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Multivariate LOESS

Smoothing over multiple predictor dimensions simultaneously.

Overview

Standard LOESS operates on a single predictor \(x\). Setting dimensions > 1 extends the neighbourhood search and local polynomial fit into an \(n\)-dimensional predictor space, enabling surface smoothing over spatial grids, time–altitude combinations, and similar multi-predictor datasets.

Multivariate LOESS

Dimensions Use Case Input Shape
1 Time series, 1D signal (default) x: 1-D array
2 Spatial surface, 2-predictor model x: n × 2 matrix
3+ High-dimensional regression x: n × d matrix

Computational cost

Neighbourhood search scales with \(d\) dimensions. For dimensions ≥ 3 keep fraction small and consider increasing delta to activate interpolation.

1D — Standard (Default)

Single predictor. No configuration required.

x <- seq(0, 10, length.out = 200)
y <- sin(x) + rnorm(200, sd = 0.2)
model <- Loess(fraction = 0.3)
result <- model$fit(x, y)

import numpy as np
import fastloess as fl

x = np.linspace(0, 10, 200)
y = np.sin(x) + np.random.normal(0, 0.2, 200)
model = fl.Loess(fraction=0.3)
result = model.fit(x, y)

let model = Loess::new()
    .fraction(0.3)
    .build()?;
let result = model.fit(&x, &y)?;

model = Loess(; fraction=0.3)
result = fit(model, x, y)

const model = new Loess({ fraction: 0.3 });
const result = model.fit(x, y);

const model = new Loess({ fraction: 0.3 });
const result = model.fit(x, y);

fastloess::Loess model({ .fraction = 0.3 });
auto result = model.fit(x, y).value();

2D — Spatial Surface

Two predictors (e.g., latitude/longitude, time/altitude). Pass an \(n \times 2\) matrix as x.

# n × 2 predictor matrix
coords <- cbind(lat, lon)
model <- Loess(dimensions = 2L, fraction = 0.3)
result <- model$fit(coords, z)

import numpy as np
import fastloess as fl

# x is an (n, 2) array flattened to 1D (Python binding requires flat input)
x2d = np.column_stack([lat, lon]).ravel()
model = fl.Loess(dimensions=2, fraction=0.3)
result = model.fit(x2d, z)

let model = Loess::new()
    .dimensions(2)
    .fraction(0.3)
    .build()?;
let result = model.fit(&x2d, &z)?;

# x is an (n, 2) matrix of predictors
x2d = hcat(lat, lon)
model = Loess(; dimensions=2, fraction=0.3)
result = fit(model, x2d, z)

// x is a flat Float64Array of length n*2, row-major
const model = new Loess({ dimensions: 2, fraction: 0.3 });
const result = model.fit(x2d, z);

const model = new Loess({ dimensions: 2, fraction: 0.3 });
const result = model.fit(x2d, z);

// x is an (n × 2) row-major matrix
fastloess::LoessOptions d2_opts;
d2_opts.dimensions = 2;
d2_opts.fraction = 0.3;
fastloess::Loess model(d2_opts);
auto result = model.fit(x2d, z).value();

3D and Higher

Three or more predictors. The neighbourhood radius grows in each additional dimension, so a larger fraction (or smaller dataset) is typically needed.

predictors <- cbind(x1, x2, x3)   # n × 3
model <- Loess(dimensions = 3L, fraction = 0.5)
result <- model$fit(predictors, y)

x3d = np.column_stack([x1, x2, x3]).ravel()   # (n*3,) flat
model = fl.Loess(dimensions=3, fraction=0.5)
result = model.fit(x3d, y)

let model = Loess::new()
    .dimensions(3)
    .fraction(0.5)
    .build()?;
let result = model.fit(&x3d, &y)?;

# x is an (n, 3) matrix of predictors
x3d = hcat(x1, x2, x3)
model = Loess(; dimensions=3, fraction=0.5)
result = fit(model, x3d, y)

const model = new Loess({ dimensions: 3, fraction: 0.5 });
const result = model.fit(x3d, y);

const model = new Loess({ dimensions: 3, fraction: 0.5 });
const result = model.fit(x3d, y);

fastloess::LoessOptions d3_opts;
d3_opts.dimensions = 3;
d3_opts.fraction = 0.5;
fastloess::Loess model(d3_opts);
auto result = model.fit(x3d, y).value();

Distance Metrics for Multivariate Data

When dimensions > 1 you can also control how inter-point distances are computed.

Metric Description When to Use
"normalized" Each dimension scaled to unit range (default) Predictors on different scales
"euclidean" Raw Euclidean distance Predictors already on same scale
"minkowski:p" Generalised Minkowski (\(L_p\)) norm Custom distance geometry
"weighted" Per-dimension weighted Euclidean Domain-specific importance

See Parameters for the full list of options per language.