Spatial analysis of shots in MLS: A model for expected goals and fractal dimensionality

2.1Existing Literature

As aforementioned the notion of expected goals is not new. Soccer practitioners have developed their own models to estimate the expected goals from the shots taken from a team. The vast majority of these models utilizes features related to the distance from the goal line Bertin (2015), Chappas (2013), 11tegen11 (2014) in order to calculate the probability of a shot leading to a goal. Lucey et al. (2015) were the first to use player tracking data in order to include game context related features in their expected goals model. One of the gaps in the existing literature is that the goal probabilities obtained from the developed models are not (properly) evaluated. Typically the model evaluation includes accuracy figures and/or the average prediction error between the actual goals scored and the expected goals based on the shot model. Our current work fills this gap by highlighting an appropriate way to evaluate the probability output from a shot model. While we develop our model and evaluations using only features from shot charts - we do not have access to detailed tracking data - the probability evaluation approach we describe can be used for more elaborate models such as the one presented by Lucey et al. (2015).

In another direction, Vilar et al. (2013) focus on the analysis of local dynamics and show that local player numerical dominance is key to defensive stability and offensive opportunity, while Bar-Eli et al. (2007), Bar-Eli and Azar (2009) further collected information from 286 penalty kicks from professional leagues in Europe and South America and analyzed the decisions made by the penalty kickers and the goal keepers. Their main conclusion is that from a statistical standpoint, it seems to be more advantageous for a goal keeper to defend the penalty kick by remaining in the goal’s center. Furthermore, Di Salvo et al. (2007) analyzed the motion of 200 soccer players from 20 games of the Spanish Premier League and 10 games of the Champions League and found that the different positional roles demand for different work intensities.

Finally, similar line of research exists in hockey analytics where models for expected goals are developed in a similar manner Eric (2012), Johnson (2016), Perry (2016). In these models again the main features that determine the shot quality include the location of the player taking the shot as well as the type of the shot (e.g., wrist shot, slap shot, deflection, etc.).

2.2Dataset

For our study we compiled a set of 1,115 (non-penalty) shots from 99 MLS games during the 2016 regular season. The data were collected by watching and manually tagging the (x,y) coordinates of shots on target in each Major League Soccer game between March 6 - May 18, 2016. John Burn-Murdoch’s soccer pitch tracker was used to tag each shot’s location, outcome, game state, assist type, and to define the phase of play from which the shot came (i.e. set piece, open play, etc.) Burn-Murdoch (2015). The final annotated dataset includes the following tuple for each shot:

<ID, Player, Team, Opponent,  Loc.X, Loc.Y, Outcome, Assist Type, Shot Type, Play Type, Angle, Distance>

Figure 1 graphically depicts the shot chart for the whole dataset, while Fig. 2 visualizes separately the goals and missed shots as well as their spatial density. Before describing our model for the goal probability for each shot, we perform some basic analysis of the shots with respect to the features included in the shot tuple described above. Table 1 presents our results. As we can see there are not any statistically significant differences in the baseline goal probabilities when focusing on the different values for a given feature. For example, an open play and a set-piece play both have the same baseline probability of a goal. The same is true for a left, right or header shot. The only slightly statistically significant difference is observed when a player creates the shot on his own (i.e., self assist), in which case there is a lower baseline probability of scoring a goal. However, as we will see in the following section, these features do not bear any significant modeling power for the probability of success for a shot. Figure 3 further depicts the probability of a shot ending up to a goal as a function of the distance from the goal line. As one might have expected the further away the shot is taken from the goal line the less probable is to result into a goal.

Fig. 2

A location density for the shots that resulted in a goal (left figure) and the ones that were misses (right figure).

3.1Goal Probability Regression Model

In this section we will present our model for the probability of a shot leading to a goal, which will form the basis of our expected goals calculations. In particular, let us denote with G_s a binary random variable that represents whether shot s resulted in a goal (G_s = 1) or not (G_s = 0). Every shot s is associated with a feature vector z_s that captures various attributes of the shot that we will describe shortly. We will use logistic regression to model the random variable G_s. The output of the model y provides us with the probability y = Pr [G_s = 1]. Hence, simply put, y is the probability of scoring a goal with shot s. The logistic regression model for G is given by:

The coefficient vector a includes the weights for each individual element of the input vector z, and is estimated using the shot data. The value of coefficient a_i quantifies the relation of feature z_i with the probability of scoring a goal (when keeping the rest of the factors included in the model constant), while the corresponding p-value quantifies its statistical significance. The features we use as the input for our model include:

Fig. 3

The goal percentage of shots as a function of distance from the goal line.

