Applied vignette · custom distributions · 0.6.0+

Build a bounded sensor-error law

A sensor reports errors only between −2 and 2 units. Small errors are common and large errors taper exponentially. That domain statement is enough to build a reusable probability distribution.

Available from 0.6.0

1 · Translate the story

Use a kernel, not a finished density

Let the unnormalized shape be exp(-abs(error) / 0.55) on [−2, 2]. It is nonnegative, symmetric, bounded, and has finite area. We deliberately omit its normalizing constant.

\[g(x)=e^{-|x|/0.55}\,\mathbf 1_{[-2,2]}(x),\qquad f(x)=g(x)\big/\int_{-2}^{2}g(t)\,dt.\]

2 · Construct once

Declare the formula and support

import jdistlib.DiagnosticPreset;
import jdistlib.NumericalContinuousDistribution;

NumericalContinuousDistribution error =
    NumericalContinuousDistribution.builder()
        .kernel(x -> Math.exp(-Math.abs(x) / 0.55))
        .support(-2.0, 2.0)
        .singularities(0.0)
        .diagnosticPreset(DiagnosticPreset.THOROUGH)
        .build();

double normalizer = error.getNormalizationConstant();

The point at zero is not mathematically singular, but declaring it as an integration breakpoint tells the numerical routine where the derivative changes abruptly.

3 · Ask domain questions

The custom law now has the familiar API

// Probability of meeting a ±0.5-unit accuracy requirement.
double withinTolerance =
    error.cumulative(0.5, true, false)
  - error.cumulative(-0.5, true, false);

// Central interval containing 95% of modeled errors.
ProbabilityInterval central = error.probabilityInterval(0.95);

// Repeatable simulation for a test or report.
error.setRandomEngine(new MersenneTwister(20260826L));
double[] simulatedErrors = error.random(1_000);

4 · Inspect and test

Use structural and data checks together

DistributionAnalysis report = error.analyzeDistribution();
if (report.hasErrors()) {
    System.err.println(report);
}

double[] observedErrors = {
    -0.82, -0.31, -0.12, -0.03, 0.04, 0.15, 0.37, 0.73
};
double[] cvm = DistributionTest.cramer_von_mises_test(
    observedErrors, error, 1_999, new MersenneTwister(7L));

The construction report checks numerical behavior of the formula. The Cramér–von Mises test asks whether observations are compatible with this fully specified distribution. They answer different questions, so keep both.

Do not tune and test on the same data without adjustment. If 0.55 or the support was selected from observedErrors, refit those choices inside every bootstrap replicate or validate on held-out data.

5 · Know what was assumed

The support is a scientific claim

This model assigns exactly zero probability outside [−2, 2]. That is appropriate only if the physical or recording process truly enforces those bounds. If extreme errors are merely rare, use infinite support and justify the tail shape instead.

Go from example to production

Read the full custom-distribution guide for log-kernels, discrete weights, mixtures, transforms, sampling strategies, and troubleshooting.