The modelling framework developed in this chapter, a (relaxed-)LASSO-HMM, allows for implicit variable selection in the state-dependent process of HMMs. The perfor-mance of the variable selection is first investigated in a simulation study, indicating that the relaxed-LASSO-HMM with the corresponding tuning parameter selected by the BIC is the best-performing fitting scheme considered.
For the analysis of a hot shoe effect, we fit both LASSO-HMMs and relaxed-LASSO-HMMs to data on penalty kicks in the German Bundesliga. Factors potentially
6.6 Discussion 91
affecting the performance of penalty-takers, such as the current score of the match, are included in the predictor. In addition, dummy variables for the penalty-takers as well as for the goalkeepers are included. Our results suggest two states with different levels of performance, and shed some light on exceptionally performing players such as Jean-Marie Pfaff, a former goalkeeper of Bayern Munich, who has been selected by our fitting schemes.
A clear limitation of the real data application considered is the problem of self selection. Since the manager (or the team) can decide which player has to take the penalty, players who have been rather unsuccessful in the past may not take penalty kicks anymore. However, several teams have demonstrated in the past that they rely on and trust in a certain player for taking penalty kicks, regardless of the outcome of the kick. Whereas penalty kicks in football yield to a time series due to the way in which penalties take place, the corresponding time intervals between actions are irregular. Although our data cover all attempts in the German Bundesliga, there are sometimes several month between two attempts. Moreover, some players might be involved in penalty kicks in matches from other competitions such as the UEFA Champions League, the UEFA European Cup, or in matches with their national teams.
From this perspective, the time series of Bundesliga penalties could be considered as partly incomplete for some players.
From a methodological point, the number of states selected (i.e., N =2) may be too coarse for modelling the underlying form of a player. Considering a continuously varying underlying state variable instead may be more realistic, since gradual changes in a player’s form could then be captured. This could be achieved by considering models with an underlying continuous state process, where regularised estimation approaches are a first point for further research. The motivation in this chapter for N =2 states, however, was to approximate the potential psychological states in a simple manner for ease of interpretation, e.g., in the sense of hot (“player is confident”) or cold (“player is nervous”) states. Moreover, our main focus was to show the usefulness of our method developed in a rather simple setting with two states. Our results should, hence, be treated with caution regarding the existence of a potential hot shoe effect. Further points for future research include regularisation approaches in HMMs where not only the intercept (as considered here), but also the parameters βj are allowed to depend on the current state. Regularisation in this model formulation could be taken into account by applying so-called fused LASSO techniques (see, e.g., Gertheiss and Tutz,
92 A regularised hidden Markov model for analysing the ‘hot shoe’ in football
2010), where the parameters could either be shrunk to zero or to the same size for all states considered.
The modelling framework developed here can easily be tailored to other applications, where implicit variable selection in HMMs is desired. For the application considered in this chapter, i.e. an analysis of a potential hot hand/hot shoe effect, other sports such as basketball or hockey could be analysed. Potential covariates — whose corresponding effects are penalised — in these sports cover the shot type, shot origin, and game score, to name but a few.
7 A copula-based multivariate hidden
Markov model for modelling momentum in football
7.1 Introduction
Sports commentators and fans frequently use vocabulary such as “momentum”, “mo-mentum shift”, or related terms to refer to change points in the dynamics of a match.
Usage of such terms is typically associated with situations during a match where an event — such as a shot hitting the woodwork in a football match — seems to change the dynamics of the match, e.g. in a sense that a team which prior to the event had been pinned back in its own half suddenly seems to dominate the match. A promi-nent example is the 2005 Champions League final between Milan and Liverpool, within which Liverpool was trailing by three goals after the first half, but fought back after half time and eventually won by penalty shootout.
Despite the widespread belief in momentum shifts in sports, it is not always clear to what extent perceived shifts in the momentum are genuine. From the literature on the “hot hand” — i.e. research on serial correlation in human performances — it is well known that most people do not have a good intuition of randomness, and in particular tend to overinterpret streaks of success and failure, respectively (see, e.g., Kahneman, 2011; Thaler and Sunstein, 2009). It is thus to be expected that many perceived momentum shifts are in fact cognitive illusions in the sense that the observed shift in a competition’s dynamics is driven by chance only.
Momentum shifts have been investigated in qualitative psychological studies, e.g. by interviewing athletes, who reported momentum shifts during matches (see, e.g., Jones and Harwood, 2008; Richardson et al., 1988). Fuelled by the rapidly growing amount of freely available sports data, quantitative studies have investigated the drivers of ball possession in football (Lago-Peñas and Dellal, 2010), the detection of main playing
94 A copula-based multivariate hidden Markov model for modelling momentum in football
styles and tactics (Diquigiovanni and Scarpa, 2018; Gonçalves et al., 2017) and the effects of momentum on risk-taking (Lehman and Hahn, 2013). In some of the existing studies, e.g. in Lehman and Hahn (2013), momentum is not investigated in a purely data-driven way, but rather pre-defined as winning several matches in a row.
In this chapter, we analyse potential momentum shifts within football matches.
Specifically, we investigate the potential occurrence of momentum shifts by analysing minute-by-minute bivariate summary statistics from the German Bundesliga using HMMs. The corresponding data is described in Section 7.2. Within the HMMs, we consider copulas to allow for within-state dependence of the variables considered.
The corresponding methodology is presented in Section 7.3. Our results, which are presented in Section 7.4, suggest states which can be tied to different levels of control in a match. In addition, we investigate the causes of momentum shifts, e.g. the cur-rent score of the match. This type of insight could be of great interest to managers, bookmakers and sports fans.