Limit of the Maximum Random Permutation Set Entropy

TL;DR

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.

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.

Figures (5)

• Figure 1: The connection between Shannon entropy shannon1948mathematical, Deng entropy deng2016deng, and RPS entropy chen2023entropy. The envelope is the function in the logarithmic function in maximum entropy expression.
• Figure 2: The relationship between RPS, DSET and probability theory.
• Figure 3: The envelope of Shannon entropy, Deng entropy and RPS entropy.
• Figure 4: The trend of maximum Shannon entropy, Deng entropy, RPS entropy and the proposed approximation of maximum RPS entropy when $N$ changes.
• Figure 5: The trend of relative error and absolute error of the proposed approximation of maximum RPS entropy when $N$ changes. The line chart is denoted as the relative error while the bar chart is denoted as the absolute error.

Theorems & Definitions (15)

• Theorem 1: Maximum Deng entropykang2019maximum
• Theorem 2: Maximum RPS entropy deng2022maximum
• Example 1: The envelope of Shannon entropy and Deng entropy
• Theorem 3: Envelope of entropy
• Lemma 1: sum of permutation
• proof : Proof of Lemma \ref{['le.sum_of_permutation']}
• Lemma 2: Approximation of $S(N)$
• proof : Proof of Lemma \ref{['le.appropriation']}
• Lemma 3: Limit of $\sum_{u=1}^{N} \frac{u!}{(N-u)!}$
• proof : Proof of Lemma \ref{['le.limit']}