Сильная согласованность в задачах восстановления зависимостей по интервальным данным

Abstract

Для задачи восстановления зависимостей по данным с интервальной неопределённостью вводится понятие сильной согласованности данных и параметров. Даётся его содержательная интерпретация. Показывается, что получающаяся усиленная формулировка задачи сводится к исследованию непустоты и дальнейшему оцениванию так называемого допускового множества решений для интервальной системы уравнений, построенной по обрабатываемым данным. The data fitting problem is a popular and practically important problem in which a functional dependency between “input” and “output” variables is to be constructed from the given empirical data. Real-life data are almost always inaccurate, and we have to deal with the measurement uncertainty. Traditionally, when processing the measurement results, models of probability theory are used, which are not always adequate to the situations under study. An alternative way to describe data inaccuracy is to use methods of interval analysis, based on specifying interval bounds of the measurement results. Data fitting problems under interval uncertainty are being solved for about half a century. Most studies in this field rely on the concept of compatibility between parameters and measurement data in which any measurement result is a kind of a large point “inflated” to a box (rectangular parallelepiped with facets parallel to the coordinate axes). That the graph of the constructed function passes through such a “point” means a nonempty intersection of the graph with the box. However, in some problems, this natural concept turns out to be unsatisfactory. In this work, for the data fitting under interval uncertainty, we introduce the concept of strong compatibility between data and parameters. It is adequate to the situations when measurements of input and output variables are broken in time, and we strive to uniformly take into account the interval results of output measurements. The paper gives a practical interpretation of the new concept. It is shown that the modified formulation of the problem reduces to recognition and further estimation of the so-called tolerable solution set to interval systems of equations constructed from the processed data.