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Random Walk in Random Permutation Set Theory

Jiefeng Zhou, Zhen Li, Yong Deng

TL;DR

This work bridges Random Permutation Set Theory (RPST) with stochastic processes by constructing RPST-generated random vectors and a two-dimensional random walk. It defines RPST-based distributions $P_{RPS}(n|N)$ and $P_{Per}(n|N)$ and introduces a Random Walk Generator to produce trajectories that, under a scaling $RW_{n,N}(t)=\frac{\sqrt{\varrho}}{N\sqrt{N}}\frac{1}{\sqrt{n}}\sum_{i=1}^{\lfloor nt\rfloor} \vec{V_i}$, converge to a Wiener process as $N,n\to\infty$, with zero-mean independent increments and variance scaling guided by $\varrho$ and $N$. The results demonstrate Gaussian-like behavior for RPST-derived walks and establish a limit to Brownian motion, thereby expanding RPST’s applicability to random-walk frameworks. This coupling suggests practical routes for uncertainty reasoning in physical and biological systems where stochastic-walk models are employed. The work lays groundwork for applying RPST-driven random walks in epidemiology, finance, and machine learning, leveraging ordered information to enrich stochastic modelling.

Abstract

Random walk is an explainable approach for modeling natural processes at the molecular level. The Random Permutation Set Theory (RPST) serves as a framework for uncertainty reasoning, extending the applicability of Dempster-Shafer Theory. Recent explorations indicate a promising link between RPST and random walk. In this study, we conduct an analysis and construct a random walk model based on the properties of RPST, with Monte Carlo simulations of such random walk. Our findings reveal that the random walk generated through RPST exhibits characteristics similar to those of a Gaussian random walk and can be transformed into a Wiener process through a specific limiting scaling procedure. This investigation establishes a novel connection between RPST and random walk theory, thereby not only expanding the applicability of RPST, but also demonstrating the potential for combining the strengths of both approaches to improve problem-solving abilities.

Random Walk in Random Permutation Set Theory

TL;DR

This work bridges Random Permutation Set Theory (RPST) with stochastic processes by constructing RPST-generated random vectors and a two-dimensional random walk. It defines RPST-based distributions and and introduces a Random Walk Generator to produce trajectories that, under a scaling , converge to a Wiener process as , with zero-mean independent increments and variance scaling guided by and . The results demonstrate Gaussian-like behavior for RPST-derived walks and establish a limit to Brownian motion, thereby expanding RPST’s applicability to random-walk frameworks. This coupling suggests practical routes for uncertainty reasoning in physical and biological systems where stochastic-walk models are employed. The work lays groundwork for applying RPST-driven random walks in epidemiology, finance, and machine learning, leveraging ordered information to enrich stochastic modelling.

Abstract

Random walk is an explainable approach for modeling natural processes at the molecular level. The Random Permutation Set Theory (RPST) serves as a framework for uncertainty reasoning, extending the applicability of Dempster-Shafer Theory. Recent explorations indicate a promising link between RPST and random walk. In this study, we conduct an analysis and construct a random walk model based on the properties of RPST, with Monte Carlo simulations of such random walk. Our findings reveal that the random walk generated through RPST exhibits characteristics similar to those of a Gaussian random walk and can be transformed into a Wiener process through a specific limiting scaling procedure. This investigation establishes a novel connection between RPST and random walk theory, thereby not only expanding the applicability of RPST, but also demonstrating the potential for combining the strengths of both approaches to improve problem-solving abilities.
Paper Structure (20 sections, 4 theorems, 34 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 4 theorems, 34 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sample space and mass function
  4. Random permutation set theory
  5. Random permutation set
  6. RPS entropy
  7. Random walk
  8. General random walk
  9. Wiener process
  10. Explore random walk in Random Permutation Set
  11. Generating random variables
  12. Simulating random variables for random walk
  13. Generating random walk with random variables
  14. Conclusion
  15. Proof of deriving random walk from RPST
  16. ...and 5 more sections

Key Result

Lemma 1

(Expected value of a random variable). The expected value of a random variable generated with RVG is zero, namely,

Figures (8)

  • Figure 1: Variance and mean value of $V_i$ with respect to $V_x$ and $V_y$.
  • Figure 2: Visualization of random variables generation with n ranging from 1 to 12 in 20,000 simulations. Each point on the graph represents a possible random variable, with the color indicating the frequency of occurrence.
  • Figure 3: Histogram of random variables with respect to $V_x$ and $V_y$.
  • Figure 4: Discrete probability distribution of $P_{Per}(n|N)$ and $P_{RPS}(n|N)$ with $N=10$.
  • Figure 5: Visualization of random walk from distribution $P_{Per}(n|N)$ and $P_{RPS}(n|N)$, where color map is showing the time steps.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 9 more