Time Series Analysis

LOESS for trend extraction and temporal smoothing.

Overview

Time series data often contains noise, seasonality, and trends. LOESS provides flexible trend extraction without parametric assumptions.

Basic Trend Extraction

fraction = 0.1 sizes the neighbourhood as 10% of the data at each evaluation point — narrow enough to follow a slowly varying trend without smearing periodic variation. Three robustness iterations down-weight noise spikes so they cannot bias the fitted curve; this is especially important when the signal-to-noise ratio is low or when occasional outliers are expected.

const { Loess } = require('fastloess-wasm');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, (xi, i) => Math.sin(xi) + (((i * 7 + 3) % 17) / 17 - 0.5) * 0.6);

const model = new Loess({
    fraction: 0.1,
    iterations: 3
});
const result = model.fit(x, y);
console.log("y[0]:", result.y[0].toFixed(4));

y[0]: -0.0964

Detrending

Remove trend to analyze residual patterns.

Setting outputs: ["residuals"] stores observed − smoothed alongside the smooth. A slightly wider fraction = 0.3 produces a smoother baseline trend, so short-duration oscillations end up in the residuals rather than being absorbed into the trend component. The residual series is then ready for spectral analysis, seasonality detection, or change-point methods.

const { Loess } = require('fastloess-wasm');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, (xi, i) => Math.sin(xi) + (((i * 7 + 3) % 17) / 17 - 0.5) * 0.6);

const model = new Loess({
    fraction: 0.3,
    iterations: 3,
    outputs: ["residuals"]
});
const result = model.fit(x, y);
console.log("y[0]:", result.y[0].toFixed(4), "residual[0]:", result.residuals[0].toFixed(4));

y[0]: 0.0281 residual[0]: -0.2222

Forecasting with Prediction Intervals

Prediction intervals widen the uncertainty band to include both the uncertainty in the fitted curve (confidence interval) and the expected scatter of new observations around it. fraction = 0.2 offers a balance between local detail and stable interval width — too small a fraction produces jagged interval edges; too large a fraction underestimates local variance near turning points.

const { Loess } = require('fastloess-wasm');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, (xi, i) => Math.sin(xi) + (((i * 7 + 3) % 17) / 17 - 0.5) * 0.6);

const model = new Loess({
    fraction: 0.2,
    iterations: 3,
    intervals: { prediction : 0.95 }
});
const result = model.fit(x, y);
console.log("Prediction lower[0]:", result.prediction_lower[0].toFixed(4));

Prediction lower[0]: -0.5069

Handling Missing Data

LOESS naturally handles irregular time sampling:

const { Loess } = require('fastloess-wasm');

const n = 100;
const tIrregular = Float64Array.from({ length: n }, (_, i) => i * 1.0 + (i * 31 % 10) * 0.1).sort((a, b) => a - b);
const yIrregular = Float64Array.from(tIrregular, t => 10 + 0.3 * t + 2.0 * Math.sin(t * 0.1));
const model = new Loess({ fraction: 0.2 });
const result = model.fit(tIrregular, yIrregular);
console.log("y[0]:", result.y[0].toFixed(4));

y[0]: 10.8052

Multi-Scale Analysis

Use different fractions to extract features at different scales:

const { Loess } = require('fastloess-wasm');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, (xi, i) => Math.sin(xi) + (((i * 7 + 3) % 17) / 17 - 0.5) * 0.6);

const trends = [0.05, 0.2, 0.5].map(f => {
    const model = new Loess({ fraction: f });
    const result = model.fit(x, y);
    return result.y;
});
console.log("Trend y[0] values:", trends.map(t => t[0].toFixed(4)));

Trend y[0] values: [ '-0.1050', '-0.0262', '0.1118' ]

Gene Expression Time Course

Biological application:

const { Loess } = require('fastloess-wasm');

const n = 24;
const hours = Float64Array.from({ length: n }, (_, i) => i);
const expression = Float64Array.from(hours, h => 5 + 3 * Math.sin(h * Math.PI / 12) + (h % 3) * 0.2);
const model = new Loess({ fraction: 0.3, iterations: 3, outputs: ["diagnostics"] });
const result = model.fit(hours, expression);

console.log("R2:", result.diagnostics?.r_squared);

R2: 0.9864121418777019

Choosing Fraction for Time Series

Data Type	Recommended Fraction	Rationale
Daily data (years)	0.3–0.5	Capture annual trends
Hourly data (days)	0.1–0.2	Capture daily patterns
Sensor data (minutes)	0.05–0.1	Preserve short-term features
Noisy data	Higher	Reduce noise impact
Clean data	Lower	Preserve detail