A Bayesian hidden Markov model for assessing the hot hand phenomenon in basketball shooting performance

Sports data analytics is a relevant topic in applied statistics that has been growing in importance in recent years. In basketball, a player or team has a hot hand when their performance during a match is better than expected or they are on a streak of making consecutive shots. This phenomenon has generated a great deal of controversy with detractors claiming its non-existence while other authors indicate its evidence. In this work, we present a Bayesian longitudinal hidden Markov model that analyses the hot hand phenomenon in consecutive basketball shots, each of which can be either missed or made. Two possible states (cold or hot) are assumed in the hidden Markov chains of events, and the probability of success for each throw is modelled by considering both the corresponding hidden state and the distance to the basket. This model is applied to a real data set, the Miami Heat team in the season 2005-2006 of the USA National Basketball Association. We show that this model is a powerful tool for assessing the overall performance of a team during a match or a season, and, in particular, for quantifying the magnitude of the team streaks in probabilistic terms

Data from: Markov switching autoregressive models for interpreting vertical movement data with application to an endangered marine apex predator

1.Time series of animal movement obtained from bio-loggers are becoming widely used across all taxa. These data are nowadays of high quality, combining high resolution with precision, as the tags are able to collect for longer times and store larger quantities of data. Due to their nature, high-frequency data sequences often pose non-trivial problems in time series analysis: non-linearity, non-Normality, non-stationarity, and long memory. These issues can be tackled by modelling the data sequence as a realization of a stochastic regime switching process. 2. We suggest a novel Markov switching autoregressive model where the hidden Markov chain is non-homogeneous, with time-varying transition probabilities, whose dynamics depend on the dynamics of some contemporary categorical covariates. 3. To illustrate the use of the method, we apply it to the depth profiles of four individuals of flapper skate (Dipturus cf. intermedia) in order to identify swimming behaviours. Individual time series were obtained from data storage tags that recorded pressure every two minutes. The environmental covariates used were lunar phase (a proxy for the spring-neap tidal cycle), lunar cycle, and diel cycle. For all individuals two states (or regimes) were always selected (the autoregressive order was either three or four), representing different regimes of animal activity, i.e., state 1 for resting or horizontal swimming or slow vertical movement; state 2 for fast ascending and descending. The cycle of the four lunar phases was the only environmental covariate that explained the hidden state dynamics in all individuals, whereas lunar cycle was selected for two individuals and diel cycle for one only. 4. The method is an efficient approach to fit one-dimensional tag data using categorical environmental covariates, and to classify the observations into a small number of states representing individual behaviours of tagged individuals

Seasonal autoregressions with regime switching

Markov switching autoregressive models (MSARMs) are efficient tools to analyse non-linear and non-normal time series. A special MSARM with a hidden state-dependent seasonal component is proposed here to analyse periodic time series. We present a complete Metropolis-within-Gibbs algorithm for constraint identification, for model choice and for the estimation of the unknown parameters and the latent data. These three consecutive steps are developed tackling the problem of the hidden states labeling, by means of random permutation sampling and constrained permutation sampling. The missing observations occurring within the observed series and the future values are respectively estimated and forecasted considering them as unknown parameters. We illustrate our methodology with an example about the dynamics of an air pollutant

Reversible Jump MCMC Methods and Segmentation Algorithms in Hidden Markov Models

We consider hidden Markov models with an unknown number of regimes for the segmentation of the pixel intensities of digital images that consist of a small set of colours. New reversible jump Markov chain Monte Carlo algorithms to estimate both the dimension and the unknown parameters of the model are introduced. Parameters are updated by random walk Metropolis\u2013Hastings moves, without updating the sequence of the hidden Markov chain. The segmentation (i.e. the estimation of the hidden regimes) is a further aim and is performed by means of a number of competing algorithms. We apply our Bayesian inference and segmentation tools to digital images, which are linearized through the Peano\u2013Hilbert scan, and perform experiments and comparisons on both synthetic images and a real brain magnetic resonance image