Nonparametric inference in hidden Markov models using P-splines

TL;DR

This paper introduces a nonparametric method for estimating state-dependent distributions in hidden Markov models using P-splines, allowing for more flexible and parsimonious modeling of complex time series data.

Contribution

It develops a novel P-spline based approach for nonparametric inference in HMMs, addressing limitations of parametric distribution assumptions.

Findings

More parsimonious models with fewer states fit the data well

Nonparametric approach captures complex data features better

Application to whale diving speeds demonstrates effectiveness

Abstract

Hidden Markov models (HMMs) are flexible time series models in which the distributions of the observations depend on unobserved serially correlated states. The state-dependent distributions in HMMs are usually taken from some class of parametrically specified distributions. The choice of this class can be difficult, and an unfortunate choice can have serious consequences for example on state estimates, on forecasts and generally on the resulting model complexity and interpretation, in particular with respect to the number of states. We develop a novel approach for estimating the state-dependent distributions of an HMM in a nonparametric way, which is based on the idea of representing the corresponding densities as linear combinations of a large number of standardized B-spline basis functions, imposing a penalty term on non-smoothness in order to maintain a good balance between goodness-of-fit and smoothness. We illustrate the nonparametric modeling approach in a real data application concerned with vertical speeds of a diving beaked whale, demonstrating that compared to parametric counterparts it can lead to models that are more parsimonious in terms of the number of states yet fit the data equally well.