Limit of the Maximum Random Permutation Set Entropy
This work provides a theoretical insight into the maximum entropy of RPS, which is incremental as it builds on prior entropy definitions for this set type.
The paper tackles the problem of computing the maximum entropy for Random Permutation Sets (RPS), a generalization of evidence theory, by deriving and proving the limit of the envelope of RPS entropy as N approaches infinity, converging to e × (N!)^2, and shows that the proposed method greatly reduces computational complexity compared to existing methods.
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.