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.

我们研究了足球比赛动态中变化点（通常称为“动量转变”）的潜在发生。为此，我们使用隐藏的马尔可夫模型（HMM）对德甲联赛的比赛中每分钟的统计数据进行建模。为了允许变量的状态内依赖，我们使用copulas制定了变量的状态依赖分布。对于考虑的德甲数据，我们发现拟合的HMM包含可以解释为对比赛显示不同级别控制的团队的状态。我们的建模框架可以进行与动量变化原因和团队战术相关的推断，这对于经理，博彩公司和体育迷来说非常有趣。