Reliable uncertainties in indirect measurements

TL;DR

This paper introduces a mathematically rigorous, intuitive method for data fitting that combines interval calculus with probability and statistics to reliably estimate parameters and uncertainties without assuming specific distribution shapes.

Contribution

It presents a novel approach that integrates interval calculations with statistical notions, enabling reliable parameter estimation without distribution assumptions or small uncertainties.

Findings

Method effectively handles outliers.

Provides reliable parameter values and uncertainties.

Operates without distribution shape assumptions.

Abstract

In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such an area in searched parameters' space that generates simulated curve crossing as many acquired experimental points as possible, but at least half of them. Such a task is pretty easy to attack with interval calculations. The problem is, however, that interval calculations operate on guaranteed intervals, that is on pairs of numbers determining minimal and maximal values of measured quantity while in vast majority of cases our measured quantities are expressed rather as a pair of two other numbers: the average value and its standard deviation. Here we propose the combination of interval calculus with basic notions from probability and statistics. This approach makes possible to obtain the results in familiar form as reliable values of searched parameters, their standard deviations, and their correlations as well. There are no assumptions concerning the probability density distributions of experimental values besides the obvious one that their variances are finite. Neither the symmetry of uncertainties of experimental distributions is required (assumed) nor those uncertainties have to be `small.' As a side effect, outliers are quietly and safely ignored, even if numerous.