LOESS Project

The fastest, most robust, and most feature-complete language-agnostic LOESS (Locally Estimated Scatterplot Smoothing) implementation for Rust, Python, R, Julia, JavaScript, C++, and WebAssembly.

[!IMPORTANT]

The loess-project contains a complete ecosystem for LOESS smoothing:

• loess-rs - Core single-threaded Rust implementation with no_std support
• fastLoess - Parallel Rust wrapper with ndarray integration
• R bindings - extendr-based R binding
• Python bindings - PyO3-based Python binding
• Julia bindings - Native Julia binding with C FFI
• JavaScript bindings - Node.js binding
• WebAssembly bindings - WASM binding
• C++ bindings - Native C++ binding with CMake integration

Installation

[!NOTE]

Currently available for R, Python, Rust, Julia, Node.js, WebAssembly, and C++. See the Installation Guide for detailed installation instructions.

Documentation

[!NOTE]

📚 View the full documentation

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) |

[!TIP] Note: For a LOWESS implementation, use lowess-project.

Why this package?

Speed

The loess project beats the competition in terms of speed, whether in single-threaded or multi-threaded parallel execution. It is typically 5–20x faster than R's loess in serial mode, and up to 200x faster on large datasets with parallel execution.

For more details on the performance comparison, see the Benchmarks page.

Robustness

This implementation is more robust than R's loess due to two key design choices:

MAD-Based Scale Estimation:

For robustness weight calculations, this crate uses Median Absolute Deviation (MAD) for scale estimation:

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

In contrast, R's loess uses the median of absolute residuals (MAR):

s = median(|r_i|)

• MAD is a breakdown-point-optimal estimator—it remains valid even when up to 50% of data are outliers.
• The median-centering step removes asymmetric bias from residual distributions.
• MAD provides consistent outlier detection regardless of whether residuals are centered around zero.

Boundary Padding:

This crate applies a range of different boundary policies at dataset edges:

• Extend: Repeats edge values to maintain local neighborhood size.
• Reflect: Mirrors data symmetrically around boundaries.
• Zero: Pads with zeros (useful for signal processing).
• NoBoundary: Original Cleveland behavior

R's loess does not apply boundary padding, which can lead to:

• Biased estimates near boundaries due to asymmetric local neighborhoods.
• Increased variance at the edges of the smoothed curve.

Features

A variety of features, supporting a range of use cases:

| Feature | This package | R (stats) | |-------------------------------|:-------------:|:----------------:| | Polynomial Degree | 5 (0–4) | 2 (1 or 2) | | Kernel | 7 options | only Tricube | | Robustness Weighting | 3 options | only Bisquare | | Scale Estimation | 3 options | only MAR | | Distance Metric | 6 options | normalized only | | Boundary Padding | 4 options | no padding | | Zero Weight Fallback | 3 options | no | | Auto Convergence | yes | no | | Online Mode | yes | no | | Streaming Mode | yes | no | | Confidence Intervals | yes | no | | Prediction Intervals | yes | no | | Diagnostics (RMSE, R², AIC) | yes | no | | Cross-Validation | 2 options | no | | Parallel Execution | yes | no | | no-std Support | yes | no |

Validation

All implementations are numerical twins of R's loess:

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

API Reference

R:

Loess(
    fraction = 0.67,
    iterations = 3L,
    weight_function = "tricube",
    robustness_method = "bisquare",
    zero_weight_fallback = "use_local_mean",
    boundary_policy = "extend",
    scaling_method = "mad",
    confidence_intervals = NULL,
    prediction_intervals = NULL,
    return_diagnostics = FALSE,
    return_residuals = FALSE,
    return_robustness_weights = FALSE,
    cv_fractions = NULL,
    cv_method = "kfold",
    cv_k = 5L,
    auto_converge = NULL,
    parallel = TRUE
)$fit(x, y)

# Result structure:
result$x,
result$y,
result$standard_errors,
result$confidence_lower,
result$confidence_upper,
result$prediction_lower,
result$prediction_upper,
result$residuals,
result$robustness_weights,
result$diagnostics,
result$iterations_used,
result$fraction_used,
result$cv_scores

Python:

from fastloess import Loess

model = Loess(
    fraction=0.67,
    iterations=3,
    weight_function="tricube",
    robustness_method="bisquare",
    zero_weight_fallback="use_local_mean",
    boundary_policy="extend",
    scaling_method="mad",
    confidence_intervals=None,
    prediction_intervals=None,
    return_diagnostics=False,
    return_residuals=False,
    return_robustness_weights=False,
    cv_fractions=None,
    cv_method="kfold",
    cv_k=5,
    auto_converge=None,
    parallel=True
)
result = model.fit(x, y)

# Result structure:
result.x,
result.y,
result.standard_errors,
result.confidence_lower,
result.confidence_upper,
result.prediction_lower,
result.prediction_upper,
result.residuals,
result.robustness_weights,
result.diagnostics,
result.iterations_used,
result.fraction_used,
result.cv_scores

Rust:

Loess::new()
    .fraction(0.67)
    .iterations(3)
    .weight_function("tricube")
    .robustness_method("bisquare")
    .zero_weight_fallback("use_local_mean")
    .boundary_policy("extend")
    .scaling_method("mad")
    .confidence_intervals(0.95)
    .prediction_intervals(0.95)
    .return_diagnostics()
    .return_residuals()
    .return_robustness_weights()
    .cv_fractions(vec![0.3, 0.5, 0.7])
    .cv_method("kfold")
    .cv_k(5)
    .cv_seed(123)
    .auto_converge(1e-4)
    .parallel(true)             // fastLoess only
    .build()?;

let result = model.fit(&x, &y)?;

