Markov-switching generalized additive models

Abstract

We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.

Movement responses to environment: fast inference of variation among southern elephant seals with a mixed effects model

“…While the linear mixed effects approach allows flexible combinations of fixed and random effects, there is scope for further enhancement. In many cases parametric, linear fixed effects may not adequately capture the complexity of movement - environment relationships and a nonparametric approach using penalised splines may improve inference (Langrock et al, 2017). Given the serial dependence structure of telemetry data, the unstructured covariance matrix we used for the random effects could be replaced with a first-order autoregressive covariance struc-ture (Brooks et al, 2017).…”

Bayesian hidden Markov modelling using circular‐linear general projected normal distribution

“…Using MCMC simulations posterior circular mean and concentration can be derived, as well as the circular‐linear correlation. Of course, different areas of application can be considered for the proposed approach, for example, animal movement modelling (Langrock et al ) and driving behavior (Jackson et al ., .…”

Spline‐based nonparametric inference in general state‐switching models

“…Current approaches include cross‐validation, model selection criteria that take into account the penalization (thus estimating the effective degrees of freedom, rather than simply counting parameters, to measure model complexity), or subjective selection based on visual inspection of fitted models. Regarding the former two (formal) methods, we found that although they mostly produce reasonable values for the λ i , they are nevertheless somewhat unstable and sometimes fail completely (Langrock, Kneib, et al, ; Langrock et al, ). Overall, smoothing parameter selection remains a challenging task in these model classes.…”

“…In a series of papers, we recently proposed the use of spline smoothing techniques for nonparametric inference within state‐switching models (Langrock, Michelot, Sohn, & Kneib, ; Langrock, Kneib, et al, ; Langrock, Kneib, Glennie, & Michelot, ; Adam, Mayr, & Kneib, ). The resulting classes of models combine two powerful tools, namely, the forward algorithm for efficient likelihood evaluation and penalized B‐splines (i.e., P‐splines; Eilers and Marx, ) for nonparametric inference, to allow for relatively straightforward and computationally tractable maximum penalized likelihood estimation within state‐switching models.…”