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

PLoS One. 2023 Mar 30;18(3):e0282578. doi: 10.1371/journal.pone.0282578. eCollection 2023.

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 Rényi, Tsallis and Kaniadakis statistics. In this regard, we analytically analyze 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.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Algorithms*
  • Likelihood Functions
  • Normal Distribution

Associated data

  • figshare/10.6084/m9.figshare.21878394.v1

Grants and funding

This work was supported by the Fundação Norte-Rio-Grandense de Pesquisa e Cultura - FUNPEC [grant number 512019 - FUNPEC/UFRN/PETROBRAS/INVERSÃO MULTIMETRICA]; J.M. de Araújo received financial support from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [grant no. 313431/2018-3]; and G. Corso received financial support from the CNPq [grant number 307907/2019-8].