A Multivariate Hidden Markov Model for the Identification of Sea Regimes from Incomplete Skewed and Circular Time Series

The identification of sea regimes from environmental multivariate times series is complicated by the mixed linear–circular support of the data, by the occurrence of missing values, by the skewness of some variables, and by the temporal autocorrelation of the measurements. We address these issues simultaneously by a hidden Markov approach, and segment the data into pairs of toroidal and skew-elliptical clusters by means of the inferred sequence of latent states. Toroidal clusters are defined by a class of bivariate von Mises densities, while skew-elliptical clusters are defined by mixed linear models with positive random effects. The core of the classification procedure is an EM algorithm accounting for missing measurements, unknown cluster membership, and random effects as different sources of incomplete information. Moreover, standard simulation routines allow for the efficient computation of bootstrap standard errors. The proposed procedure is illustrated for a multivariate marine time series, and identifies a number of wintertime regimes in the Adriatic Sea.

[1]  H. Teicher.  Identifiability of Mixtures of Product Measures , 1967 .

[2]  S. Yakowitz,et al.  On the Identifiability of Finite Mixtures , 1968 .

[3]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[4]  R. Redner,et al.  Mixture densities, maximum likelihood, and the EM algorithm , 1984 .

[5]  D E Weeks,et al.  Efficient computation of lod scores: genotype elimination, genotype redefinition, and hybrid maximum likelihood algorithms , 1989, Annals of human genetics.

[6]  Odd M. Faltinsen,et al.  Sea loads on ships and offshore structures , 1990 .

[7]  Peter Guttorp,et al.  A Hidden Markov Model for Space‐Time Precipitation , 1991 .

[8]  P C Molenaar,et al.  Confidence intervals for hidden Markov model parameters. , 2000, The British journal of mathematical and statistical psychology.

[9]  Gérard Govaert,et al.  Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[11]  Peter C. M. Molenaar,et al.  Fitting hidden Markov models to psychological data , 2002, Sci. Program..

[12]  Harshinder Singh,et al.  Probabilistic model for two dependent circular variables , 2002 .

[13]  Christophe Biernacki,et al.  Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models , 2003, Comput. Stat. Data Anal..

[14]  S. Sahu,et al.  A new class of multivariate skew distributions with applications to bayesian regression models , 2003 .

[15]  Kang-Ren Jin,et al.  Case Study: Modeling of Sediment Transport and Wind-Wave Impact in Lake Okeechobee , 2004 .

[16]  Eric Moulines,et al.  Inference in hidden Markov models , 2010, Springer series in statistics.

[17]  A. Sterl,et al.  A New Nonparametric Method to Correct Model Data: Application to Significant Wave Height from the ERA-40 Re-Analysis , 2005 .

[18]  C. Guedes Soares,et al.  Analysis of sea waves and wind from X-band radar , 2005 .

[19]  Nikolaos Limnios,et al.  Maximum likelihood estimation for hidden semi-Markov models , 2006 .

[20]  A. Munk,et al.  Hidden Markov models for circular and linear-circular time series , 2006, Environmental and Ecological Statistics.

[21]  Marc Prevosto,et al.  Survey of stochastic models for wind and sea state time series , 2007 .

[22]  K. Mardia,et al.  Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data , 2007, Biometrics.

[23]  Kanti V. Mardia,et al.  A multivariate von mises distribution with applications to bioinformatics , 2008 .

[24]  Jan Bulla,et al.  Computational issues in parameter estimation for stationary hidden Markov models , 2008, Comput. Stat..

[25]  K. Shimizu,et al.  Dependent models for observations which include angular ones , 2008 .

[26]  H. Kapitza,et al.  Interaction of waves, currents and tides, and wave-energy impact on the beach area of Sylt Island , 2009 .

[27]  W. Zucchini,et al.  Hidden Markov Models for Time Series: An Introduction Using R , 2009 .

[28]  Tsung I. Lin,et al.  Maximum likelihood estimation for multivariate skew normal mixture models , 2009, J. Multivar. Anal..

[29]  Luigi Cavaleri,et al.  Wind and wave predictions in the Adriatic Sea , 2009 .

[30]  Marc G. Genton,et al.  Multivariate log‐skew‐elliptical distributions with applications to precipitation data , 2009 .

[31]  Gilles Celeux,et al.  Combining Mixture Components for Clustering , 2010, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[32]  L. Hamilton.  Characterising spectral sea wave conditions with statistical clustering of actual spectra , 2010 .

[33]  Jan Bulla,et al.  hsmm - An R package for analyzing hidden semi-Markov models , 2010, Comput. Stat. Data Anal..

[34]  Qiang Zhang,et al.  Multivariate Discrete Hidden Markov Models for Domain-Based Measurements and Assessment of Risk Factors in Child Development , 2010, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[35]  Salvatore Ingrassia,et al.  Degeneracy of the EM algorithm for the MLE of multivariate Gaussian mixtures and dynamic constraints , 2011, Comput. Stat. Data Anal..

[36]  R. Langrock,et al.  Hidden Markov models with arbitrary state dwell-time distributions , 2011, Comput. Stat. Data Anal..

[37]  Adrian Wing-Keung Law,et al.  Wave-induced drift of small floating objects in regular waves , 2011 .

[38]  A. Maruotti.  Mixed Hidden Markov Models for Longitudinal Data: An Overview , 2011 .

[39]  Gordon Reikard,et al.  Forecasting ocean waves: Comparing a physics-based model with statistical models , 2011 .

[40]  Pierre Ailliot,et al.  Markov-switching autoregressive models for wind time series , 2012, Environ. Model. Softw..

[41]  F. Lagona,et al.  Model-based clustering of multivariate skew data with circular components and missing values , 2012 .

[42]  Victor H. Lachos,et al.  Multivariate mixture modeling using skew-normal independent distributions , 2012, Comput. Stat. Data Anal..

[43]  F. Lagona,et al.  Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data , 2013 .

[44]  Francesco Lagona,et al.  A Latent-Class Model for Clustering Incomplete Linear and Circular Data in Marine Studies , 2021 .