loess-rs

A high-performance implementation of LOESS (Locally Estimated Scatterplot Smoothing) in Rust. This crate provides a robust, production-ready implementation with support for confidence intervals, multiple kernel functions, and optimized execution modes.

Important

For parallelization or ndarray support, use fastLoess.

How LOESS works

LOESS creates smooth curves through scattered data using local weighted neighborhoods:

LOESS vs. LOWESS

Feature	LOESS (This Crate)	LOWESS
Polynomial Degree	Linear, Quadratic, Cubic, Quartic	Linear (Degree 1)
Dimensions	Multivariate (n-D support)	Univariate (1-D only)
Flexibility	High (Distance metrics)	Standard
Complexity	Higher (Matrix inversion)	Lower (Weighted average/slope)

LOESS can fit higher-degree polynomials for more complex data:

LOESS can also handle multivariate data (n-D), while LOWESS is limited to univariate data (1-D):

Tip

Note: For a simple, lightweight, and fast LOWESS implementation, use lowess crate.

Features

• Robust Statistics: IRLS with Bisquare, Huber, or Talwar weighting for outlier handling.
• Multidimensional Smoothing: Support for n-D data with customizable distance metrics (Euclidean, Manhattan, etc.).
• Flexible Fitting: Linear, Quadratic, Cubic, and Quartic local polynomials.
• Uncertainty Quantification: Point-wise standard errors, confidence intervals, and prediction intervals.
• Optimized Performance: Interpolation surface with Tensor Product Hermite interpolation and streaming/online modes for large or real-time datasets.
• Parameter Selection: Built-in cross-validation for automatic smoothing fraction selection.
• Flexibility: Multiple weight kernels (Tricube, Epanechnikov, etc.) and no_std support (requires alloc).
• Validated: Numerical twin of R's stats::loess with exact match (< 1e-12 diff).

Performance

Benchmarked against R's loess. Achieves 4.4×–53× faster performance across all tested scenarios. No regressions observed.

Summary

Category	Matched	Median Speedup	Mean Speedup
Polynomial Degrees	2	21.67×	21.67×
Fraction	6	13.36×	19.70×
Iterations	6	16.35×	15.90×
Pathological	4	16.14×	16.27×
Dimensions	3	7.98×	8.01×
Scalability	2	5.86×	5.86×
Genomic	2	5.27×	5.27×
Financial	3	4.88×	5.97×
Scientific	3	4.46×	5.28×

Top 10 Performance Wins

Benchmark	Rust	R	Speedup
fraction_0.67	0.83ms	44.80ms	53.71×
fraction_0.5	1.22ms	32.25ms	26.50×
degree_quadratic	0.75ms	19.21ms	25.52×
high_noise	1.60ms	34.68ms	21.68×
iterations_1	0.80ms	15.88ms	19.94×
iterations_0	0.75ms	13.68ms	18.25×
degree_linear	0.78ms	13.92ms	17.81×
clustered	1.15ms	19.74ms	17.17×
iterations_2	0.97ms	16.63ms	17.13×
iterations_3	1.13ms	17.56ms	15.57×

Check Benchmarks for detailed results and reproducible benchmarking code.

Robustness Advantages

This implementation includes several robustness features beyond R's loess:

MAD-Based Scale Estimation

Uses MAD-based scale estimation for robustness weight calculations:

s = median(|r_i - median(r)|)

MAD is a breakdown-point-optimal estimator—it remains valid even when up to 50% of data are outliers, compared to the median of absolute residuals used by some other implementations.

Median Absolute Residual (MAR), which is the default Cleveland's choice, is also available through the scaling_method parameter.

Configurable Boundary Policies

R's loess uses asymmetric windows at data boundaries, which can introduce edge bias. This implementation offers configurable boundary policies to mitigate this:

• Extend (default): Pad with constant values for symmetric windows
• Reflect: Mirror data at boundaries (best for periodic data)
• Zero: Pad with zeros (signal processing applications)
• NoBoundary: Original R behavior (no padding)

Boundary Degree Fallback

When using Interpolation mode with higher polynomial degrees (Quadratic, Cubic), vertices outside the tight data bounds can produce unstable extrapolation. This implementation offers a configurable boundary degree fallback:

• true (default): Reduce to Linear fits at boundary vertices (more stable)
• false: Use full requested degree everywhere (matches R exactly)

Validation

The Rust loess-rs crate is a numerical twin of R's loess implementation:

Aspect	Status	Details
Accuracy	✅ EXACT MATCH	Max diff < 1e-12 across all scenarios
Consistency	✅ PERFECT	20/20 scenarios pass with strict tolerance
Robustness	✅ VERIFIED	Robust smoothing matches R exactly

Check Validation for detailed scenario results.

Installation

Add this to your Cargo.toml:

[dependencies]
loess-rs = "0.2"

For no_std environments:

[dependencies]
loess-rs = { version = "0.2", default-features = false }

Quick Start

use loess_rs::prelude::*;

fn main() -> Result<(), LoessError> {
    let x = vec![1.0, 2.0, 3.0, 4.0, 5.0];
    let y = vec![2.0, 4.1, 5.9, 8.2, 9.8];

    // Build and fit model
    let result = Loess::new()
        .fraction(0.5)      // Use 50% of data for each local fit
        .iterations(3)      // 3 robustness iterations
        .adapter(Batch)
        .build()?
        .fit(&x, &y)?;

    println!("{}", result);
    Ok(())
}

Summary:
  Data points: 5
  Fraction: 0.5

Smoothed Data:
       X     Y_smooth
  --------------------
    1.00     2.00000
    2.00     4.10000
    3.00     5.90000
    4.00     8.20000
    5.00     9.80000

