Interval straight line fitting

TL;DR

This work introduces an interval-analytic approach to linear data fitting with uncertainties in both variables, replacing optimization with constraint satisfaction to produce guaranteed interval enclosures for the slope $a$ and intercept $b$ in the model $y = a x + b$. The core method, box slicing, iteratively narrows a search box by transforming the problem into inequalities and applying probing/rejection tests to obtain tight, validated bounds without relying on probabilistic assumptions. The paper formalizes solution types (united, tolerable, controllable) and demonstrates the method on a synthetic dataset, showing that interval enclosures can match or exceed the predictive reliability of classical LSQ while remaining robust to outliers and capable of handling uncertainty in both coordinates. It further discusses practical extensions, including outlier detection and locating asymptotic lines, and emphasizes memory efficiency and the absence of arbitrary data weighting. Overall, the interval box-slicing framework offers guaranteed, data-uncertainty-aware parameter bounds with practical applicability in experimental sciences.

Abstract

I consider the task of experimental data fitting. Unlike the traditional approach I do not try to minimize any functional based on available experimental information, instead the minimization problem is replaced with constraint satisfaction procedure, which produces the interval hull of solutions of desired type. The method, called 'box slicing algorithm', is described in details. The results obtained this way need not to be labeled with confidence level of any kind, they are simply certain (guaranteed). The method easily handles the case with uncertainties in one or both variables. There is no need for, always more or less arbitrary, weighting the experimental data. The approach is directly applicable to other experimental data processing problems like outliers detection or finding the straight line, which is tangent to the experimental curve.

Figures (2)

• Figure 3: United (U) vs. tolerable (T) and controllable (C) solutions for two experimental points. Uncertain data are shown as light grey rectangles. Solid lines bound regions, marked with appropriate letters, where the straight lines belonging to each solution set can be drawn. For united solutions only the dark shadowed region is inaccessible.
• Figure 4: The bounding box denotes initial search domain in the plane $ab$. The target, slightly oversized, is a rectangle located near the center of figure. Two remaining lines, solid and dashed, divide the initial box into parts, for which some conditions are, or are not, met. The region bounded by dashed line is labelled as I, the region bounded by solid line has the label II. The details concerning the remaining domains III --- VII are given in text.