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Bayesian cylindrical data modeling using Abe-Ley mixtures

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Abstract
This paper proposes a Metropolis-Hastings algorithm based on Markov chain Monte Carlo sampling, to estimate the parameters of the Abe-Ley distribution, which is a recently proposed Weibull-Sine-Skewed-von Mises mixture model, for bivariate circular-linear data. Current literature estimates the parameters of these mixture models using the expectation-maximization method, but we will show that this exhibits a few shortcomings for the considered mixture model. First, standard expectation-maximization does not guarantee convergence to a global optimum, because the likelihood is multi-modal, which results from the high dimensionality of the mixture's likelihood. Second, given that expectation-maximization provides point estimates of the parameters only, the uncertainties of the estimates (e.g., confidence intervals) are not directly available in these methods. Hence, extra calculations are needed to quantify such uncertainty. We propose a Metropolis-Hastings based algorithm that avoids both shortcomings of expectation-maximization. Indeed, Metropolis-Hastings provides an approximation to the complete (posterior) distribution, given that it samples from the joint posterior of the mixture parameters. This facilitates direct inference (e.g., about uncertainty, multi-modality) from the estimation. In developing the algorithm, we tackle various challenges including convergence speed, label switching and selecting the optimum number of mixture components. We then (i) verify the effectiveness of the proposed algorithm on sample datasets with known true parameters, and further (ii) validate our methodology on an environmental dataset (a traditional application domain of Abe-Ley mixtures where measurements are function of direction). Finally, we (iii) demonstrate the usefulness of our approach in an application domain where the circular measurement is periodic in time. (C) 2018 Elsevier Inc. All rights reserved.
Keywords
CIRCULAR-CIRCULAR DATA, MARGINAL LIKELIHOOD, DISTRIBUTIONS, Cylindrical data, Cylindrical mixture probabilistic model, Abe-Ley, distribution, WeiSSVM distribution, Metropolis-Hastings, Bayesian, inference

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Citation

Please use this url to cite or link to this publication:

Chicago
Sadeghianpourhamami, Nasrin, Dries Benoit, Dirk Deschrijver, and Chris Develder. 2019. “Bayesian Cylindrical Data Modeling Using Abe-Ley Mixtures.” Applied Mathematical Modelling 68: 629–642.
APA
Sadeghianpourhamami, N., Benoit, D., Deschrijver, D., & Develder, C. (2019). Bayesian cylindrical data modeling using Abe-Ley mixtures. APPLIED MATHEMATICAL MODELLING, 68, 629–642.
Vancouver
1.
Sadeghianpourhamami N, Benoit D, Deschrijver D, Develder C. Bayesian cylindrical data modeling using Abe-Ley mixtures. APPLIED MATHEMATICAL MODELLING. 2019;68:629–42.
MLA
Sadeghianpourhamami, Nasrin et al. “Bayesian Cylindrical Data Modeling Using Abe-Ley Mixtures.” APPLIED MATHEMATICAL MODELLING 68 (2019): 629–642. Print.
@article{8603555,
  abstract     = {This paper proposes a Metropolis-Hastings algorithm based on Markov chain Monte Carlo sampling, to estimate the parameters of the Abe-Ley distribution, which is a recently proposed Weibull-Sine-Skewed-von Mises mixture model, for bivariate circular-linear data. Current literature estimates the parameters of these mixture models using the expectation-maximization method, but we will show that this exhibits a few shortcomings for the considered mixture model. First, standard expectation-maximization does not guarantee convergence to a global optimum, because the likelihood is multi-modal, which results from the high dimensionality of the mixture's likelihood. Second, given that expectation-maximization provides point estimates of the parameters only, the uncertainties of the estimates (e.g., confidence intervals) are not directly available in these methods. Hence, extra calculations are needed to quantify such uncertainty. We propose a Metropolis-Hastings based algorithm that avoids both shortcomings of expectation-maximization. Indeed, Metropolis-Hastings provides an approximation to the complete (posterior) distribution, given that it samples from the joint posterior of the mixture parameters. This facilitates direct inference (e.g., about uncertainty, multi-modality) from the estimation. In developing the algorithm, we tackle various challenges including convergence speed, label switching and selecting the optimum number of mixture components. We then (i) verify the effectiveness of the proposed algorithm on sample datasets with known true parameters, and further (ii) validate our methodology on an environmental dataset (a traditional application domain of Abe-Ley mixtures where measurements are function of direction). Finally, we (iii) demonstrate the usefulness of our approach in an application domain where the circular measurement is periodic in time. (C) 2018 Elsevier Inc. All rights reserved.},
  author       = {Sadeghianpourhamami, Nasrin and Benoit, Dries and Deschrijver, Dirk and Develder, Chris},
  issn         = {0307-904X},
  journal      = {APPLIED MATHEMATICAL MODELLING},
  language     = {eng},
  pages        = {629--642},
  title        = {Bayesian cylindrical data modeling using Abe-Ley mixtures},
  url          = {http://dx.doi.org/10.1016/j.apm.2018.11.039},
  volume       = {68},
  year         = {2019},
}

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