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

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 family of distributions for modelling the multivariate observation vectors conditional on the states. For example, the marginal data distribution may not immediately reveal the within-state distributional form, and also the different variables observed over time may operate on different supports, rendering the common approach of using a multivariate normal distribution inadequate. Here, we explore a nonparametric estimation of the state-dependent distributions within a bivariate HMM based on tensor-product B-splines. In two simulation studies, we show the feasibility of our modelling approach and demonstrate the potential pitfalls of inappropriate choices of parametric distributions. To illustrate the practical applicability, we present a case study using 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.