Football is a globally popular sport and its financial and social impact attracts researchers whose main aim is to increase comprehension of training and match-play performance (1–3). Thanks to technological and analytical evolutions, new performance-oriented research perspectives have emerged from the analysis of player performance. Both training and match data, collected from football players using global navigation satellite systems (4) and markerless optical tracking systems (5), have become more plentiful and increasingly accurate. The information enables the development of advanced statistical and machine learning approaches to help analyze and subsequently optimize football performance and attempt predictions of match outcomes.

A popular performance metric in football is expected goals (xG). This metric represents the probability of a shot resulting in a goal. It was first introduced in football by Green (6) with the aim of identifying the key factors underpinning how goals are scored and has become a valuable objective measure of an individual player’s performance that can also be extended to the team level (7–10). To date, xG models typically account for spatio-temporal information, such as the time, the distance and angle between the player and the goal at the shooting time, the type of shot, and the preceding event (i.e., the last action such as a low pass or a high pass). Beyond these data, previous studies have reported different approaches and model architectures (11–16). Modeling xG requires multiple features as the complexity of the task and its variability calls into question its predictability when using a restricted set of game features (e.g., the goal distance) (15). However, Umami et al. (16) reported that a logistic regression using only a few features (the distance and angle to the goal, and whether the shot is headed or not) provided convincing results. Alternatively, more complex architectures have been employed to attempt to better estimate xG (14). In effect, the authors in the latter study compared a logistic regression with non-linear ensemble learning algorithms (random forest and adaptive boosting) to predict the match score by summing the estimated xG of each shot opportunity. According to their results, the random forest algorithm provided the best model performance. In another study, Anzer and Bauer (11) used advanced features such as the height of the ball when the shot was attempted and analysis of the player’s movement at the time of the shot. By comparing several supervised machine learning models, a gradient boosting model that accounted for the type of shot (header, leg kick, and direct free kick) provided the best performance in predicting the number of goals scored.

Most studies that have attempted to model xG have focused on fitting the best model using non-linear and complex model architectures, at the expense of model interpretability. These studies aimed to achieve the best performance in predicting that the shot will be converted into a goal. However, complex models are difficult to interpret. There are methods for explaining such models, such as the use of Shapley values (17, 18), but these approaches are still criticized today (19, 20), questioning the ability of one to fully master the complexity of Shapley value calculations. There are a few studies that have focused on xG modeling while preserving interpretability. To build a model that identifies key factors influencing xG, Decroos and Davis (13) and Bransen and Davis (12) proposed the use of a generalized additive model (GAM). The studies show that GAMs provide comparable results to a more complex gradient boosting model while retaining the advantage of interpretability. To further improve the interpretability of the model and since logistic regression has proved effective in modeling xG (16), one should consider the relationships between features and xG to be linear and consequently should utilize a generalized linear model instead of a GAM. One also assumes that soccer games evolve and that patterns are slightly different across seasons and competitions. As such, the present authors made a choice to investigate a Bayesian framework with mixed effects. Very recently, Scholtes and Karakuş (21) also proposed a hierarchical Bayesian approach to model xG. They used this model to determine whether the individual player or their positional role impacted xG. The main strength of their work was the identification of specific player abilities throughout an interpretable model and a rigorous assessment of prediction uncertainty. However, the potential of the Bayesian framework was arguably not fully exploited and further research is warranted.

In the present article, an alternative prior specification method is proposed to model xG: an interpretable Bayesian generalized linear mixed-effects model (22). This model might achieve better estimation quality, and its benefits are potentially numerous. First, the linear structure makes the model very easy to interpret by analyzing the model’s coefficients. Second, the inclusion of random effects means that intra- and inter-player/team variability can be considered. This enables interpretation of the strength of players and/or teams in specific game situations. Third, using the Bayesian framework enables the utilization of limited training data while incorporating expert prior knowledge. Such a model is not new and is used in other fields and applications. Yet, it was unseen in xG modeling until the recent work of Scholtes and Karakuş (21) and also demonstrates the aforementioned advantages. In our study, we propose to further exploit the Bayesian contribution with a transfer learning strategy to build a highly informative prior. The prior is built using past competition, rather than relying on a hand-constructed prior, which is inevitably less informative. A study on the impact of prior choice on the posterior was presented using the Wasserstein Impact Measure (WIM) (23), showing the benefits of choosing the prior we propose (more informative prior and better predictive performance). Furthermore, a practical model explainability of outcomes is provided through Shapley Additive exPlanations estimates (SHAP), which can, according to the literature (24), benefit football analysts and coaches in their decision-making processes.

