Estimation of parameters in multivariate wrapped models for data on a p-torus

Nodehi, Anahita; Golalizadeh, Mousa; Maadooliat, Mehdi; Agostinelli, Claudio

doi:10.1007/s00180-020-01006-x

Cited by 16 publications

(21 citation statements)

References 23 publications

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“…As previously stated in the "Introduction" [28] provided effective iterative algorithms to fit a multivariate Wrapped distribution on the p-dimensional torus. Here, robust estimation is achieved by a suitable modification of their CEM algorithm, consisting in a weighting step before performing the M-step, in which data-dependent weights are evaluated according to (6) yielding a WLEE ( 8) to be solved in the M-step.…”

Section: A Weighted Cem Algorithmmentioning

confidence: 99%

“…Algorithms based on the Expectation-Maximization (EM) method have been used by [15] for parameter estimation in autoregressive models of Wrapped Normal distributions and by [10], [32] and [14] in a Bayesian framework according to a data augmentation approach to estimate the missing unobserved wrapping coefficients. An innovative estimation strategy based on EM and Classification EM algorithms has been discussed in [28]. In order to perform maximum likelihood estimation, the wrapping coefficients are treated as latent variables.…”

Section: Introductionmentioning

confidence: 99%

“…In the M-step, the conditional expectation of ( 1) is maximized with respect to Ω. The reader is pointed to [28] for computational details about such maximization problem for the multivariate Wrapped Normal distribution.…”

Section: Introductionmentioning

confidence: 99%

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Robust estimation for multivariate wrapped models

Saraceno

Agostinelli

Greco

2021

METRON

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A weighted likelihood technique for robust estimation of multivariate Wrapped distributions of data points scattered on a $$p-$$ p - dimensional torus is proposed. The occurrence of outliers in the sample at hand can badly compromise inference for standard techniques such as maximum likelihood method. Therefore, there is the need to handle such model inadequacies in the fitting process by a robust technique and an effective downweighting of observations not following the assumed model. Furthermore, the employ of a robust method could help in situations of hidden and unexpected substructures in the data. Here, it is suggested to build a set of data-dependent weights based on the Pearson residuals and solve the corresponding weighted likelihood estimating equations. In particular, robust estimation is carried out by using a Classification EM algorithm whose M-step is enhanced by the computation of weights based on current parameters’ values. The finite sample behavior of the proposed method has been investigated by a Monte Carlo numerical study and real data examples.

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Section: A Weighted Cem Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Robust estimation for multivariate wrapped models

Saraceno

Agostinelli

Greco

2021

METRON

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“…The convolution of two wrapped normal variables is also wrapped normal (Jammalamadaka & SenGupta, 2001), but the multivariate von Mises distribution requires a rather complex estimation procedure (Mardia & Voss, 2014). The maximum likelihood parameter estimation in multivariate von Mises is also still an open problem (Nodehi et al, 2018). Assuming that the error in equation 1follows the multivariate wrapped normal distribution with zero mean and variance-covariance matrix C, the likelihood function p(d|m) is as expressed in equation 13.…”

Section: Bayesian Frameworkmentioning

confidence: 99%

Bayesian Inversion of Wrapped Satellite Interferometric Phase to Estimate Fault and Volcano Surface Ground Deformation Models

González

2020

Preprint

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Phase unwrapping is the process of recovering the absolute phase from unambiguous wrapped phase values that are measured modulo 2pi rad. From a mathematical point of view, phase unwrapping is an inverse problem, however, it is ill-posed and notoriously difficult to solve in the presence of noise. Meanwhile, phase unwrapping errors severely impact the estimation of earthquake and volcano source parameters using interferometric observations, therefore avoiding phase unwrapping completely is desirable.&#160;A potential solution to avoid the unwrapping error issue completely, is to carry out a geophysical inversion directly on the wrapped phase observations. To overcome the need for phase unwrapping, we propose a novel approach that we can invert directly the interferometric wrapped phase, circumventing the ill-posed phase unwrapping processing step. This approach includes (1) a downsampling algorithm, (2) a method to estimate the covariance function of the wrapped phase, (3) an appropriate misfit function between the observed and the simulated wrapped phase. We also assess the uncertainties of source parameters within a Bayesian approach, and finally we test the robustness of the inversion methodology in multiple simulations including variable decorrelation and atmospheric noise simulations.We demonstrate the proposed methodology on synthetic cases with variable noise and one real earthquake case. We show that the method is robust in challenging noise scenarios. We also show an improvement with the Bayesian approach in performance with respect to similar previous methods, resulting in avoiding any influence of seed starting models, and escaping local minima. We study the impact of a small percentage of incorrectly unwrapped phase observations in current state-of-the-art methods, and show that the presence of a small fraction of unwrapping errors affect strongly the estimation process. We conclude that in the cases where phase unwrapping is difficult or even impossible, the proposed inversion methodology with wrapped phase will provide an alternative approach to assess earthquake and volcano source model parameters.

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“…The convolution of two wrapped normal variables is also wrapped normal (Jammalamadaka & SenGupta, 2001), but the multivariate von Mises distribution requires a rather complex estimation procedure (Mardia & Voss, 2014). The maximum likelihood parameter estimation in multivariate von Mises is also still an open problem (Nodehi et al, 2018). Assuming that the error in equation ( 1) follows the multivariate wrapped normal distribution with zero mean and variance-covariance matrix C, the likelihood function p(d|m) is as expressed in equation ( 13).…”

Section: Bayesian Frameworkmentioning

confidence: 99%

Bayesian Inversion of Wrapped Satellite Interferometric Phase to Estimate Fault and Volcano Surface Ground Deformation Models

González

2020

JGR Solid Earth

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Bayesian inference and an improved downsampling method is used to determine earthquake and volcano source parameters using a popular geodetic observation method, satellite radar interferometry. The main novelty of the proposed approach is that the interferometric wrapped phase can be directly inverted, circumventing the ill‐posed phase unwrapping processing step. Phase unwrapping errors severely affect the estimation of earthquake and volcano source parameters using interferometric observations. Therefore, it is desirable to avoid phase unwrapping completely. To overcome the need for phase unwrapping, we propose a downsampling algorithm and a method to estimate the covariance function of the wrapped phase and establish an appropriate misfit function between the observed and simulated wrapped phase. Uncertainties in source parameters are assessed with a Bayesian approach, and finally, the robustness of the inversion methodology is tested in multiple simulations including variable decorrelation and atmospheric noise simulations. The method is shown to be robust in challenging noise scenarios. It features an improvement in performance with the Bayesian approach, compared to similar previous methods, avoiding any influence of seed starting models and escaping local minima. The impact of a small percentage of incorrectly unwrapped phase observations in current state‐of‐the‐art methods is shown to strongly affect the estimation process. We conclude that in the cases where phase unwrapping is difficult or even impossible, the proposed inversion methodology with wrapped phase will provide an alternative approach to assess earthquake and volcano source model parameters.

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Estimation of parameters in multivariate wrapped models for data on a p-torus

Cited by 16 publications

References 23 publications

Robust estimation for multivariate wrapped models

Robust estimation for multivariate wrapped models

Bayesian Inversion of Wrapped Satellite Interferometric Phase to Estimate Fault and Volcano Surface Ground Deformation Models

Bayesian Inversion of Wrapped Satellite Interferometric Phase to Estimate Fault and Volcano Surface Ground Deformation Models

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