Flexible unimodal density estimation in hidden Markov models

TL;DR

This paper introduces a shape-constrained spline method for unimodal density estimation in hidden Markov models, enhancing interpretability and stability over fully flexible nonparametric approaches.

Contribution

It proposes a novel unimodality constraint on spline coefficients within HMMs, balancing flexibility, interpretability, and parsimony for state-dependent density estimation.

Findings

More stable density estimates in simulations

Improved model interpretability

Effective application to narwhal dive data

Abstract

1. Hidden Markov models (HMMs) are powerful tools for modelling time-series data with underlying state structure. However, selecting appropriate parametric forms for the state-dependent distributions is often challenging and can lead to model misspecification. To address this, P-spline-based nonparametric estimation of state-dependent densities has been proposed. While offering great flexibility, these approaches can result in overly complex densities (e.g. bimodal) that hinder interpretability. 2. We propose a straightforward method that builds on shape-constrained spline theory to enforce unimodality in the estimated state-dependent densities through enforcing unimodality of the spline coefficients. This constraint strikes a practical balance between model flexibility, interpretability, and parsimony. 3. Through two simulation studies and a real-world case study using narwhal (Monodon monoceros) dive data, we demonstrate the proposed approach yields more stable estimates compared to fully flexible, unconstrained models improving model performance and interpretability. 4. Our method bridges a key methodological gap, by providing a parsimonious HMM framework that balances the interpretability of parametric models with the flexibility of nonparametric estimation. This provides ecologists with a powerful tool to derive ecologically meaningful inference from telemetry data while avoiding the pitfalls of overly complex models.