Limit of the Maximum Random Permutation Set Entropy

TL;DR

This paper introduces the concept of the entropy envelope for Random Permutation Sets (RPS), derives its limit as the set size grows infinitely large, and demonstrates a significant reduction in computational complexity for entropy calculation.

Contribution

It defines the entropy envelope for RPS, derives its limit as size approaches infinity, and offers a more efficient method for entropy computation compared to existing approaches.

Findings

The limit of the RPS entropy envelope converges to e times the square of N factorial as N approaches infinity.

The proposed method significantly reduces computational complexity for entropy calculation.

Numerical examples confirm the efficiency and accuracy of the entropy envelope.

Abstract

The Random Permutation Set (RPS) is a new type of set proposed recently, which can be regarded as the generalization of evidence theory. To measure the uncertainty of RPS, the entropy of RPS and its corresponding maximum entropy have been proposed. Exploring the maximum entropy provides a possible way of understanding the physical meaning of RPS. In this paper, a new concept, the envelope of entropy function, is defined. In addition, the limit of the envelope of RPS entropy is derived and proved. Compared with the existing method, the computational complexity of the proposed method to calculate the envelope of RPS entropy decreases greatly. The result shows that when $N \to \infty$, the limit form of the envelope of the entropy of RPS converges to $e \times (N!)^2$, which is highly connected to the constant $e$ and factorial. Finally, numerical examples validate the efficiency and conciseness of the proposed envelope, which provides a new insight into the maximum entropy function.