Random Permutation Set

Abstract

For exploring the meaning of the power set in evidence theory, a possible explanation of power set is proposed from the view of Pascal’s triangle and combinatorial number. Here comes the question: what would happen if the combinatorial number is replaced by permutation number? To address this issue, a new kind of set, named as random permutation set (RPS), is proposed in this paper, which consists of permutation event space (PES) and permutation mass function (PMF). The PES of a certain set considers all the permutation of that set. The elements of PES are called the permutation events. PMF describes the chance of a certain permutation event that would happen. Based on PES and PMF, RPS can be viewed as a permutation-based generalization of random finite set. Besides, the right intersection (RI) and left intersection (LI) of permutation events are presented. Based on RI and LI, the right orthogonal sum (ROS) and left orthogonal sum (LOS) of PMFs are proposed. In addition, numerical examples are shown to illustrate the proposed conceptions. The comparisons of probability theory, evidence theory, and RPS are discussed and summarized. Moreover, an RPS-based data fusion algorithm is proposed and applied in threat assessment. The experimental results show that the proposed RPS-based algorithm can reasonably and efficiently deal with uncertainty in threat assessment with respect to threat ranking and reliability ranking.

Maximum Entropy of Random Permutation Set

“…• According to Y. Deng (2021), if each permutation event is limited to containing just one element, P ES(Θ) will degenerate into the sample space in probability theory: Ω = {{X}, {Y }}. Based on this sample space, calculate the maximum Shannon entropy:…”

“…Deng, 2020b), which is further generalized into information volume (J. Deng & Deng, 2021;Y. Deng, 2020a;Q.…”

“…In 2021, Y. Deng (2021) proposed Random permutation set (RPS), which is a novel set consisting of permutation event space (PES) and permutation mass function (PMF). Some basic definitions of RPS are given as follows.…”

“…To solve this problem, Y. Deng (2021) proposed a new type of set, called Random permutation set (RPS), which consists of permutation event space (PES) and permutation mass function (PMF). The PES of a certain set considers all the permutations of that set.…”

An Evidential Software Risk Evaluation Model

“…To solve these problems, a data-driven risk assessment model, based on Dempster-Shafer evidence theory (DST) [20,21], Deng entropy [22] and risk matrix [23], is proposed. Due to effectively deal with uncertain information, DST is widely used in decisionmaking [24][25][26], risk analysis [27], information fusion [28,29], uncertainty measurements [30], fault diagnosis [31][32][33], time-series [34], IoT applications [35] and many other fields [36,37]. Since most experts prefer to express their opinions with linguistic information, such as good, better, best, bad, worse, worst, DST can effectively deal with uncertain information about linguistic expressions involved in risk evaluation [38,39].…”