Spline‐based nonparametric inference in general state‐switching models

State‐switching models combine immense flexibility with relative mathematical simplicity and computational tractability and, as a consequence, have established themselves as general‐purpose models for time series data. In this paper, we provide an overview of ways to use penalized splines to allow for flexible nonparametric inference within state‐switching models, and provide a critical discussion of the use of corresponding classes of models. The methods are illustrated using animal acceleration data and energy price data.

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

[2]  R. Huisman,et al.  Regime Jumps in Electricity Prices , 2001 .

[3]  I. Macdonald,et al.  Numerical Maximisation of Likelihood: A Neglected Alternative to EM? , 2014 .

[4]  H. Holzmann,et al.  Nonparametric identification of hidden Markov models , 2014 .

[5]  S Schliehe-Diecks,et al.  On the application of mixed hidden Markov models to multiple behavioural time series , 2012, Interface Focus.

[6]  David Raubenheimer,et al.  Modeling Time Series of Animal Behavior by Means of a Latent‐State Model with Feedback , 2008, Biometrics.

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

[8]  Roland Langrock,et al.  Nonparametric inference in hidden Markov models using P‐splines , 2013, Biometrics.

[9]  Iain L. MacDonald,et al.  Hidden Markov Models for Time Series: An Introduction Using R (2nd Edition) , 2017 .

[10]  Massimo Piccardi,et al.  Hidden Markov Models with Kernel Density Estimation of Emission Probabilities and their Use in Activity Recognition , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[11]  G. Heller,et al.  Flexible Regression and Smoothing: Using Gamlss in R , 2017 .

[12]  Hajo Holzmann,et al.  Hidden Markov models with state-dependent mixtures: minimal representation, model testing and applications to clustering , 2015, Stat. Comput..

[13]  Roland Langrock,et al.  Semiparametric stochastic volatility modelling using penalized splines , 2013, Comput. Stat..

[14]  Ci-Ren Jiang,et al.  A Functional Approach to Deconvolve Dynamic Neuroimaging Data , 2014, Journal of the American Statistical Association.

[15]  K. Mauff,et al.  Residency, Habitat Use and Sexual Segregation of White Sharks, Carcharodon carcharias in False Bay, South Africa , 2013, PloS one.

[16]  L. Baum,et al.  A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains , 1970 .

[17]  Jan Bulla,et al.  Stylized facts of financial time series and hidden semi-Markov models , 2006, Comput. Stat. Data Anal..

[18]  M. Bebbington.  Identifying volcanic regimes using Hidden Markov Models , 2007 .

[19]  A. Maruotti,et al.  A Multivariate Hidden Markov Model for the Identification of Sea Regimes from Incomplete Skewed and Circular Time Series , 2012 .

[20]  A. Mysterud,et al.  Seasonality, weather and climate affect home range size in roe deer across a wide latitudinal gradient within Europe. , 2013, The Journal of animal ecology.

[21]  S. Wood,et al.  Generalized Additive Models: An Introduction with R , 2006 .

[22]  S. Wood.  Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models , 2004 .

[23]  Juliet S Lamb,et al.  Influence of density‐dependent competition on foraging and migratory behavior of a subtropical colonial seabird , 2017, Ecology and evolution.

[24]  Anders Nielsen,et al.  TMB: Automatic Differentiation and Laplace Approximation , 2015, 1509.00660.

[25]  Roland Langrock,et al.  Analysis of animal accelerometer data using hidden Markov models , 2016, 1602.06466.

[26]  Sylvia Frühwirth-Schnatter,et al.  Finite Mixture and Markov Switching Models , 2006 .

[27]  Roland Langrock,et al.  Markov-switching generalized additive models , 2014, Stat. Comput..

[28]  Tatiana Xifara,et al.  A coupled hidden Markov model for disease interactions , 2012, Journal of the Royal Statistical Society. Series C, Applied statistics.

[29]  Jiqiang Guo,et al.  Stan: A Probabilistic Programming Language. , 2017, Journal of statistical software.

[30]  R. Altman.  Mixed Hidden Markov Models , 2007 .

[31]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[32]  Göran Kauermann,et al.  Density estimation and comparison with a penalized mixture approach , 2012, Comput. Stat..

[33]  Michael Eichler,et al.  Fitting semiparametric Markov regime-switching models to electricity spot prices , 2012 .

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

[35]  D L Borchers,et al.  Using Hidden Markov Models to Deal with Availability Bias on Line Transect Surveys , 2013, Biometrics.

[36]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[37]  Chang-Jin Kim,et al.  Estimation of Markov regime-switching regression models with endogenous switching , 2003 .

[38]  Giampiero Marra,et al.  Practical variable selection for generalized additive models , 2011, Comput. Stat. Data Anal..

[39]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[40]  G. Kauermann.  A note on smoothing parameter selection for penalized spline smoothing , 2005 .

[41]  Stéphane Robin,et al.  Hidden Markov Models with mixtures as emission distributions , 2012, Statistics and Computing.

[42]  S. Wood,et al.  Smoothing Parameter and Model Selection for General Smooth Models , 2015, 1511.03864.

[43]  Roland Langrock,et al.  Combining hidden Markov models for comparing the dynamics of multiple sleep electroencephalograms , 2013, Statistics in medicine.

[44]  S. Wood.  Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models , 2011 .

[45]  Y. Guédon.  Estimating Hidden Semi-Markov Chains From Discrete Sequences , 2003 .

[46]  Roland Langrock,et al.  Multi-scale Modeling of Animal Movement and General Behavior Data Using Hidden Markov Models with Hierarchical Structures , 2017, Journal of Agricultural, Biological and Environmental Statistics.

[47]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[48]  Jennifer Pohle,et al.  Selecting the Number of States in Hidden Markov Models: Pragmatic Solutions Illustrated Using Animal Movement , 2017, Journal of Agricultural, Biological and Environmental Statistics.

[49]  Jennifer Pohle,et al.  Selecting the Number of States in Hidden Markov Models - Pitfalls, Practical Challenges and Pragmatic Solutions , 2017 .

[50]  Roland Langrock,et al.  moveHMM: an R package for the statistical modelling of animal movement data using hidden Markov models , 2016 .

[51]  Thomas Kneib,et al.  Gradient boosting in Markov-switching generalized additive models for location, scale, and shape , 2017 .

[52]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

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