Matrix operations in Random Permutation Set

Abstract

In D-S evidence theory, belief and commonality functions are well-defined and deeply researched. However, these concepts have not been extended to Random Permutation Set (RPS) because of the difficulties in giving a suitable definition of containment relationships of permutations. In this paper, we addressed this issue by defining both ordered and unordered containment relationships. Based on the defined containment relationships, the corresponding belief function and commonality functions for RPS are derived naturally. Besides, the plausibility function for RPS is also proposed following the definition in D-S evidence theory. Furthermore, the transformations between them and permutation mass functions are discussed and implemented by matrix converters. Thanks to the properties of commonality functions, the combination process in RPS, including left and right orthogonal sum, are both implemented by matrix operations. Moreover, based on matrix operations, the reverse transformation of combination, that is, the decomposition of fusion results given one information source is achieved for the first time. Numerical examples are listed to illustrate the calculation process. The proposed matrix operations provide a way to implement and accelerate information transformations, combinations and decompositions.