Following the presentation of the data and the features used to evaluate xG (Section 2.1), we will define our xG model (Section 2.2) and then present the results (Section 3). Thereafter, we will engage in a discussion of the results and benefits of the model within the context of the existing literature (Section 4).

The comparison of the performance of the proposed model (i.e., a Bayesian mixed effect logistic regression, namely, Bayesian xG) with that of StatsBomb’s and Scholtes’ models is reported in Table 2. Considering all competitions, the results show that model performances differed slightly, but similar performance was obtained with the Bayesian approach and the StatsBomb model. Scholtes’ model seemed to suffer from less informative priors when fitted on small data samples. This is supported by a lower AUC score than the other models, and the xG predictions tend to be symptomatically higher (higher recall with a smaller threshold).

Having obtained a reasonable model, we will now illustrate the advantages of the proposed approach and focus on a given international football competition: the FIFA World Cup 2022. First, to underline the impact of the prior used for the Bayesian xG model compared to the prior proposed by Scholtes, we calculated the WIM. This measure is used to compare two distributions, and in the Bayesian framework, to compare two posterior distributions obtained with distinct priors. As suggested in the original paper, we computed the WIM using the uniform prior. The aim of the approach is to evaluate the quantity of information provided by the prior: the higher the WIM between a given prior and the uniform prior, the more informative the prior is. The WIM was computed using the samples obtained with the MCMC algorithm, and resulted in a WIM almost four times higher for the Bayesian xG prior against the uniform prior than for the Scholtes prior against the uniform prior (Table 3).

To focus on the Bayesian xG model, the maximum a posteriori parameter estimation showed that the distance feature (i.e., the distance between the shot location and the center of the goal) had the greatest negative influence on the predicted xG (see Table 4). Furthermore, the optimal model structure retained for xG prediction includes a random intercept ωα and two random slopes (on angle and closest_opponent parameters, denoted by ωβ5 and ωβ7, respectively). Since ωβ5 and ωβ7 differ from 0, they highlight team-specific traits and relationships between these predictors and the predicted xG. Accordingly, such a model structure allows for consideration of inter-team variability. The univariate posterior and prior distribution for the fixed effects showed the differences between the Bayesian model calibrated on the 2022 FIFA World Cup and the global model fitted on other competitions (see Figure 3). For this competition, the features nb_opponent_on_traj, angle, and body_part_head had a particularly higher negative impact than the global model. This means that in this competition and situation, the players were less likely to score than usual. However, the optimal model retained a random slope on the angle feature, meaning that some teams were significantly better than others to deal with the bad angles.

Computing SHAP values allowed us to explain the model predictions and, for a given shot, to interpret the impact of each feature’s value on the prediction. In addition, we can examine poorly predicted shots to understand where the failure arises from. For example, Figure 4 presents two poorly predicted shots and their respective feature contributions. The first shot has a very high xG value because it is close, central, and there is no opponent on its trajectory. However, the pressure exerted by the nearest opponent likely exceeded the model’s prediction, resulting in a missed shot. The second shot was challenging, characterized by its considerable distance and lateral displacement, with two opposing players present on its trajectory. Nevertheless, the player successfully converted his attempt into a goal.

The present xG mixed-effect model, based on a dataset from the 2022 FIFA World Cup, provided an opportunity to study teams’ goal scoring through strengths and weaknesses in an interpretable and explainable way. The results are arguably of major interest to coaching practitioners, sports scientists, and researchers interested in determining the influence of player and team actions on scoring goals and match outcomes. For practical usage, the models were implemented in a Streamlit web application (33).

Using transfer learning, our model further advances the recent work of Scholtes and Karakuş (21). It can notably undergo an initial pre-training phase on a more extensive and historical dataset and a secondary training phase on a smaller, recent dataset to refine its predictive capabilities. The application of transfer learning within a Bayesian framework yielded results that were comparable to those of a commercial xG model (developed by StatsBomb), despite the utilization of a reduced dataset (see Table 2 for a summary). Furthermore, the interpretability of our linear model is a notable advantage as each model parameter can be analyzed and compared with one another. For instance, the distance from the goal had the most significant influence on xG predictions (over the shot angle) and should, consequently, be considered in any training drills and in-game tactical decision-making.

