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

Gustavo Z Dos Santos Lima1, João V T de Lima2, João M de Araújo2

  • 1School of Science and Technology, Federal University of Rio Grande do Norte, Natal, RN, Brazil.

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|March 30, 2023
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Summary
This summary is machine-generated.

This study introduces generalized Gaussian distributions for robust data inversion, overcoming limitations of traditional methods when dealing with outlier data. The findings suggest these new methods offer improved performance and efficiency in geophysical applications.

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Area of Science:

  • Geophysics
  • Data Science
  • Statistical Modeling

Background:

  • Conventional data-driven inversion relies on Gaussian statistics, which are sensitive to measurement outliers.
  • Outliers in geophysical data can significantly degrade the accuracy of inversion results.
  • A need exists for robust statistical frameworks to handle noisy and outlier-corrupted data in inversion problems.

Purpose of the Study:

  • To develop and analyze maximum likelihood estimators using generalized Gaussian distributions (Rényi, Tsallis, Kaniadakis) for robust data inversion.
  • To assess the outlier-resistance of these generalized statistical approaches via influence function analysis.
  • To formulate inverse problems using objective functions derived from these estimators and demonstrate their efficacy in a geophysical context.

Main Methods:

  • Maximum likelihood estimation using generalized Gaussian distributions (Rényi, Tsallis, Kaniadakis).
  • Analytical assessment of outlier-resistance using the influence function.
  • Formulation of inverse problems with objective functions tailored to generalized statistics.
  • Application to a geophysical inverse problem with high-noise, spike-contaminated data.

Main Results:

  • Generalized methodologies demonstrate superior outlier-resistance compared to conventional Gaussian approaches.
  • Optimal data inversion performance is achieved when the entropic index is linked to objective functions inversely proportional to error amplitude.
  • Under specific conditions, Rényi, Tsallis, and Kaniadakis statistics become equivalent and highly resistant to outliers.

Conclusions:

  • Generalized Gaussian statistics provide a robust framework for data-driven inversion, particularly in the presence of outliers.
  • The proposed methods offer improved accuracy and computational efficiency for geophysical inverse problems.
  • The equivalence of generalized statistics under outlier-prone conditions simplifies inversion and reduces computational cost.