A copula-based multivariate hidden Markov model for modelling momentum in football

Abstract

We investigate the potential occurrence of change points—commonly referred to as “momentum shifts”—in the dynamics of football matches. For that purpose, we model minute-by-minute in-game statistics of Bundesliga matches using hidden Markov models (HMMs). To allow for within-state dependence of the variables, we formulate multivariate state-dependent distributions using copulas. For the Bundesliga data considered, we find that the fitted HMMs comprise states which can be interpreted as a team showing different levels of control over a match. Our modelling framework enables inference related to causes of momentum shifts and team tactics, which is of much interest to managers, bookmakers, and sports fans.