Flexible Latent‐State Modelling of Old Faithful's Eruption Inter‐Arrival Times in 2009

Abstract

SummaryThis paper is concerned with the analysis of a time series comprising the eruption inter‐arrival times of the Old Faithful geyser in 2009. The series is much longer than other well‐documented ones and thus gives a more comprehensive insight into the dynamics of the geyser. Basic hidden Markov models with gamma state‐dependent distributions and several extensions are implemented. In order to better capture the stochastic dynamics exhibited by Old Faithful, the different non‐standard models under consideration seek to increase the flexibility of the basic models in various ways: (i) by allowing non‐geometric distributions for the times spent in the different states; (ii) by increasing the memory of the underlying Markov chain, with or without assuming additional structure implied by mixture transition distribution models; and (iii) by incorporating feedback from the observation process on the latent process. In each case it is shown how the likelihood can be formulated as a matrix product which can be conveniently maximized numerically.