Pseudo-likelihood Estimator of the Bivariate Von-Mises Cosine Model

Document Type : Original Paper


Department of Statistics, University of Tarbiat Modares


Directional statistics are very useful tools to model the phenomenon that are characterized by the angles. Recently, various disciplines including biology, astronomy, meteorology and bioinformatics have paid attention to use these distributions. Particularly, it was shown in biological researches that there are two pair angles describing, relatively, the complete geometrical and spatial structures of a protein in the three dimensional space. There is a distribution, called bivariate Von-Mises, to represent the position of the atoms based upon the values of these angles in a probabilistic manner. In this paper, considering an especial case of this density (cosine model), the properties of distribution including the numbers of modes and its approximation by the bivariate normal distribution are first studied. Then, to estimate the parameters using the pseudo-likelihood method is described. The theoretical materials are evaluated in simulation studies and then the application of the cosine model in a real example is presented.


Main Subjects

[1] Mardia‎, ‎K‎. ‎V and Jupp‎, ‎P.‎ ‎(2000),‎ Directional Statistics‎, ‎John Wiley and Sons‎, ‎New York‎.
[2] Mardia‎, ‎K‎. ‎V.‎, ‎Taylor‎, ‎C‎. ‎C‎. ‎and Subramanian‎, ‎G‎. ‎K‎. ‎(2003)‎, Applications of Circular Distribution to Conformational Angles in Proteins‎, In  Proceedings of the 22nd LASR Workshop, edited byKV Mardia‎, ‎RG Aykroyd and MJ‎ Langdon, 149-152. Leeds University Press‎.
[3] Mardia‎, ‎K‎. ‎V.‎, ‎Kent‎, ‎J‎. ‎T‎. ‎and Taylor‎, ‎C‎. ‎C‎. ‎(2010)‎, Matching Unlabelled‎ Configurations and Protein Bioinformatics. Research Report STAT10-01,University of Leeds.
[4] Mardia‎, ‎K‎. ‎V‎. ‎(1972)‎, Statistics of Directional Data‎, ‎Academic Press‎, ‎London‎.
[5] Mardia‎, ‎K‎. ‎V‎. ‎(1975)‎, ‎Statistics of Directional Data‎, Journal of the Royal Statistical Society Series B (Methodological), 37‎, ‎349-393‎.
[6] Singh‎, ‎H.‎, ‎Hnizdo‎, ‎V‎. ‎and Demchuk‎, ‎E‎. ‎(2002)‎, Probabilistic Model for Two Dependent Circular Variable‎, Bioinformatics‎, 89‎, ‎719-723‎.
[7] Mardia‎, ‎K‎. ‎V.‎, ‎Hughes‎, ‎G‎., ‎ Taylor‎, ‎C‎. ‎C‎.  ‎and Subramanian‎. ‎G‎. ‎K‎. ‎(2007b)‎, Bivariate Von Mises Densities for Angular Data with‎ Application to Protein Bioinformatics‎, Annals of Statistics‎, 35‎, ‎166-180‎.
[8] Mardia‎, ‎K.V.‎, ‎Hughes‎, ‎G.‎, ‎Taylor‎, ‎C.C‎. ‎and Singh‎, ‎H‎. ‎(2008)‎. ‎A Multivariate Von‎ Mises Distribution‎ with Applications to Bioinformatics‎, Canadian Journal of Statistics‎, 36‎, ‎99-109‎.
[9] Mardia‎, ‎K‎. ‎V‎. ‎and Voss‎, ‎J‎. ‎(2014)‎, ‎Some Fundamental Properties of a Multivariate‎ Von Mises Distribution‎, Communications in Statistics-Theory and Methods, 43‎, ‎1132-1144‎.
[10] Hamelryck‎, ‎T.‎, ‎Mardia‎, ‎K‎. ‎V‎. ‎and Ferkinghoff-Borg‎, ‎J‎. ‎(2012)‎, ‎Bayesian Methodsin Structural Bioinformatics,Statistics for Biology and Health, ‎Springer-Verlag‎, ‎Heidelberg‎.
[11] Abramowitz‎, ‎M and Stegun‎, ‎I‎. ‎A‎. ‎(1965)‎, Handbook of Mathematical Functions‎‎, ‎Dover.
[12] Besag‎, ‎J‎. ‎(1975)‎, ‎Statistical Analysis of Non-Lattice Data‎, The Statistician‎, 24, ‎179-195‎.
[13] Mardia‎, ‎K‎. ‎V.‎, ‎Kent‎, ‎J‎. ‎T.‎, ‎Hughes‎, ‎G‎. ‎and Taylor‎, ‎C‎. ‎C‎. ‎(2009)‎, Maximum Likelihood Estimation Using Composite Likelihoods for‎ Closed Exponential Families‎, Biometrika‎, ‎21-30‎.
[14] Robert‎, ‎C‎. ‎P‎. ‎and Casella‎, ‎G‎. ‎(2004)‎, Monte Carlo Statistical Methods‎, ‎Springer‎, New York‎.
[15] Ross‎, ‎S‎, ‎M‎. ‎(2006)‎, Simulation‎, ‎(4th edition‎), ‎Academic Press‎, ‎Amsterdam‎.
[16] Bourne E‎. ‎B‎. ‎and Weissig H‎. ‎E‎. ‎(2009)‎, Structural Bioinformatics‎, ‎(2nd edition)‎, ‎John Wiley and Sons‎, ‎New Jersey‎.
[17] Ramakrishan‎, ‎C‎. ‎and Ramachandran‎, ‎G‎. ‎N‎. ‎(1965)‎. ‎Stereochemical Criteria for‎ Polypeptide and Protein Chain Conformation, Biophysical Journal,  5, ‎909-933‎.‎
‎[18] Mardia‎, ‎K‎. ‎V.‎, ‎Taylor‎, ‎C‎. ‎C‎. ‎and Subramanian‎, ‎G‎. ‎K‎. ‎(2007a)‎, Protein Bioinformatics and Mixtures of Bivariate Von Mises Distribution for‎ Angular Data‎, Biometrics‎, 63‎, ‎502-512‎.