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Estimation of parameters in multivariate wrapped models for data on a p-torus

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Abstract

Multivariate circular observations, i.e. points on a torus arise frequently in fields where instruments such as compass, protractor, weather vane, sextant or theodolite are used. Multivariate wrapped models are often appropriate to describe data points scattered on p-dimensional torus. However, the statistical inference based on such models is quite complicated since each contribution in the log-likelihood function involves an infinite sum of indices in \({\mathbb {Z}}^p\), where p is the dimension of the data. To overcome this problem, for moderate dimension p, we propose two estimation procedures based on Expectation-Maximisation and Classification Expectation-Maximisation algorithms. We study the performance of the proposed techniques on a Monte Carlo simulation and further illustrate the advantages of the new procedures on three real-world data sets.

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References

  • Agostinelli C (2007) Robust estimation for circular data. Comput Stat Data Anal 51(12):5867–5875

    Article  MathSciNet  Google Scholar 

  • Agostinelli C, Lund U (2017) R package circular: circular statistics (version 0.4-93). CA: Department of Mathematics, University of Trento, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA. https://r-forge.r-project.org/projects/circular/

  • Agostinelli C, Leung A, Yohai VJ, Zamar RH (2015) Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination. TEST 24(3):441–461

    Article  MathSciNet  Google Scholar 

  • Baba Y (1981) Statistics of angular data: wrapped normal distribution model. Proc Inst Stat Math 28:41–54 (in Japanese)

    MathSciNet  MATH  Google Scholar 

  • Batschelet E (1981) Circular statistics in biology. Academic Press, NewYork

    MATH  Google Scholar 

  • Breckling J (1989) The analysis of directional time series: applications to wind speed and direction. Lecture notes in statistics, vol 61. Springer, Berlin

    Book  Google Scholar 

  • Celeux G, Govaert G (1992) A classification EM algorithm for clustering and two stochastic versions. Comput Stat Data Anal 14:315–332

    Article  MathSciNet  Google Scholar 

  • Coles S (1998) Inference for circular distributions and processes. Stat Comput 8:105–113

    Article  Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc B 39(1):1–38

    MathSciNet  MATH  Google Scholar 

  • Eltzner B, Huckermann S, Mardia KV (2018) Torus principal component analysis with applications to RNA structure. Ann Appl Stat (in press)

  • Ferrari C (2009) The wrapping approach for circular data bayesian modeling. Ph.D. thesis, Alma Mater Studiorum Universit di Bologna. Dottorato di ricerca in Metodologia statistica per la ricerca scientifica, 21 Ciclo

  • Fisher NI (1987) Problem with the current definition of the standard deviation of wind direction. J Clim Appl Meteorol 26:1522–1529

    Article  Google Scholar 

  • Fisher NI, Lee AJ (1994) Time series analysis of circular data. J R Stat Soc B 56:327–339

    MathSciNet  MATH  Google Scholar 

  • Jammalamadaka SR, SenGupta A (2001) Topics in circular statistics, multivariate analysis, vol 5. World Scientific, Singapore

    Book  Google Scholar 

  • Johnson RA, Wehrly T (1978) Some angular-linear distributions and related regression models. J Am Stat Assoc 73:602–606

    Article  MathSciNet  Google Scholar 

  • Kent JT (1978) Limiting behaviour of the von Mises-Fisher distribution. Math Proc Camb Philos Soc 84:531–536

    Article  MathSciNet  Google Scholar 

  • Mardia KV (1972) Statistics of directional data. Academic Press, London

    MATH  Google Scholar 

  • Mardia KV (2010) Bayesian analysis for bivariate von Mises distributions. J Appl Stat 37:515–528

    Article  MathSciNet  Google Scholar 

  • Mardia KV, Jupp PE (2000) Directional statistics. Wiley, New York

    MATH  Google Scholar 

  • Mardia KV, Voss J (2014) Some fundamental properties of a multivariate von Mises distribution. Commun Stat Theory Methods 43:1132–1144

    Article  MathSciNet  Google Scholar 

  • Mardia KV, Taylor CC, Subramaniam GK (2007) Protein bioinformatics and mixtures of bivariate von Mises distributions for angular data. Biometrics 63:505–512

    Article  MathSciNet  Google Scholar 

  • Mardia KV, Hughes G, Taylor CC, Singh H (2008) A multivariate von Mises distribution with applications to bioinformatics. Can J Stat 1:99–109

    Article  MathSciNet  Google Scholar 

  • Najibi SM, Maadooliat M, Zhou L, Huang JZ, Gao X (2017) Protein structure classication and loop modeling using multiple Ramachandran distributions. Comput Struct Biotechnol J 15:243–254

    Article  Google Scholar 

  • Oldfield TJ, Hubbard RE (1994) Analysis of \(C_{\alpha }\) geometry in protein structures. Proteins 18:324–337

    Article  Google Scholar 

  • Pinheiro JC, Bates DM (1996) Unconstrained parameterizations for variance–covariance matrices. Stat Comput 6(3):289–296

    Article  Google Scholar 

  • R Core Team (2019) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  • Ravindran P, Ghosh S (2011) Bayesian analysis of circular data using wrapped distributions. J Stat Theory Pract 5:547–561

    Article  MathSciNet  Google Scholar 

  • Stephens MA (1963) Random walk on a circle. Biometrika 50:385–390

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is supported in part by a grant BS-1395-01-07 from the Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.

The authors thank Stephan Huckemann and Benjamin Eltzner for providing the RNA data set. We would also like to thank the editor, and two referees for their constructive and thoughtful comments which helped us tremendously in improving the manuscript.

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Correspondence to Mousa Golalizadeh.

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Nodehi, A., Golalizadeh, M., Maadooliat, M. et al. Estimation of parameters in multivariate wrapped models for data on a p-torus. Comput Stat 36, 193–215 (2021). https://doi.org/10.1007/s00180-020-01006-x

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