Builder Methods

All builder parameters have sensible defaults. You only need to specify what you want to change.

use loess_rs::prelude::*;

Loess::new()
    // Smoothing span (0, 1] - default: 0.67
    .fraction(0.5)

    // Polynomial degree - default: Linear
    .degree(Quadratic)

    // Number of dimensions - default: 1
    .dimensions(2)

    // Distance metric - default: Euclidean
    .distance_metric(Manhattan)

    // Robustness iterations - default: 3
    .iterations(5)

    // Kernel selection - default: Tricube
    .weight_function(Epanechnikov)

    // Robustness method - default: Bisquare
    .robustness_method(Huber)

    // Boundary handling - default: Extend
    .boundary_policy(Reflect)

    // Boundary degree fallback - default: true
    .boundary_degree_fallback(true)

    // Confidence intervals (Batch only)
    .confidence_intervals(0.95)

    // Prediction intervals (Batch only)
    .prediction_intervals(0.95)

    // Include diagnostics
    .return_diagnostics()
    .return_residuals()
    .return_robustness_weights()

    // Cross-validation (Batch only)
    .cross_validate(KFold(5, &[0.3, 0.5, 0.7]).seed(123))

    // Auto-convergence
    .auto_converge(1e-4)

    // Interpolation settings
    .surface_mode(Interpolation)

    // Interpolation cell size - default: 0.2
    .cell(0.2)

    // Execution mode
    .adapter(Batch)

    // Build the model
    .build()?;

Result Structure

pub struct LoessResult<T> {
    /// Sorted x values (independent variable)
    pub x: Vec<T>,

    /// Smoothed y values (dependent variable)
    pub y: Vec<T>,

    /// Point-wise standard errors of the fit
    pub standard_errors: Option<Vec<T>>,

    /// Confidence interval bounds (if computed)
    pub confidence_lower: Option<Vec<T>>,
    pub confidence_upper: Option<Vec<T>>,

    /// Prediction interval bounds (if computed)
    pub prediction_lower: Option<Vec<T>>,
    pub prediction_upper: Option<Vec<T>>,

    /// Residuals (y - fit)
    pub residuals: Option<Vec<T>>,

    /// Final robustness weights from outlier downweighting
    pub robustness_weights: Option<Vec<T>>,

    /// Detailed fit diagnostics (RMSE, R^2, Effective DF, etc.)
    pub diagnostics: Option<Diagnostics<T>>,

    /// Number of robustness iterations actually performed
    pub iterations_used: Option<usize>,

    /// Smoothing fraction used (optimal if selected via CV)
    pub fraction_used: T,

    /// RMSE scores for each fraction tested during CV
    pub cv_scores: Option<Vec<T>>,
}

Tip

Using with ndarray: While the result struct uses Vec<T> for maximum compatibility, you can effortlessly convert any field to an Array1 using Array1::from_vec(result.y).

Streaming Processing

For datasets that don't fit in memory:

let mut processor = Loess::new()
    .fraction(0.3)
    .iterations(2)
    .adapter(Streaming)
    .chunk_size(1000)
    .overlap(100)
    .build()?;

// Process data in chunks
let result1 = processor.process_chunk(&chunk1_x, &chunk1_y)?;
let result2 = processor.process_chunk(&chunk2_x, &chunk2_y)?;

// Finalize to get remaining buffered data
let final_result = processor.finalize()?;

Online Processing

For real-time data streams:

let mut processor = Loess::new()
    .fraction(0.2)
    .iterations(1)
    .adapter(Online)
    .window_capacity(100)
    .build()?;

// Process points as they arrive
for i in 1..=10 {
    let x = i as f64;
    let y = 2.0 * x + 1.0;
    if let Some(output) = processor.add_point(&[x], y)? {
        println!("Smoothed: {:.2}", output.smoothed);
    }
}

Parameter Selection Guide

Fraction (Smoothing Span)

• 0.1-0.3: Fine detail, may be noisy
• 0.3-0.5: Moderate smoothing (good for most cases)
• 0.5-0.7: Heavy smoothing, emphasizes trends
• 0.7-1.0: Very smooth, may over-smooth
• Default: 0.67 (Cleveland's choice)

Robustness Iterations

• 0: No robustness (fastest, sensitive to outliers)
• 1-3: Light to moderate robustness (recommended)
• 4-6: Strong robustness (for contaminated data)
• 7+: Diminishing returns

Polynomial Degree

• Constant: Local weighted mean (smoothing only)
• Linear (default): Standard LOESS, good bias-variance balance
• Quadratic: Better for peaks/valleys, higher variance
• Cubic/Quartic: Specialized high-order fitting

Kernel Function

• Tricube (default): Best all-around, Cleveland's original choice
• Epanechnikov: Theoretically optimal MSE
• Gaussian: Maximum smoothness, no compact support
• Uniform: Fastest, least smooth (moving average)

Boundary Policy

• Extend (default): Pad with constant values
• Reflect: Mirror data at boundaries (for periodic/symmetric data)
• Zero: Pad with zeros (signal processing)
• NoBoundary: Original Cleveland behavior

Note: For nD data, Extend defaults to NoBoundary to preserve regression accuracy.

Examples

cargo run --example batch_smoothing
cargo run --example online_smoothing
cargo run --example streaming_smoothing

MSRV

Rust 1.85.0 or later (2024 Edition).

Contributing

Contributions are welcome! Please see CONTRIBUTING.md for guidelines.

License

Licensed under either of

• Apache License, Version 2.0 (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

References

• Cleveland, W.S. (1979). "Robust Locally Weighted Regression and Smoothing Scatterplots". Journal of the American Statistical Association.
• Cleveland, W.S. & Devlin, S.J. (1988). "Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting". Journal of the American Statistical Association.