Abstract
Both existing models for non-symmetric distributions on 3-dimensional rotations and their associated one-sample inference methods have serious limitations in terms of both interpretability and ease of use. Based on the intuitively appealing Uniform Axis- Random Spin (UARS) construction of Bingham, Nordman, and Vardeman (2009) for symmetric families of distributions, we propose new highly interpretable and tractable classes of non-symmetric distributions that are derived from mixing UARS distributions. These have an appealing Preferred Axis-Random Spin (PARS) construction and (unlike existing models) directly interpretable parameters. Non-informative one-sample Bayes inference in these models is a direct generalization of UARS methods introduced in Bingham, Vardeman, and Nordman (2009), where credible levels were found to be essentially equivalent to frequentist coverage probabilities. We apply the new models and inference methods to a problem in biomechanics, where comparison of model parameters provides meaningful comparisons for the nature of movement about the calcaneocuboid joint of three different primate subjects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, K. A., and Greiner, T. M. (2012). “A Procedure to Refine Joint Kinematic Assessments: Functional Alignment,” Computer Methods in Biomechanics and Biomedical Engineering, 15, 487–500.
Bingham, M. A., Nordman, D. J., and Vardeman, S. B. (2009), “Modeling and Inference for Measured Crystal Orientations and a Tractable Class of Symmetric Distributions for Rotations in Three Dimensions,” Journal of the American Statistical Association, 104, 1385–1397.
— (2010), “Finite-Sample Investigation of Likelihood and Bayes Inference for the Symmetric von Mises-Fisher Distribution,” Computational Statistics and Data Analysis, 5, 1317–1327.
Bingham, M. A., Vardeman, S. B., and Nordman, D. J. (2009), “Bayes One-Sample and One-Way Random Effects Analyses for 3-D Orientations With Application to Materials Science,” Bayesian Analysis, 4, 607–630.
Bunge, H. J. (1982), Texture Analysis in Materials Science, London: Butterworth.
Chang, T., and Bingham, C. (1996), “Bayesian Spherical Regression,” in Bayesian Analysis in Statistics and Econometrics, eds. D. A. Berry, K. M. Chaloner, and J. K. Geweke, New York: Wiley.
Downs, T. D. (1972), “Orientation Statistics,” Biometrika, 59, 665–676.
Fisher, N. I., Lewis, T., and Embleton, B. J. J. (1987), Statistical Analysis of Spherical Data, Cambridge: Cambridge University Press.
Haddou, M., Rivest, L.-P., and Pierrynowski, M. (2010), “A Nonlinear Mixed Effects Directional Model for the Estimation of the Rotation Axes of the Human Ankle,” Annals of Applied Statistics, 4, 1892–1912.
Hielscher, R., Schaeben, H., and Siemes, H. (2010), “Orientation Distribution Within a Single Hematite Crystal,” Mathematical Geosciences, 42, 359–375.
Jupp, P. E., and Mardia, K. V. (1979), “Maximum Likelihood Estimators for the Matrix Von Mises-Fisher and Bingham Distributions,” Annals of Statistics, 7, 599–606.
Khatri, C. G., and Mardia, K. V. (1977), “The Von Mises-Fisher Matrix Distribution in Orientation Statistics,” Journal of the Royal Statistical Society, Series B, 39, 95–106.
León, C. A., Massé, J.-C., and Rivest, L.-P. (2006), “A Statistical Model for Random Rotations,” Journal of Multivariate Analysis, 97, 412–430.
Mardia, K. V., and Jupp, P. E. (2000), Directional Statistics, Chichester: Wiley.
Matthies, S., Muller, J., and Vinel, G. W. (1988), “On the Normal Distribution in the Orientation Space,” Textures and Microstructures, 10, 77–96.
Miles, R. E. (1965), “On random rotations in R 3,” Biometrika, 52, 636–639.
Nikolayev, D. I., and Savyolova, T. I. (1997), “Normal Distribution on the Rotation Group SO(3),” Textures and Microstructures, 29, 201–233.
Prentice, M. J. (1986), “Orientation Statistics Without Parametric Assumptions,” Journal of the Royal Statistical Society, Series B, 48, 214–222.
Rivest, L.-P. (2001), “A Directional Model for the Statistical Analysis of Movement in Three Dimensions,” Biometrika, 88, 779–791.
— (2005), “A Correction for Axis Misalignment in the Joint Angle Curves Representing Knee Movement in Gait Analysis,” Journal of Biomechanics, 38, 1604–1611.
Rivest, L.-P., Baillargeon, S., and Pierrynowski, M. (2008), “A Directional Model for the Estimation of the Rotation Axes of the Ankle Joint,” Journal of the American Statistical Association, 103, 1060–1069.
Watson, G. S. (1983), Statistics on Spheres, New York: Wiley.
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Bingham, M.A., Nordman, D.J. & Vardeman, S.B. Bayes Inference for a Tractable New Class of Non-symmetric Distributions for 3-Dimensional Rotations. JABES 17, 527–543 (2012). https://doi.org/10.1007/s13253-012-0107-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13253-012-0107-9