.1: RhoFunction losses, MAD scale, location and regression M-estimation, and the sampling-theory API. high-breakdown regression (S, MM, LTS/FAST-LTS, Theil–Sen), the full loss set (Cauchy, Welsch andrews, Hampel), extra scales (Qn, Sn, S-scale, Huber Proposal 2), trimmed/Winsorized means and Hodges–Lehmann and seeded RNG for the randomized fits. Multivariate location/scatter (FAST-MCD, OGK, M-scatter, Tyler) and the robust-Mahalanobis / χ² outlier map.

A plain M-estimator has ~0 breakdown against leverage: it bounds the influence of a large residual, but a high-leverage outlier can still carry the fit. On the starsCYG data (the stars_mm example) both OLS and Huber return a negative slope through a physically positive relationship, because four giant stars sit at high leverage with small residuals; MMEstimator recovers the positive slope and drives the four giants to zero weight.

Planned (v0.2+): robust PCA, cellwise-outlier tooling, an optional LAPACK backend, and a pyrobust PyO3 binding.

License

Dual-licensed under either MIT or Apache-2.0, at your option.