Penalized estimation of flexible hidden Markov models for time series of counts

Abstract

Hidden Markov models are versatile tools for modeling sequential observations, where it is assumed that a hidden state process selects which of finitely many distributions generates any given observation. Specifically for time series of counts, the Poisson family often provides a natural choice for the state-dependent distributions, though more flexible distributions such as the negative binomial or distributions with a bounded range can also be used. However, in practice, choosing an adequate class of (parametric) distributions is often anything but straightforward, and an inadequate choice can have severe negative consequences on the model's predictive performance, on state classification, and generally on inference related to the system considered. To address this issue, we propose an effectively nonparametric approach to fitting hidden Markov models to time series of counts, where the state-dependent distributions are estimated in a completely data-driven way without the need to select a distributional family. To avoid overfitting, we add a roughness penalty based on higher-order differences between adjacent count probabilities to the likelihood, which is demonstrated to produce smooth probability mass functions of the statedependent distributions. The feasibility of the suggested approach is assessed in * Corresponding author; email: timo.adam@uni-bielefeld.de. a simulation experiment, and illustrated in two real-data applications, where we model the distribution of i) major earthquake counts and ii) acceleration counts of an oceanic whitetip shark (Carcharhinus longimanus) over time.

Uncovering ecological state dynamics with hidden Markov models

“…Driven by the influx of new biotelemetry sensor technology, HMMs have also been used to analyse the sequences of dives of marine animals (Hart et al ., 2010; Quick et al ., 2017; DeRuiter et al ., 2017; van Beest et al ., 2019). The remote collection of activity data at potentially very high temporal resolutions using accelerometers is another emerging application area (Diosdado et al ., 2015; Leos‐Barajas et al ., 2017b; Papastamatiou et al ., 2018a,b; Adam et al ., 2019b). These HMM formulations are conceptually very similar to the movement model outlined above, with the state process corresponding to behavioural modes (or at least proxies thereof), and the activity data represented by the state‐dependent process.…”

Flexible estimation of the state dwell-time distribution in hidden semi-Markov models

“…λ i = λ j for i = j is possible. A common way to select the smoothing parameters is via cross validation (see Langrock et al, 2015;Adam et al, 2019). Second, the difference order m influences the shape of d i (r), especially when λ i becomes large.…”

Control charts for monitoring a Poisson hidden Markov process

“…It should be noted that any other count model could be used for the HMM as well, also different models for different states. Recently, Adam et al 15 even developed an estimation approach for a nonparametric count HMM. A brief summary of further stochastic properties as well as approaches for the parameter estimation, forecasting, and decoding of the hidden states is provided in Appendix .…”