The Bayesian framework has several advantages. Building an efficient predictive model in a frequentist way implies training a model over a large enough dataset (34). In their study, Robberechts and Davis (35) concluded that five seasons of data were needed to fit an accurate frequentist and non-parametric xG model. Since the Bayesian inference comes with prior distributions, fewer training observations are needed to fit a model correctly. In addition, our approach also addressed confidentiality issues. Identifying a prior distribution implies learning a first function that approximates xG using a separated data sample (in our case, to use another competition to build the baseline model). Through transfer learning, models trained on multiple competitions can be reused for the targeted competition, reducing the computation time and improving model performances. Furthermore, retraining the baseline model on the competition of interest helps identify differences and similarities between the competitions.

In this work, we focused on open-play phases only, while overall, football performance should be modeled from all playing phases. However, the method is transferable to any playing phases where models could be built separately or combined. As aforementioned, player characteristics have significant importance in any football performance modeling. Player-specific data might improve model accuracy and outperform other commercial xG models. Gender has also shown some importance, as mentioned by Bransen and Davis (12). Considering gender in xG modeling or building gender or youth-specific models is recommended. Since football data on the women’s game are generally less abundant, the proposed approach based on transfer learning should address this issue as an optimal model would be built for a given competition while benefiting from broader information.

Even though sports performance modeling remains challenging due to its inherent complexity, analyzing players’ and teams’ characteristics provides essential information for this task. In this study, we considered individuals as teams since the data did not allow us to consider each player individually. Hence, the cross-random effects from the mixed logistic regression highlighted singularities between football teams, particularly regarding the body_part_head and nb_opponent_traj features. However, one may note that player-level information could significantly increase the performance of xG predictions, as players have strengths and weaknesses of their own. Beyond this, the model could allow for adapting a pre-game team strategy or making on-field decisions, to optimize the efficiency of the team.

Expert knowledge can be included in the model as priors. Nevertheless, if the number of observations is insufficient to construct the prior or if the prior information reflects extreme parameter values, there is a risk of bias and identification failure of posterior distributions (36–38). This phenomenon can be observed in our dataset when a comparison is made between the results of Scholtes’ model and the Bayesian xG model that we propose. Indeed, fitting the model by competition results in a smaller dataset than that employed in the aforementioned work. Furthermore, the selection of priors has led to a model with a higher xG prediction than that of StatsBomb and our own model. Conversely, when a sufficient number of observations have been accumulated, the incorporation of expert knowledge as a prior enables coaches to exert direct influence on predictions and decisions, thereby facilitating the provision of valuable real-time feedback. To illustrate, a coach’s knowledge of the game could be incorporated into the prior to facilitate meaningful insight into the players. Knowledge of the strengths of the starting team he has chosen, or of the opposing team, could lead to a manual shift in the value of the parameters influenced. Coaches could then readily utilize the model’s parameters to optimize in-game strategies, such as identifying the optimal shooting distance for individual players and offering practical insights for enhancing overall team performance.

The proposed method may also be of particular interest for modeling expected goals on target (xGOT). xGOT is a post-shot metric that uses the position at which the ball enters the goal and whether it is saved or scored. The xGOT is primarily employed to assess a team’s finishing proficiency by comparing xG with xGOT and to evaluate the performance of goalkeepers according to the quality of the shot. The methodology delineated in this paper also permits the identification of scenarios wherein specific teams exhibit superior finishing proficiency or more efficacious goalkeeping and the characterization of these scenarios. However, although our approach is generalizable to many cases, it is possible that predictive quality may be reduced in tasks that are too complex, such as expected threat. Maintaining high interpretability and, therefore, simple models for such tasks would potentially lead to a greater loss of predictive quality.

The development of interpretable and, more widely, explainable artificial intelligence represents a pivotal area of research within the field of computer science and subsequently sports science. This approach facilitates the extraction of novel insights from complex sports data, thereby empowering practitioners to make well-informed decisions (39). Our approach, based on a Bayesian mixed logistic regression model, is aligned with the principles of reproducibility and interpretability. Furthermore, it achieves comparable predictive performance to that of more complex models, despite the utilization of a limited sample of competition data. It also addresses practical concerns such as the identification of team strengths and weaknesses, and could be further extended to model xG from individual characteristics in a straightforward, accessible, and reliable manner.