On the average uncertainty for systems with nonlinear coupling

TL;DR

This paper introduces a generalized measure of average uncertainty for nonlinear systems by transforming entropy functions into the probability domain, revealing new relationships for various distributions and proposing a coupled entropy function.

Contribution

It defines a new generalized average uncertainty based on weighted means and introduces a coupled entropy function for nonlinear systems.

Findings

Weighted geometric mean of probabilities relates to distribution density.

Generalized mean of distributions equals density at specific points.

Coupled entropy function accounts for nonlinear coupling effects.

Abstract

The increased uncertainty and complexity of nonlinear systems have motivated investigators to consider generalized approaches to defining an entropy function. New insights are achieved by defining the average uncertainty in the probability domain as a transformation of entropy functions. The Shannon entropy when transformed to the probability domain is the weighted geometric mean of the probabilities. For the exponential and Gaussian distributions, we show that the weighted geometric mean of the distribution is equal to the density of the distribution at the location plus the scale, i.e. at the width of the distribution. The average uncertainty is generalized via the weighted generalized mean, in which the moment is a function of the nonlinear source. Both the Renyi and Tsallis entropies transform to this definition of the generalized average uncertainty in the probability domain. For the generalized Pareto and Student's t-distributions, which are the maximum entropy distributions for these generalized entropies, the appropriate weighted generalized mean also equals the density of the distribution at the location plus scale. A coupled entropy function is proposed, which is equal to the normalized Tsallis entropy divided by one plus the coupling.