[Paper Review] Interval straight line fitting

M. Gutowski|ArXiv.org|Aug 23, 2001

Advanced Numerical Analysis Techniques7 references3 citations

TL;DR

This paper proposes a constraint satisfaction approach using interval analysis to compute guaranteed, tight bounds for linear model parameters (slope and intercept) in experimental data fitting. Instead of minimizing error functions, it employs a box slicing algorithm to compute the interval hull of all solutions consistent with measurement uncertainties, ensuring rigorous, non-probabilistic bounds without confidence levels or assumptions about error distributions.

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.

Motivation & Objective

• To address the limitations of traditional regression methods that rely on probabilistic assumptions, such as normal error distributions and confidence levels, which are often unjustified in real experimental data.
• To provide a deterministic, guaranteed enclosure of the true linear model parameters (slope and intercept) despite measurement uncertainties in both x and y variables.
• To eliminate reliance on arbitrary choices such as weighting schemes, confidence levels, or asymptotic approximations common in least squares or least absolute deviation methods.
• To enable robust detection of outliers and identification of asymptotic or tangent lines in experimental data without prior assumptions about data structure.

Proposed method

• The method replaces minimization of error functions with a constraint satisfaction problem, solving for intervals of slope (a) and intercept (b) such that each measured data point lies within the interval-defined line.
• It uses the box slicing algorithm to compute the interval hull of all solutions satisfying the system of equations: a*x_j + b ∈ y_j for all j, where x_j and y_j are intervals.
• The approach is based on interval arithmetic and the inclusion principle, ensuring that the resulting parameter intervals are guaranteed to contain the true values.
• For outlier detection, points not satisfying the interval equation system are identified as outliers without prior suspicion.
• For finding asymptotic or tangent lines, the method incrementally adds data points and monitors when the interval hull stops shrinking or becomes empty, indicating the best-fit line.
• The method is computationally frugal and terminates reliably, even with up to 50% outliers, due to the finite nature of the constraint satisfaction process.

Experimental results

Research questions

• RQ1 How can we compute guaranteed, non-probabilistic bounds for linear model parameters when both x and y measurements have uncertainties?
• RQ2 Can we detect outliers in experimental data without prior assumptions or subjective thresholds?
• RQ3 How can we reliably determine the parameters of a line tangent to a curve based on experimental data?
• RQ4 What is a robust alternative to least squares or least absolute deviation fitting that avoids assumptions about error distributions?
• RQ5 Can we systematically identify the asymptotic behavior of a process from experimental data using interval analysis?

Key findings

• The box slicing algorithm computes the interval hull of all solutions satisfying the interval linear system, providing guaranteed bounds for slope and intercept without confidence levels.
• The method produces conservative but certain parameter intervals, ensuring that the true values are enclosed with mathematical certainty.
• Outliers are automatically detected as data points that do not satisfy the interval equation system with the computed parameter intervals.
• The method is robust and can handle up to 50% of outliers in the data set, as demonstrated in limited tests.
• For finding asymptotic lines, the algorithm identifies the stable interval hull by incrementally adding data points and stopping when the hull no longer shrinks or becomes empty.
• The approach requires no assumptions about error distributions, weighting schemes, or confidence levels, making it suitable for digital measurement data with discrete, non-Gaussian uncertainties.