Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes

Xiumei Chen,Shuai Wang,Lihui Wang,Yufu Ning,Hong Huang

doi:10.3390/sym14010014

Xiumei Chen, Shuai Wang + Show 3 more

Open Access

https://doi.org/10.3390/sym14010014

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Abstract

In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results.

Highlights

We are devoted to studying indeterminacy through uncertainty theory
The uncertain measure is defined for modeling the belief degree of an uncertain event and uncertain variable for depicting the quantity with uncertainty
Similar to Theorem 3, the operational law for the inverse uncertainty distribution of the monotone function of uncertain processes was presented in Liu [16]

Summary

Introduction

“Indeterminacy is absolute, while determinacy is relative” (Liu [1]). It seems that real decisions are usually made in the context of indeterminacy. For more research-based answers, interested readers can refer to [2] While encountering these cases, uncertainty theory is a legitimate approach to model the belief degree by treating the indeterminate quantity as an uncertain variable. Compared with density function in probability theory, inverse uncertainty distribution in uncertainty theory is a convenient and useful tool to calculate the expected value and variance of uncertain variables. It is necessary to complete this proof By using this conjecture, Yao([21]) provided a formula for calculating the inverse uncertainty distribution of the time integral.

Preliminaries

The Proof for Inverse Uncertainty Distribution of Uncertain Process

Theorems for Inverse Uncertainty Distribution of the Monotone Function of

Conclusions