2024 , Volume 29, № 4, p.71-94

The purpose of the paper is to present a simple and natural approach to reconstructing linear functional dependencies from non-enclosing data with interval uncertainty. It denotes interval data that is not guaranteed to contain the true values of the measured quantities, and therefore must be processed significantly differently than interval data that is certain to contain true values (enclosing). From the very definition of non-enclosing interval data it follows that they should be considered, rather, as integral objects without any internal structure, since it does not make sense for their point elements to require satisfaction of two-sided interval constraints, etc.

For this reason, the construction of functional dependencies from non-enclosing interval data should be performed on the basis of approaches that find the best approximation of the intervals under consideration without resorting to their internal content. This can be done, for example, using the approximation theory. In the present study, solving the line fitting problem is reduced to finding the minimum deviation of the graph of the constructed function from the interval data boxes.

The properties of the deviation functional for the most popular vector norms, which can be used to determine the distance between points, are investigated. It is shown that, under some conditions on the norm, the deviation functional is a convex polyhedral function. Its minimum can be efficiently found using existing non-smooth optimization methods. In particular, the paper presents a free program implemented by the authors for computing this minimum.

In conclusion, the work provides numerical examples demonstrating the behavior of the new technique in various situations, as well as its comparison with methods for solving the problem of line fitting from enclosing interval data. Finally, correlations with methods of Symbolic Data Analysis are discussed in detail.