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Original Articles

Bayesian Inference for a New Class of Distributions on Equivalence Classes of Three-Dimensional Orientations With Applications to Materials Science

, &
Pages 214-224 | Received 01 Oct 2012, Published online: 18 Apr 2016

References

  • Arnold, R., Jupp, P. (2013), Statistics of Orthogonal Axial Frames, Biometrika, 100, 571–586.
  • Bachmann, F., Hielscher, R., Jupp, P.E., Pantleon, W., Schaeben, H., Wegert, E. (2010), Inferential Statistics of Electron Backscatter Diffraction Data From Within Individual Crystalline Grains, Journal of Applied Crystallography, 43, 1338–1355.
  • Basson, F. (1997), Probabilities of Random Disorientation Axes in Cubic Polycrystals, Journal of Applied Crystallography, 30, 102–106.
  • ——— (1998), Response to Morawiec’s (1998) “Comment on Probabilities of Random Disorientation Axes in Cubic Polycrystals, Journal of Applied Crystallography, 31, 486–487.
  • Bingham, C. (1974), An Antipodally Symmetric Distribution on the Sphere, The Annals of Statistics, 2, 1201–1225.
  • Bingham, M.A., Nordman, D.J., Vardeman, S.B. (2009a), 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, 54, 1317–1327.
  • Bingham, M.A., Vardeman, S.B., Nordman, D.J. (2009b), Bayes One-Sample and One-Way Random Effects Analyses for 3-D Orientations with Applications to Materials Science, Bayesian Analysis, 4, 607–630.
  • Bunge, H.J. (1982), Texture Analysis in Materials Science, London: Butterworths.
  • Camano-Garcia, G. (2006), “Statistics on Stiefel Manifolds,” Ph.D. thesis, Ames, IA: Iowa State University.
  • Chang, T., and Bingham, C. (1996), Bayesian Spherical Regression,” Bayesian Analysis in Statistics and Econometrics, New York: Wiley.
  • Chang, T., Rivest, L.-P. (2001), M-Estimation for Location and Regression Parameters in Group Models: A Case Study Using Stiefel Manifolds, The Annals of Statistics, 29, 784–814.
  • Chikuse, Y. (2003), Statistics on Special Manifolds, New York: Springer.
  • Downs, T.D. (1972), Orientation Statistics, Biometrika, 59, 665–676.
  • Gelman, C.A., Carlin, J., Stern, H., and Rubin, D. (2004), Bayesian Data Analysis, Boca Raton: Chapman & Hall.
  • Handscomb, D.C. (1958), On the Random Disorientation of Two Cubes, Canadian Journal of Mathematics, 10, 85–88.
  • Hielscher, R., Schaeben, H., Siemes, H. (2010), Orientation Distribution Within a Single Hematite Crystal, Mathematical Geosciences, 42, 359–375.
  • Jupp, P.E., Mardia, K.V. (1979), Maximum Likelihood Estimators for the Matrix Von Mises–Fisher and Bingham Distributions, The Annals of Statistics, 7, 599–606.
  • Khatri, C.G., Mardia, K.V. (1977), The Von Mises–Fisher Matrix Distribution in Orientation Statistics, Journal of the Royal Statistical Society, Series B, 39, 95–106.
  • Kunze, K., Schaeben, H. (2004), The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis, Mathematical Geology, 36, 917–943.
  • ——— (2005), Ideal Patterns of Crystallographic Preferred Orientation and Their Representation by the von Mises - Fisher Matrix or Bingham Quaternion Distribution, Materials Science Forum, 495–497, 295–300.
  • León, C.A., Massé, J.-C., Rivest, L.-P. (2006), A Statistical Model for Random Rotations, Journal of Multivariate Analysis, 97, 412–430.
  • Mackenzie, J.K. (1958), Second Paper on Statistics Associated With the Random Disorientation of Cubes, Biometrika, 45, 229–240.
  • Mackenzie, J.K., Thomson, M.J. (1957), Some Statistics Associated with the Random Disorientation of Cubes, Biometrika, 44, 205–210.
  • Mardia, K., and Jupp, P. (2000), Directional Statistics ( vol. 494), New York: Wiley.
  • Matthies, S., Muller, J., Vinel, G.W. (1988), On the Normal Distribution in the Orientation Space, Textures and Microstructures, 10, 77–96.
  • Matthies, S., Vinel, G.W., and Helming, K. (1987), Standard Distribution in Texture Analysis: Maps for the Case of Cubic–Orthorhombic Symmetry ( vol. 1), Berlin: Akademie.
  • Miles, R. (1965), On Random Rotations in R3, Biometrika, 52, 636–639.
  • Morawiec, A. (1995), Misorientation-Angle Distribution of Randomly Oriented Symmetric Objects, Journal of Applied Crystallography, 28, 289–293.
  • ——— (1996), Distributions of Rotation Axes for Randomly Oriented Symmetric Objects, Journal of Applied Crystallography, 29, 164–169.
  • ——— (1998), Comment on “Probabilities of Random Disorientation Axes in Cubic Polycrystals by Basson, Journal of Applied Crystallography, 31, 486–486.
  • ——— (2004), Orientations and Rotations: Computations in Crystallographic Textures, Berlin: Springer.
  • Prentice, M.J. (1986), Orientation Statistics Without Parametric Assumptions, Journal of the Royal Statistical Society, Series B, 48, 214–222.
  • Qiu, Y., Nordman, D., Vardeman, S. (2014), One-Sample Bayes Inference for Existing Symmetric Distributions on 3-d Rotations, Computational Statistics and Data Analysis, to appear.
  • Rancourt, D., Rivest, L.-P., Asselin, J. (2000), Using Orientation Statistics to Investigate Variations in Human Kinematics, Journal of the Royal Statistical Society, Series C, 49, 81–94.
  • Randle, V. (2003), Microtexture Determination and Its Applications, London: Maney for the Institute of Materials, Minerals and Mining.
  • Savyolova, T.I., Nikolayev, D.I. (1995), Normal Distribution on the Rotation Group SO(3), Textures and Microstructures, 29, 201–233.
  • Watson, G.S. (1983), Statistics on Spheres, New York: Wiley.

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