On the Maximum Entropy Negation of a Complex-Valued Distribution

Abstract

In real applications of artificial and intelligent decision-making systems, how to represent the knowledge involved with uncertain information is still an open issue. The negation method has great significance to address this issue from another perspective. However, it has the limitation that can be used only for the negation of the probability distribution. In this article, therefore, we propose a generalized model of the traditional one, so that it can have more powerful capability to represent the knowledge, and uncertainty measure. In particular, we first define a vector representation of complex-valued distribution. Then, an entropy measure is proposed for the complex-valued distribution, called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> entropy. In this context, a transformation function to acquire the negation of the complex-valued distribution is exploited on the basis of the newly defined <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> entropy. Afterward, the properties of this negation function are analyzed, and investigated, as well as some special cases. Finally, we study the negation function on the view from the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> entropy. It is verified that the proposed negation method for the complex-valued distribution is a scheme with a maximal entropy.