A Direct Link between Rényi–Tsallis Entropy and Hölder’s Inequality—Yet Another Proof of Rényi–Tsallis Entropy Maximization

Tanaka, Hisaaki; Nakagawa, Masazumi; Oohama, Yasutada

doi:10.3390/e21060549

Cited by 9 publications

(8 citation statements)

References 38 publications

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“…From the maximum entropy principle (MEP) for the Rényi α -entropy, several statistical distributions have emerged to model and describe a wide variety of complex systems, such as power-law decay in Hamiltonian systems [ 62 ] and statistical inference [ 63 ]. In this work, we consider an optimal probability function which is derived from the maximization of the α -entropy subject to the normalization condition: and the unity variance In this regard, the α -generalized Gaussian probability distribution (or α -Gaussian distribution) is a distribution function resulting from the MEP for the Rényi α -entropy ( Eq (16) ) subject to normalization condition ( Eq (18) ) and the unity variance ( Eq (19) ) [ 64 – 66 ], which is given by: where [ y ] + = max{0, y } and Z α is the normalizing constant given by [ 65 ]: for , in which Γ represents the Gamma Function.…”

Section: Methodsmentioning

confidence: 99%

Full-waveform inversion based on generalized Rényi entropy using patched Green’s function techniques

et al. 2022

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The estimation of physical parameters from data analyses is a crucial process for the description and modeling of many complex systems. Based on Rényi α-Gaussian distribution and patched Green’s function (PGF) techniques, we propose a robust framework for data inversion using a wave-equation based methodology named full-waveform inversion (FWI). From the assumption that the residual seismic data (the difference between the modeled and observed data) obeys the Rényi α-Gaussian probability distribution, we introduce an outlier-resistant criterion to deal with erratic measures in the FWI context, in which the classical FWI based on l2-norm is a particular case. The new misfit function arises from the probabilistic maximum-likelihood method associated with the α-Gaussian distribution. The PGF technique works on the forward modeling process by dividing the computational domain into outside target area and target area, where the wave equation is solved only once on the outside target (before FWI). During the FWI processing, Green’s functions related only to the target area are computed instead of the entire computational domain, saving computational efforts. We show the effectiveness of our proposed approach by considering two distinct realistic P-wave velocity models, in which the first one is inspired in the Kwanza Basin in Angola and the second in a region of great economic interest in the Brazilian pre-salt field. We call our proposal by the abbreviation α-PGF-FWI. The results reveal that the α-PGF-FWI is robust against additive Gaussian noise and non-Gaussian noise with outliers in the limit α → 2/3, being α the Rényi entropic index.

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Section: Methodsmentioning

confidence: 99%

Full-waveform inversion based on generalized Rényi entropy using patched Green’s function techniques

et al. 2022

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“…Taking into account the constraints in Eqs. ( 3) and ( 4), α-entropy is maximized by the α-generalized Gaussian distribution, α-Gaussian, which is expressed in the form [49,50,33]:…”

Section: Rényi's Frameworkmentioning

confidence: 99%

“…In this work, we consider deformed Gaussian distributions associated to generalized statistical mechanics in the sense of Rényi (α-statistics) [30], Tsallis (q-statistics) [31] and Kaniadakis (κ-statistics) [32] to mitigate the undesirable effects of outliers on estimates of physical parameters. In particular, we place the objective functions based on α-, qand κ-generalizations in the broad context of the Gauss' law of error [33,34,35], see Refs. [29,36,37].…”

Section: Introductionmentioning

confidence: 99%

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

T.¹,

Silva²,

Araújo³

et al. 2022

Preprint

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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 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.

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“…In order to mitigate the effect of non-Gaussian errors, several robust formulations have been proposed in the literature. Among them, we may mention the criteria based on heavytailed probability functions, such as Student's t and Cauchy-Lorentz distributions [8,9]; hybrid functions [10][11][12]; and generalized probability distributions, such as the deformed Gaussian distributions in the context of Rényi [13][14][15], Tsallis [16][17][18][19], and Kaniadakis statistics [20][21][22]. Very recently, a connection between Jackson, Tsallis, and Hausdorff approaches in the context of generalized statistical mechanics was proposed [23].…”

Section: Introductionmentioning

confidence: 99%

Improving Seismic Inversion Robustness via Deformed Jackson Gaussian

Silva

Souza

et al. 2021

Entropy

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The seismic data inversion from observations contaminated by spurious measures (outliers) remains a significant challenge for the industrial and scientific communities. This difficulty is due to slow processing work to mitigate the influence of the outliers. In this work, we introduce a robust formulation to mitigate the influence of spurious measurements in the seismic inversion process. In this regard, we put forth an outlier-resistant seismic inversion methodology for model estimation based on the deformed Jackson Gaussian distribution. To demonstrate the effectiveness of our proposal, we investigated a classic geophysical data-inverse problem in three different scenarios: (i) in the first one, we analyzed the sensitivity of the seismic inversion to incorrect seismic sources; (ii) in the second one, we considered a dataset polluted by Gaussian errors with different noise intensities; and (iii) in the last one we considered a dataset contaminated by many outliers. The results reveal that the deformed Jackson Gaussian outperforms the classical approach, which is based on the standard Gaussian distribution.

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A Direct Link between Rényi–Tsallis Entropy and Hölder’s Inequality—Yet Another Proof of Rényi–Tsallis Entropy Maximization

Cited by 9 publications

References 38 publications

Full-waveform inversion based on generalized Rényi entropy using patched Green’s function techniques

Full-waveform inversion based on generalized Rényi entropy using patched Green’s function techniques

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

Improving Seismic Inversion Robustness via Deformed Jackson Gaussian

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