Coupled Entropy: A Goldilocks Generalization for Complex Systems

Abstract

The coupled entropy is proven to correct a flaw in the derivation of the Tsallis entropy and thereby solidify the theoretical foundations for analyzing the uncertainty of complex systems. The Tsallis entropy originated from considering power probabilities $p_i^q$ in which \textit{q} independent, identically-distributed random variables share the same state. The maximum entropy distribution was derived to be a \textit{q}-exponential, which is a member of the shape ($\kappa$), scale ($\sigma$) distributions. Unfortunately, the $q$-exponential parameters were treated as though valid substitutes for the shape and scale. This flaw causes a misinterpretation of the generalized temperature and an imprecise derivation of the generalized entropy. The coupled entropy is derived from the generalized Pareto distribution (GPD) and the Student's t distribution, whose shape derives from nonlinear sources and scale derives from linear sources of uncertainty. The Tsallis entropy of the GPD converges to one as $\kappa\rightarrow\infty$, which makes it too cold. The normalized Tsallis entropy (NTE) introduces a nonlinear term multiplying the scale and the coupling, making it too hot. The coupled entropy provides perfect balance, ranging from $\ln \sigma$ for $\kappa=0$ to $\sigma$ as $\kappa\rightarrow\infty$. One could say, the coupled entropy allows scientists, engineers, and analysts to eat their porridge, confident that its measure of uncertainty reflects the mathematical physics of the scale of non-exponential distributions while minimizing the dependence on the shape or nonlinear coupling. Examples of complex systems design including a coupled variation inference algorithm are reviewed.