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A statistical model for random rotations

Author

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  • León, Carlos A.
  • Massé, Jean-Claude
  • Rivest, Louis-Paul

Abstract

This paper studies the properties of the Cayley distributions, a new family of models for random pxp rotations. This class of distributions is related to the Cayley transform that maps a p(p-1)/2x1 vector s into SO(p), the space of pxp rotation matrices. First an expression for the uniform measure on SO(p) is derived using the Cayley transform, then the Cayley density for random rotations is investigated. A closed-form expression is derived for its normalizing constant, a simple simulation algorithm is proposed, and moments are derived. The efficiencies of moment estimators of the parameters of the new model are also calculated. A Monte Carlo investigation of tests and of confidence regions for the parameters of the new density is briefly summarized. A numerical example is presented.

Suggested Citation

  • León, Carlos A. & Massé, Jean-Claude & Rivest, Louis-Paul, 2006. "A statistical model for random rotations," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 412-430, February.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:2:p:412-430
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    References listed on IDEAS

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    1. D. Rancourt & L.‐P. Rivest & J. Asselin, 2000. "Using orientation statistics to investigate variations in human kinematics," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 49(1), pages 81-94.
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    Cited by:

    1. Bingham, Melissa A. & Nordman, Daniel J. & Vardeman, Stephen B., 2010. "Finite-sample investigation of likelihood and Bayes inference for the symmetric von Mises-Fisher distribution," Computational Statistics & Data Analysis, Elsevier, vol. 54(5), pages 1317-1327, May.
    2. Atsushi Inoue & Lutz Kilian, 2020. "The Role of the Prior in Estimating VAR Models with Sign Restrictions," Working Papers 2030, Federal Reserve Bank of Dallas.
    3. Jeon, Jeong Min & Van Keilegom, Ingrid, 2023. "Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    4. Qiu, Yu & Nordman, Daniel J. & Vardeman, Stephen B., 2014. "One-sample Bayes inference for symmetric distributions of 3-D rotations," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 520-529.
    5. Arnold, R. & Jupp, P.E. & Schaeben, H., 2018. "Statistics of ambiguous rotations," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 73-85.
    6. Stanfill, Bryan & Genschel, Ulrike & Hofmann, Heike & Nordman, Dan, 2015. "Nonparametric confidence regions for the central orientation of random rotations," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 106-116.
    7. Rau, Christian, 2013. "Bayes classifiers of three-dimensional rotations and the sphere with symmetries," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 930-935.

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