Intervals

Confidence and prediction intervals for uncertainty quantification.

Overview

Confidence and Prediction Intervals

Confidence and prediction coverage levels are independent; for example, a 90% confidence interval can be paired with a 99% prediction interval.

Type	Represents	Width	Use
Confidence	Uncertainty in mean curve	Narrow	Where is the true trend?
Prediction	Uncertainty for new points	Wide	Where will new data fall?

Confidence Intervals

Estimate uncertainty in the smoothed curve itself.

const fl = require('fastlowess');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1);

const model = new fl.Lowess({fraction: 0.5, intervals: {confidence: 0.95}});
const result = model.fit(x, y);

result.y.slice(0, 5).forEach((y, i) => {
    console.log(`x=${result.x[i].toFixed(4)}: y=${y.toFixed(4)} [${result.confidence_lower[i].toFixed(4)}, ${result.confidence_upper[i].toFixed(4)}]`);
});

x=0.0000: y=0.3325 [0.2929, 0.3722]
x=0.0635: y=0.3593 [0.3190, 0.3995]
x=0.1269: y=0.3871 [0.3462, 0.4280]
x=0.1904: y=0.4160 [0.3743, 0.4576]
x=0.2539: y=0.4458 [0.4034, 0.4881]

Prediction Intervals

Estimate where new observations might fall.

const fl = require('fastlowess');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1);

const model = new fl.Lowess({fraction: 0.5, intervals: {prediction: 0.95}});
const result = model.fit(x, y);
console.log(`Prediction bounds: [${result.prediction_lower[0]}, ${result.prediction_upper[0]}]`);

Prediction bounds: [-0.04054603685842645, 0.7056287954394038]

Both Intervals

Request both types simultaneously:

const fl = require('fastlowess');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1);

const model = new fl.Lowess({fraction: 0.5,
    intervals: {confidence: 0.95, prediction: 0.95}});
const result = model.fit(x, y);
console.log("95% CI: [" + result.confidence_lower[0].toFixed(4) + ", " + result.confidence_upper[0].toFixed(4) + "]");

95% CI: [0.2929, 0.3722]

Confidence Levels

Common levels and their z-values:

Level	z-value	Interpretation
0.90	1.645	90% of intervals contain true value
0.95	1.960	95% of intervals contain true value
0.99	2.576	99% of intervals contain true value

const fl = require('fastlowess');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1);

// 99% confidence interval
const model = new fl.Lowess({intervals: {confidence: 0.99}});
const result = model.fit(x, y);
console.log("99% CI: [" + result.confidence_lower[0].toFixed(4) + ", " + result.confidence_upper[0].toFixed(4) + "]");

99% CI: [0.3175, 0.4443]

Standard Errors

Access standard errors directly (available when intervals are computed):

const fl = require('fastlowess');

const n = 100;
const x = Float64Array.from({ length: n }, (_, i) => i * 2 * Math.PI / (n - 1));
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1);

const model = new fl.Lowess({intervals: {confidence: 0.95}});
const result = model.fit(x, y);

result.standard_errors.slice(0, 5).forEach((se, i) => {
    console.log(`Point ${i}: SE = ${se.toFixed(4)}`);
});

Point 0: SE = 0.0246
Point 1: SE = 0.0252
Point 2: SE = 0.0258
Point 3: SE = 0.0265
Point 4: SE = 0.0271

Residual Bootstrap

Set bootstrap to at least 2 to replace analytic uncertainty with residual-bootstrap refits. Batch shares its outer seed with CV; Streaming restarts the seed per combined chunk, and Online per full-update window. Online bootstrap requires update_mode: "full".

const fl = require('fastlowess');

const x = Float64Array.from({ length: 30 }, (_, i) => i * 0.1);
const y = Float64Array.from(x, xi => Math.sin(xi) + 0.1 * Math.cos(7 * xi));
const intervals = { confidence: 0.95, prediction: 0.95, bootstrap: 20 };

const batch = new fl.Lowess({ intervals, seed: 42 }).fit(x, y);
console.log("Batch SEs:", batch.standard_errors.length);

const stream = new fl.StreamingLowess({ intervals, seed: 42 }, { chunk_size: x.length });
console.log("Streaming CI present:", stream.process_chunk(x, y).confidence_lower !== null);

const online = new fl.OnlineLowess({ intervals, seed: 42 }, { min_points: 5, update_mode: "full" });
let last = null;
for (let i = 0; i < 12; i++) {
    last = online.add_point(x[i], y[i]) ?? last;
}
console.log("Online PI present:", last.prediction_lower !== null);

Batch SEs: 30
Streaming CI present: true
Online PI present: true

Availability

Feature	Batch	Streaming	Online
Confidence intervals	✓	✓	✓ (`update_mode: "full"` only)
Prediction intervals	✓	✓	✓ (`update_mode: "full"` only)
Standard errors	✓	✓	✓ (`update_mode: "full"` only)
Residual bootstrap	✓	✓	✓ (`update_mode: "full"` only)