Set-based computations

Abstract

The representation of uncertain information and inference with such information are some of the fundamental issues in uncertainty management. Conventional methods for uncertainty management usually use a single value with the exception of interval-based approaches. In interval-based approaches, an interval is used to represent the uncertain information. It is assumed that the true, possibly unknown, value lies in an interval. However, in order to use interval-based methods, there must exist an order relation on the set of data values. The main objective of this thesis is to extend single-valued and interval-valued methods by introducing a framework of set-valued computations. In this model, uncertain information is described by a set without any further restrictions. Basic issues of set-based computations are investigated. Operations on set values are defined based on the corresponding point-based (i.e., single-value-based) operations on their members. The properties of set-based computations are examined in connection to the corresponding properties of the point-based computations. Within the proposed framework, a critical analysis of a number of existing set-based computation methods is presented. This provides further evidence supporting the proposed model. To a large extent, the present study may be regarded as a more explicit re-examination of methods that have been implicitly used in many studies, using a unified notion. The results of such an investigation will be useful in establishing a framework for more systematic study of set-based computations.