Shannon entropy for imprecise and under-defined or over-defined information

TL;DR

This paper extends Shannon entropy to handle under-defined, over-defined, and imprecise information systems by normalizing data through affine transformations, enabling its application to various fuzzy and neutrosophic information types.

Contribution

It introduces a method to normalize and apply Shannon entropy to diverse non-standard information systems, including imprecise, under-defined, and over-defined data.

Findings

Shannon entropy can be adapted for neutrosophic and fuzzy information.

The approach handles imprecise and over/under-defined data effectively.

Application examples demonstrate the method's versatility.

Abstract

Shannon entropy was defined for probability distributions and then its using was expanded to measure the uncertainty of knowledge for systems with complete information. In this article, it is proposed to extend the using of Shannon entropy to under-defined or over-defined information systems. To be able to use Shannon entropy, the information is normalized by an affine transformation. The construction of affine transformation is done in two stages: one for homothety and another for translation. Moreover, the case of information with a certain degree of imprecision was included in this approach. Besides, the article shows the using of Shannon entropy for some particular cases such as: neutrosophic information both in the trivalent and bivalent case, bifuzzy information, intuitionistic fuzzy information, imprecise fuzzy information, and fuzzy partitions.