For multivariate time series driven by underlying states, hidden Markov models (HMMs) constitute a powerful framework which can be flexibly tailored to the situation at hand. However, in practice, it can be challenging to choose an adequate family of distributions for modelling the multivariate observation vectors conditional on the states. For example, the marginal data distribution may not immediately reveal the within-state distributional form, and also the different variables observed over time may operate on different supports, rendering the common approach of using a multivariate normal distribution inadequate. Here, we explore a nonparametric estimation of the state-dependent distributions within a bivariate HMM based on tensor-product B-splines. In two simulation studies, we show the feasibility of our modelling approach and demonstrate the potential pitfalls of inappropriate choices of parametric distributions. To illustrate the practical applicability, we present a case study using an HMM to model the bivariate time series comprising the lengths and angles of goalkeeper passes during the UEFA EURO 2020, investigating the effect of match dynamics on the teams’ tactics.
AilliotP, ThompsonC and ThomsonP (2009) Space–time modelling of precipitation by using a hidden Markov model and censored Gaussian distributions. Journal of the Royal Statistical Society: Series C (Applied Statistics), 58, 405–26.
2.
AlaaAM and Van Der SchaarM (2018) A hidden absorbing semi-Markov model for informatively censored temporal data: learning and inference. The Journal of Machine Learning Research, 19, 108–69.
3.
AlexandrovichG, HolzmannH and LeisterA (2016) Nonparametric identification and maximum likelihood estimation for hidden Markov models. Biometrika, 103, 423–34.
4.
AltmanR (2007) Mixed hidden Markov models: an extension of the hidden Markov model to the longitudinal data setting. Journal of the American Statistical Association, 102, 201–10.
5.
BauerP and AnzerG (2021) Data-driven detection of counterpressing in professional football. Data Mining and Knowledge Discovery, 35, 2009–49.
6.
BeumerLT, PohleJ, SchmidtNM, ChimientiM, DesforgesJ-P, HansenLH, LangrockR, PedersenSH, StelvigM and van BeestFM (2020) An application of upscaled optimal foraging theory using hidden Markov modelling: year-round behavioural variation in a large arctic herbivore. Movement Ecology, 8, 1–16.
7.
BoerMP (2023) Tensor product P-splines using a sparse mixed model formulation. Statistical Modelling, 23, 465–79.
8.
BrunelN and PieczynskiW (2005) Unsupervised signal restoration using hidden Markov chains with copulas. Signal Processing, 85, 2304–15.
9.
BullaJ, LagonaF, MaruottiA and PiconeM (2012) A multivariate hidden Markov model for the identification of sea regimes from incomplete skewed and circular time series. Journal of Agricultural, Biological, and Environmental Statistics, 17, 544–67.
10.
BureauA, ShiboskiS and HughesJP (2003) Applications of continuous time hidden Markov models to the study of misclassified disease outcomes. Statistics in Medicine, 22, 441–62.
11.
CeleuxG and DurandJ-B (2008) Selecting hidden Markov model state number with cross-validated likelihood. Computational Statistics, 23, 541–64.
12.
ChenS and FinkenstädtB (2024) Bayesian spline-based hidden Markov models with applications to actimetry data and sleep analysis. Journal of the American Statistical Association, 119, 2833–843.
13.
De GooijerJG, HenterGE and YuanA (2022) Kernel-based hidden Markov conditional densities. Computational Statistics & Data Analysis, 169, 107431. doi:https://doi.org/10.1016/j.csda.2022.107431. URLhttps://www.sciencedirect.com/science/article/pii/S0167947322000111.
14.
DecroosT and DavisJ (2019) Player vectors: characterizing soccer players’ playing style from match event streams. In Joint european conference on machine learning and knowledge discovery in databases, pages 569–84. Berlin: Springer.
15.
DecroosT, Van HaarenJ and DavisJ (2018) Automatic discovery of tactics in spatio-temporal soccer match data. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining, pages 223–32.
16.
DecroosT, van RoyM and DavisJ (2020) Soccermix: Representing soccer actions with mixture models. In Joint european conference on machine learning and knowledge discovery in databases, pages 459–74. Springer.
17.
DeRuiterS, LangrockR, SkirbutasT, GoldbogenJ, CalambokidisJ, FriedlaenderA and SouthallBL (2017) A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure. The Annals of Applied Statistics, 11, 362–92.
18.
EilersP, CurrieI and DurbánM (2006) Fast and compact smoothing on large multidimensional grids. Computational Statistics & Data Analysis, 50, 61–76.
19.
EilersPHC and MarxBD (1996) Flexible smoothing with B-splines and penalties. Statistical Science, 11, 89–121. doi: 10.1214/ss/1038425655. URLhttps://doi.org/10.1214/ss/1038425655.
20.
FahrmeirL, KneibT, LangS and MarxB (2013) Regression: models, methods and applications. Berlin: Springer-Verlag.
21.
FiecasM, FrankeJ, von SachsR and Tadjuidje KamgaingJ (2017) Shrinkage estimation for multivariate hidden Markov models. Journal of the American Statistical Association, 112, 424–35.
22.
FraleyC and RafteryAE (2002) Model-based clustering, discriminant analysis, and density estimation. Journal of the American statistical Association, 97, 611–31.
23.
HärdleWK, OkhrinO and WangW (2015) Hidden Markov structures for dynamic Copulae. Econometric Theory, 31, 981–1015.
