On the average uncertainty for systems with nonlinear coupling

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.

24 pages, including 4 figures and 1 table

References in corpus (5)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Generalized non-additive entropies and quantum entanglement
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators
• A definition of the coupled-product for multivariate coupled-exponentials

Cited by in corpus (5)

• Opinion Dynamics in Financial Markets via Random Networks
• Coupled VAE: Improved Accuracy and Robustness of a Variational Autoencoder
• Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions
• Independent Approximates enable closed-form estimation of heavy-tailed distributions
• Reduced Perplexity: A simplified perspective on assessing probabilistic forecasts