// Result structure:
pub struct LoessResult<T> {
    pub x: Vec<T>,                           // Sorted x values
    pub y: Vec<T>,                           // Smoothed y values
    pub standard_errors: Option<Vec<T>>,
    pub confidence_lower: Option<Vec<T>>,
    pub confidence_upper: Option<Vec<T>>,
    pub prediction_lower: Option<Vec<T>>,
    pub prediction_upper: Option<Vec<T>>,
    pub residuals: Option<Vec<T>>,
    pub robustness_weights: Option<Vec<T>>,
    pub diagnostics: Option<Diagnostics<T>>,
    pub iterations_used: Option<usize>,
    pub fraction_used: T,
    pub cv_scores: Option<Vec<T>>,
}

Julia:

Loess(;
    fraction=0.67,
    iterations=3,
    weight_function="tricube",
    robustness_method="bisquare",
    zero_weight_fallback="use_local_mean",
    boundary_policy="extend",
    scaling_method="mad",
    confidence_intervals=NaN,
    prediction_intervals=NaN,
    return_diagnostics=false,
    return_residuals=false,
    return_robustness_weights=false,
    cv_fractions=Float64[], # e.g. [0.3, 0.5]
    cv_method="kfold",
    cv_k=5,
    auto_converge=NaN,
    parallel=true
)

# Result structure:
result.x,
result.y,
result.standard_errors,
result.confidence_lower,
result.confidence_upper,
result.prediction_lower,
result.prediction_upper,
result.residuals,
result.robustness_weights,
result.diagnostics,
result.iterations_used,
result.fraction_used,
result.cv_scores

Node.js:

new Loess({
    fraction: 0.67,
    iterations: 3,
    weight_function: "tricube",
    robustness_method: "bisquare",
    zero_weight_fallback: "use_local_mean",
    boundary_policy: "extend",
    scaling_method: "mad",
    confidence_intervals: 0.95,
    prediction_intervals: 0.95,
    return_diagnostics: true,
    return_residuals: true,
    return_robustness_weights: true,
    cv_fractions: [0.3, 0.5, 0.7],
    cv_method: "kfold",
    cv_k: 5,
    auto_converge: 1e-4,
    parallel: true
}).fit(x, y)

// Result structure:
result.x,
result.y,
result.standard_errors,
result.confidence_lower,
result.confidence_upper,
result.prediction_lower,
result.prediction_upper,
result.residuals,
result.robustness_weights,
result.diagnostics,
result.iterations_used,
result.fraction_used,
result.cv_scores

WebAssembly:

new Loess({
    fraction: 0.67,
    iterations: 3,
    weight_function: "tricube",
    robustness_method: "bisquare",
    zero_weight_fallback: "use_local_mean",
    boundary_policy: "extend",
    scaling_method: "mad",
    confidence_intervals: 0.95,
    prediction_intervals: 0.95,
    return_diagnostics: true,
    return_residuals: true,
    return_robustness_weights: true,
    cv_fractions: [0.3, 0.5, 0.7],
    cv_method: "kfold",
    cv_k: 5,
    auto_converge: 1e-4,
    parallel: true
}).fit(x, y)

// Result structure:
result.x,
result.y,
result.standard_errors,
result.confidence_lower,
result.confidence_upper,
result.prediction_lower,
result.prediction_upper,
result.residuals,
result.robustness_weights,
result.diagnostics,
result.iterations_used,
result.fraction_used,
result.cv_scores

C++:

fastloess::LoessOptions options;
options.fraction = 0.67;
options.iterations = 3;
options.weight_function = "tricube";
options.robustness_method = "bisquare";
options.zero_weight_fallback = "use_local_mean";
options.boundary_policy = "extend";
options.scaling_method = "mad";
options.confidence_intervals = 0.95;
options.prediction_intervals = 0.95;
options.return_diagnostics = true;
options.return_residuals = true;
options.return_robustness_weights = true;
options.cv_fractions = {0.3, 0.5, 0.7};
options.cv_method = "kfold";
options.cv_k = 5;
options.auto_converge = 1e-4;
options.parallel = true;

fastloess::Loess model(options);
auto result = model.fit(x, y);

// Result structure:
result.x_vector(),
result.y_vector(),
result.standard_errors(),
result.confidence_lower(),
result.confidence_upper(),
result.prediction_lower(),
result.prediction_upper(),
result.residuals(),
result.robustness_weights(),
result.diagnostics(),
result.iterations_used(),
result.fraction_used(),
result.cv_scores()

Contributing

Contributions are welcome! Please see the Contributing Guide for more information.

Changelog

See the Changelog for a history of changes.

License

Licensed under MIT or Apache-2.0.

Citation

If you use this software in your research, please cite it using the CITATION.cff file or the BibTeX entry below:

@software{loess_project,
  author = {Valizadeh, Amir},
  title = {LOESS Project: High-Performance Locally Estimated Scatterplot Smoothing},
  year = {2026},
  url = {https://github.com/thisisamirv/loess-project},
  license = {MIT OR Apache-2.0}
}