Breakthrough in Interval Data Fitting I. The Role of Hausdorff Distance

TL;DR

This paper introduces an interval-based data fitting method utilizing Hausdorff distance, offering a rigorous alternative to traditional least squares that does not rely on assumptions about uncertainty distributions.

Contribution

It presents a novel interval-oriented fitting methodology that handles implicit equations and avoids assumptions like Gaussian errors, improving reliability in uncertain data analysis.

Findings

Method fits data with interval uncertainties without prior assumptions.

Handles implicit and explicit functional dependencies.

Provides rigorous and reliable data fitting under uncertainty.

Abstract

This is the first of two papers describing the process of fitting experimental data under interval uncertainty. Here I present the methodology, designed from the very beginning as an interval-oriented tool, meant to replace to the large extent the famous Least Squares (LSQ) and other slightly less popular methods. Contrary to its classical counterparts, the presented method does not require any poorly justified prior assumptions, like smallness of experimental uncertainties or their normal (Gaussian) distribution. Using interval approach, we are able to fit rigorously and reliably not only the simple functional dependencies, with no extra effort when both variables are uncertain, but also the cases when the constitutive equation exists in implicit rather than explicit functional form. The magic word and a key to success of interval approach appears the Hausdorff distance.