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Bayesian Inference of a Parametric Random Spheroid from its Orthogonal Projections

Author

Listed:
  • Mathieu Langlard

    (Univ Montpellier
    INRIA)

  • Fabrice Lamadie

    (Univ Montpellier)

  • Sophie Charton

    (Univ Montpellier)

  • Johan Debayle

    (Mines Saint-Etienne, CNRS, UMR 5307 LGF, Centre SPIN)

Abstract

The paper focuses on a new method for the inference of a parametric random spheroid from the observations of its 2D orthogonal projections. Such a stereological problem is well-known from the literature when the projections come from only one deterministic spheroid. Nevertheless, when the spheroid is random itself, the estimation of its distribution is not straightforward. From a theoretical viewpoint, it is shown that the semi-axes of the spheroid and the ones of the projected ellipses are linked through a random polynomial of degree two which admits two real random positive roots. The likelihood can be formulated in terms of the coefficients of the random polynomial, but is not analytically tractable. Assuming that the random spheroid is parameterized by a set of parameters θreal, an approximation of the maximum a posteriori is used to estimate θreal. The estimator is based on the so-called approximate Bayesian computation method and a kernel density technique. As an illustration, the case of a spheroids population, whose semi-major axis follows a gamma distribution and the flattening coefficient a truncated normal distribution, is studied. The numerical results demonstrate that the bias of the estimator is very low, with a reasonable variance, both for the first and the second order moments of the semi-axes. The proposed method enables to recover some 3D morphological characteristics of a population of independent and identically distributed spheroids thanks to the only observations of its projected ellipses.

Suggested Citation

  • Mathieu Langlard & Fabrice Lamadie & Sophie Charton & Johan Debayle, 2021. "Bayesian Inference of a Parametric Random Spheroid from its Orthogonal Projections," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 549-567, June.
  • Handle: RePEc:spr:metcap:v:23:y:2021:i:2:d:10.1007_s11009-020-09806-w
    DOI: 10.1007/s11009-020-09806-w
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    References listed on IDEAS

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    1. Zougab, Nabil & Adjabi, Smail & Kokonendji, Célestin C., 2014. "Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 28-38.
    2. Zhang, Xibin & King, Maxwell L. & Hyndman, Rob J., 2006. "A Bayesian approach to bandwidth selection for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3009-3031, July.
    3. M. Sköld & G. O. Roberts, 2003. "Density Estimation for the Metropolis–Hastings Algorithm," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 699-718, December.
    4. Mark A. Beaumont & Jean-Marie Cornuet & Jean-Michel Marin & Christian P. Robert, 2009. "Adaptive approximate Bayesian computation," Biometrika, Biometrika Trust, vol. 96(4), pages 983-990.
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