Nonparametric estimation of multivariate hidden Markov models using tensor-product B-splines

TL;DR

This paper introduces a nonparametric method using tensor-product B-splines for estimating emission distributions in multivariate hidden Markov models, enhancing flexibility in modeling complex time series data.

Contribution

It proposes a novel nonparametric estimation approach for multivariate HMMs that overcomes limitations of parametric assumptions, demonstrated through simulations and a real-world sports data case study.

Findings

Feasibility demonstrated through simulation studies

Potential pitfalls of parametric choices highlighted

Effective modeling of sports match dynamics shown

Abstract

For multivariate time series driven by underlying states, hidden Markov models (HMMs) constitute a powerful framework which can be flexibly tailored to the situation at hand. However, in practice it can be challenging to choose an adequate emission distribution for multivariate observation vectors. For example, the marginal data distribution may not immediately reveal the within-state distributional form, and also the different data streams may operate on different supports, rendering the common approach of using a multivariate normal distribution inadequate. Here we explore a nonparametric estimation of the emission distributions within a multivariate HMM based on tensor-product B-splines. In two simulation studies, we show the feasibility of our modelling approach and demonstrate potential pitfalls of inappropriate choices of parametric distributions. To illustrate the practical applicability, we present a case study where we use an HMM to model the bivariate time series comprising the lengths and angles of goalkeeper passes during the UEFA EURO 2020, investigating the effect of match dynamics on the teams' tactics.