On estimation of parameters for bivariate von Mises and toroidal wrapped Gaussian to-rus distributions: A simulation study
DOI:
https://doi.org/10.37376/ljst.v12i1.7060Keywords:
Bivariate von Mises Torus Distribution, Toroi-dal Wrapped Normal Torus Distribution, Max-imum Likelihood, Maximum Pseudolikelihood, Trigonometric Moments’ MethodAbstract
The bivariate von Mises sine (BvMST) and toroidal wrapped Gaussian or normal (TWNT) distributions are defined on torus and they are applicable to statistical directional analysis of orientational data in 2D. The multivariate wrapped normal distribution in -dimension has both multivariate marginal wrapped normal models and bivariate marginal wrapped normal distributions, and thus has a theoretical merit. One drawback of the wrapped normal torus (WNT) distribution is that, unlike the sine and cosine models, it does not form an exponential family and thus additional statistical and mathematical care is required to tackle the problem. The maximum likelihood (ML), maximum pseudolikelihood (MPL) and moments’ (M) methods are numerically compared with respect to their efficiency rates for estimating the corresponding parameters of the BvMST distribution. Both MPL and M methods provide good estimates relative to the estimates of the ML method under an acceptance – rejection simulation scheme with Bingham-Angular Central Gaussian (BACG) distribution as an envelope. A proposed trigonometric moments’ (TM) method is derived and utilized for the purpose of parameters estimation of the TWNT distribution as compared to the traditional maximum likelihood technique. It provides efficient estimates either in small or large simulated random samples.
Downloads
References
Baba, Y. (1981) ‘Statistics of angular data: wrapped normal distribution model’, Proceedings of the Institute of Statistical Mathematics, 28, 41 - 54.
Batschelet, E. (1981) Circular Statistics in Biology, Second Edition, New York: Academic Press.
Besag, J. (1975) ‘Statistical analysis of non-lattice data’, The Statistician, 24, 179 - 195.
Breckling, J. (1989) The Analysis of Directional Time Series: Applications to Wind Speed and Direction, First Edition, Berlin: Springer-Verlag.
Coles, S. (1998) ‘Inference for circular distributions and process’, Statistics and Computing, 8, 105 -113.
Davison, A. C. (2003) Statistical Models, First Edition, Cambridge University Press, UK.
Fisher, N. I. (1987) ‘Problem with the current definition of the standard deviation of wind direction’, Journal of Climate and Applied Meteorology, 26, 1522-1529.
Gallier, J. and Xu, D. (2013) A Guide to the Classification Theorem for Compact Surfaces: Geometry and Computing, First Edition, Springer, Heidelberg.
Jammalamadaka, S. R. and SenGupta, A. (2001) Topics in Circular Statistics, Second Edition, Singapore: World Scientific.
Jammalamadaka, S. R. and Sarma, Y. (1988) ‘A Correlation Coefficient for Angular Variables’, Statistical Theory and Data Analysis II, 349– 364.
Johnson, R. A. and Wehrly, T. (1977) ‘Measures and models for angular correlation and angular-linear correlation’, Journal of Royal Statistical Society, 39 (2), 222 - 229.
Johnson, R. A. and Wehrly, T. (1978) ‘Some angular-linear distributions and related regression models’, Journal of the American Statistical Association, 73, 602- 606.
Kent, J. T., Ganeiber, A. M. and Mardia, K. V. (2018) ‘A New Unified Approach for the Simulation of a Wide Class of Directional Distributions’, Journal of Computational and Graphical Statistics, 27(2), 291-301, Taylor & Francis Group, American Statistical Association, USA.
Kent, J. T., Mardia, K. V. and Taylor, C. C. (2008) ‘Modelling strategies for bivariate circular data’, In Gusnanto, A., Barber, S., Baxter, P.D. and Mardia, K. V. Editors, The Art and Science of Statistical Bioinformatics, 70- 74. University of Leeds Press, UK.
Kent, J. T., Mardia, K. V. and Taylor, C. C. (2009) ‘Principal component analysis for the wrapped normal torus model’, In A. Fallaize, C. J., Holmes, P. S., Mardia, K.V., Gusnanto, A., Kent, J. T. and Voss, J. Editors, Statistical Tools for Challenges in Bioinformatics, 39 - 41. University of Leeds Press, UK.
Mardia, K. V. (1972) Statistics of Directional Data, First Edition, London: Academic Press.
Mardia, K. V. (1975) ‘Statistics of directional data (with discussion)’, Journal of Royal Statistical Society, B(37), 349 – 393.
Mardia, K. V. and Jupp, P. E. (2000) Directional Statistics, First Edition, New York, John Wiley and Sons.
Mardia, K. V., Hughes, G., Taylor, C. C. and Singh, H. (2008) ‘A multivariate von Mises distribution with applications to bioinformatics’, The Canadian Journal of Statistics, 36(1), 99 – 109.
Mardia, K. V. and Patrangenaru, V. (2005) ‘Directions and projective shapes. Annals of Statistics. 33, 1666 – 1699.
Mardia, K.V., Taylor, C.C., and Subramaniam, G.K. (2007) ‘Protein Bioinformatics and Mixture of Bivariate von Mises distributions for Angular data’, Biometrics, The international Biometric Society, 63, 505 – 512.
Mardia, K. V., Taylor, C. C. and Subramaniam. G. K. (2009) ‘Bivariate von Mises densities for angular data with applications to protein bioinformatics’, Annals of Statistics, 35, 166 – 180.
Nodehi, A., Golaizadeh, M., Maadooliat, M. and Agostinelli, C. (2018) ‘Estimation of Multivariate Wrapped Models for Data in Torus’, arXiv: 1811.06007v1.
Pinkall, U. (1986) ‘Cyclides of Dupin’, In Proceedings of Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer), Braunschweig, Germany: Vieweg, 28 - 30.
Roy, A., Parui, S. K. and Roy, U. (2014) ‘SWGMM: A Semi-wrapped Gaussian Mixture Model for Clustering of Circular linear Data’, Pattern Analysis and Applications, 112 - 120.
Singh, H., Hnizdo, V. and Demchuk, E. (2002) ‘Probabilistic model for two dependent circular variables’, Biometrika, 89, 719 - 723
Wood, A. T. A. (1993) ‘Estimation of the concentration parameters of the fisher matrix distribution on SO(3) and the Bingham distribution on S^q, q≥2’, Australian Journal of Statistics, 35, 69 –79.
Zhan, X., Ma, T., Liu, S. and Shimizu, K. (2019) ‘On circular correlation for data on the torus’, Springer Link, 60, 1827 – 1847.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Libyan Journal of Science &Technology
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