24.
JacksonCH, SharplesLD, ThompsonSG, DuffySW and CoutoE (2003) Multistate Markov models for disease progression with classification error. Journal of the Royal Statistical Society: Series D (The Statistician), 52, 193–209.
25.
KarlisD and XekalakiE (2003) Choosing initial values for the EM algorithm for finite mixtures. Computational Statistics & Data Analysis, 41, 577–90.
26.
KoslikJ-O (2024) Efficient smoothness selection for nonparametric Markov-switching models via quasi restricted maximum likelihood. arXiv preprint arXiv:2411.11498.
27.
LanchantinP, Lapuyade-LahorgueJ and PieczynskiW (2011) Unsupervised segmentation of randomly switching data hidden with non-Gaussian correlated noise. Signal Processing, 91, 163–75.
28.
LangrockR, AdamT, Leos-BarajasV, MewsS, MillerDL and PapastamatiouYP (2018) Spline-based nonparametric inference in general state-switching models. Statistica Neerlandica, 72, 179–200.
29.
LangrockR, KneibT, SohnA and DeRuiterS (2015) Nonparametric inference in hidden Markov models using P-splines. Biometrics, 71, 520–28.
30.
LangrockR and ZucchiniW (2011) Hidden Markov models with arbitrary state dwell-time distributions. Computational Statistics & Data Analysis, 55, 715–24.
31.
MaruottiA, PunzoA and BagnatoL (2019) Hidden Markov and semi-Markov models with multivariate leptokurtic-normal components for robust modeling of daily returns series. Journal of Financial Econometrics, 17, 91–117.
32.
McLachlanG and PeelDA (2000) Finite mixture models. New York: Wiley.
33.
MichelotT (2022) hmmTMB: hidden Markov models with flexible covariate effects in R. arXiv preprint arXiv:2211.14139.
34.
NgôMC, Heide-JørgensenMP and DitlevsenS (2019) Understanding narwhal diving behaviour using hidden Markov models with dependent state distributions and long range dependence. PLoS Computational Biology, 15, e1006425.
35.
OrfanogiannakiK and KarlisD (2018) Multivariate Poisson hidden Markov models with a case study of modelling seismicity. Australian & New Zealand Journal of Statistics, 60, 301–22.
36.
ÖttingM and KarlisD (2023) Football tracking data: a copula-based hidden Markov model for classification of tactics in football. Annals of Operations Research, 325, 167–83.
37.
ÖttingM, LangrockR and MaruottiA (2021) A copula-based multivariate hidden Markov model for modelling momentum in football. AStA Advances in Statistical Analysis, pages 1–19.
38.
PhillipsJS, PattersonTA, LeroyB, PillingGM and NicolSJ (2015) Objective classification of latent behavioral states in bio-logging data using multivariate-normal hidden Markov models. Ecological Applications, 25, 1244–58.
39.
PiccardiM and PérezÓ (2007) Hidden Markov models with kernel density estimation of emission probabilities and their use in activity recognition. In 2007 IEEE conference on computer vision and pattern recognition, pages 1–8. IEEE.
40.
PohleJ, LangrockR, SchaarMvd, KingR and JensenFH (2021) A primer on coupled state-switching models for multiple interacting time series. Statistical Modelling, 21 (3), 264–85.
41.
PunzoA and MaruottiA (2016) Clustering multivariate longitudinal observations: the contaminated Gaussian hidden Markov model. Journal of Computational and Graphical Statistics, 25, 1097–98.
42.
RobberechtsP (2019) Valuing the art of pressing. In Proceedings of the StatsBomb innovation in football conference, pages 1–11. StatsBomb.
43.
RuppertD, WandMP and CarrollRJ (2009) Semiparametric regression during 2003–2007. Electronic Journal of Statistics, 3, 1193.
44.
SpeziaL (2010) Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes. Journal of Time Series Analysis, 31, 1–11.
45.
SrivastavaA, KunduA, SuralS and MajumdarA (2008) Credit card fraud detection using hidden Markov model. IEEE Transactions on Dependable and Secure Computing, 5, 37–48.
46.
StatsBomb (2020) StatsBombR. URLhttps://github.com/statsbomb/StatsBombR. R package version 0.1.0.
47.
van BeestFM, MewsS, ElkenkampS, SchuhmannP, TsolakD, WobbeT, BartolinoV, BastardieF, DietzR, von DorrienC, . (2019) Classifying grey seal behaviour in relation to environmental variability and commercial fishing activity: A multivariate hidden Markov model. Scientific Reports, 9, 1–14.
48.
VisserI, RaijmakersME and MolenaarP (2002) Fitting hidden Markov models to psychological data. Scientific Programming, 10, 185–99.
49.
WandMP (2003) Smoothing and mixed models. Computational Statistics, 18, 223–49.
50.
WoodSN (2017) Generalized additive models: an introduction with R. Boca Raton, FL: Chapman and Hall/CRC.
51.
ZimmermanR, CraiuRV and Leos-BarajasV (2023) Copula modeling of serially correlated multivariate data with hidden structures. Journal of the American Statistical Association, 119, 2598–609.
52.
ZucchiniW, MacDonaldIL and LangrockR (2016) Hidden Markov models for time series: an introduction using R. Boca Raton, FL: Chapman & Hall/CRC.
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