Nonextensive statistical mechanics for robust physical parameter estimation: the role of entropic index

Abstract

The problem of inferencing parameters of complex systems from measured data has been extensively studied based on the inverse problem theory. Classically, an inverse problem is formulated as an optimisation task in which the objective function is based on the maximisation of Boltzmann–Gibbs entropy under appropriate constraints. In this classical framework, errors are assumed to follow a Gaussian-behaviour. However, in many situations, the errors are non-Gaussian and therefore the classical approach is predestined to failure. To mitigate the inverse-problem sensitivity to non-Gaussian errors, we have considered in this study a q-generalised objective function within the context of nonextensive statistical mechanics. In this regard, the errors are assumed to follow the q-Gaussian distribution, which arises from the maximisation of the nonadditive Tsallis entropy. We study the robustness properties of the q-objective function, which is a generalisation of the classical objective function, through the so-called influence function. In this work, we analyse and discuss from an analytical and numerical perspective, the role of the entropic index (q) of Tsallis entropy on the effectiveness and robustness in the inference of physical parameters from with strongly noisy-data. In this perspective, we show that exists an optimum value for the entropic index at the limit $q \rightarrow 3$, which implies in $\delta _q = 3 - q \approx 0$ in the q-objective function. To validate our proposal, we consider a classic geophysical inverse problem by employing a realistic geological model and a data set contaminated by spikes.