Generalized statistics: applications to data inverse problems with outlier-resistance

TL;DR

This paper introduces generalized Gaussian-based maximum likelihood estimators for data inversion, demonstrating improved outlier resistance and robustness in noisy geophysical inverse problems, with potential computational efficiency benefits.

Contribution

It develops and analyzes outlier-resistant inverse problem methodologies using generalized Gaussian statistics, unifying different approaches and highlighting their robustness and efficiency.

Findings

Generalized Gaussian estimators improve outlier resistance in data inversion.

Optimal entropic index values enhance robustness and accuracy.

Unified approach reduces computational cost and accelerates convergence.

Abstract

The conventional approach to data-driven inversion framework is based on Gaussian statistics that presents serious difficulties, especially in the presence of outliers in the measurements. In this work, we present maximum likelihood estimators associated with generalized Gaussian distributions in the context of R\'enyi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyse the outlier-resistance of each proposal through the so-called influence function. In this way, we formulate inverse problems by constructing objective functions linked to the maximum likelihood estimators. To demonstrate the robustness of the generalized methodologies, we consider an important geophysical inverse problem with high noisy data with spikes. The results reveal that the best data inversion performance occurs when the entropic index from each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that in such a limit the three approaches are resistant to outliers and are also equivalent, which suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the fast convergence of the optimization process.