We trained the model with the shot data described in Section 2 and Table 2 presents our model. As we can see the distance from the goal and the angle of the shot are the only statistically significant features that impact the probability of a shot leading to a goal, while the type of shot, assist and play are not significant independent variables. A typical way to evaluate a logistic regression model is through its accuracy in predicting the output variable. In our case the prediction accuracy of the model is 77.2%. However, the purpose of using the above model is to calculate the expected goals based on the shots taken by a team or a player and not to predict whether a shot will result in a goal or not. In fact, the accuracy metric is rather misleading in this setting since given that only 25% of the shots end up to a goal the dataset¹1 is unbalanced Chawla et al. (2004); a deterministic model that predicts every shot to not be a goal will have a good performance with regards to accuracy - 75% in our dataset. On the contrary, the model’s accuracy of the predicted probability is our main evaluation criterion and not the model’s accuracy at predicting a goal. In order to evaluate the predicted probability, ideally we would like to have the same shot repeated several times. For example, consider a shot with a 30% probability of score. Then if the shot was taken 100 times - under the same conditions - one should expect approximately 30 of them to end up to a goal. However, this is obviously not possible to perform and hence, we rely in the following approach. In particular we will use all the shots in our dataset. If the predicted probabilities were accurate, when considering all the shots given a probability of x% being a goal, then x% of these shots should end up to a goal. Given the continuous nature of the probabilities we quantize them into groups that cover a 5% probability range. For higher predicted probabilities (> 0.7) we quantize the data to a 10% probability range given the small sample size within this range. Figure 4 presents on the y-axis the fraction of shots that ended up to a goal, while the x-axis corresponds to the predicted probability of goal. In order to avoid any biases in the estimation of the goal probability of a shot we used the leave-one-out approach, that is, for every shot s we used the rest of the data to train a model and then predict the goal probability for s. The blue line corresponds to the linear fit, while the shaded area is the 95% confidence interval of the fit. The red line is the y = x line (ideal case, i.e., perfect probability estimation). As we can see this line falls within the confidence interval, which translates to an accurate probability inference. The corresponding linear regression provides a slope with a 95% confidence interval of [0.77, 1.1] (R² = 0.93) and a non-significant intercept (0.03, p-value = 0.39), which to reiterate means that we cannot reject the hypothesis that our data fall on the line y = x where the slope is equal to 1 and the intercept is 0.

Fig. 4

Our logistic regression model is providing accurate goal probabilities. For a given set of shots with predicted goal probability π the fraction of shots that resulted in a goal is also (approximately) equal to π.

Another metric that has been traditionally used in the literature to evaluate the performance of a probabilistic prediction is the Brier score β Brier (1950). In the case of a binary probabilistic prediction the Brier score is calculated as:

In conclusion, the model for the goal probability is accurate and well-calibrated (compared to the baseline climatology model).

3.2Expected Goals Model

Using the above model we can now estimate the number of expected goals of a team/player within a game or a span of games. Let us consider the set of shots ST that team T has taken during the time span of interest. The shots that the team has taken are independent but not identically distributed since each shot i has a specific probability of success given the attributes of the shot, π_i, which is given by the above model. Therefore, we can model a sequence of shots taken by team T through a Poisson binomial distribution. A Poisson binomial distribution is the sum of independent (but not necessarily identically distributed) Bernoulli trials. Hence, the mean μ and variance σ² of the distribution are given by:

Fig. 5

Offensive and defensive efficiency of MLS teams as captured by the expected goals model.

Using the expected goals as calculated above one can classify teams based on how much better/worse they actually perform over the expectation. This evaluation can be made for both the offense and the defense of a team. Formally, we have the following definitions:

Fig. 6

Our expected goals model can help us classify players to “underperformers” and “overperformers”.

3.3Fractal Dimension and Offensive Efficiency

In what follows we want to examine whether there is any relationship between the spatial distribution of the shots taken and the offensive efficiency of the teams. In order to achieve this we need a metric that concretely describes the spatial distribution of a team’s shot in a condensed manner (e.g., through a single number). For that matter we will borrow the notion of fractal dimension from fractal theory. In particular, let us consider a set of points S . With C (r) being the fraction of pairs of points from S that have distance smaller or equal to r, S behaves like a fractal with intrinsic fractal dimension D₂ in the range of scales r₁ to r₂ iff:

An infinitely complicated set S would exhibit the above scaling over all the possible ranges of r. However, real objects are finite and hence, Equation (7) holds only over a specific range of scales. For example, a cloud of points uniformly distributed in the unit square, has intrinsic dimension D₂ = 2, for the range of scales [r_min, 1], where r_min is the smallest distance among the pairs of S .

Fig. 7

The empirical cumulative distribution of the MLS teams’ shot chart fractal dimension.

Fig. 8

A location density for the shots that resulted in a goal (left figure) and the ones that were misses (right figure).

In our case, the set S is the shot chart of the team T under examination. One of the things that the fractal dimension of T, D₂ (T), reveals is how uniformly its shots are distributed. A small value for D₂ (T) essentially translates to a team that exploits a small area in the field for its attack (more accurately its shots), while a high value for D₂ (T) describes a team whose attack is more uniformly distributed over the field. Figure 7 depicts the distribution of the fractal dimension for all the teams based on our dataset. As we can see there are teams that exhibit very small fractal dimensionality (as small as 0.2), while there are teams with dimensionality as high as 1.6. In fact, there seems to be a rapid increase in the cumulative distribution right around the median of the values of fractal dimensions, that is, 1.35. We thus split the teams into two groups, that is, teams with fractal dimension at the top 50-th percentile (i.e., greater than 1.35) and those at the bottom 50-th percentile. We then compare the offensive efficiency of the two groups and we find that the offensive efficiency of the teams in the bottom 50-th percentile of D₂ is approximately 6.7%, while that of the teams in the top 50-th percentile is approximately -9.3%. This translates to an average of 16% difference (p-value < 0.05) in the offensive efficiency of the two groups of teams. Furthermore, overall there is a moderate correlation (ρ = -0.36) between the offensive efficiency and D₂.

What these results seem to imply is that teams with higher offensive efficiency tend to utilize a smaller area on the field with regards to their shots. This might seem counter intuitive, since one might expect that taking shots from a variety of locations would stretch the defense more, create more open spaces and hence, create better situations for scoring. However, the opposite is true and one of the reasons might be the fact that a team by having its shots uniformly distributed over the space, will end up taking many low quality shots. Furthermore, the defenses are aware of the low chance of making a shot from a long distance and hence, the stretching of the defense is less than expected. Of course, the tactical strategy followed by a team’s offensive or defensive unit - and of course the spatial distribution of the team on the field - depends a lot on the talent on the team. For example, a defense that has to face a team with talented offensive players might want to give them space in the backfield in order to avoid one-to-one situations near the penalty area that can be a miss-match for the defense. This will end up in a situation similar to the one observed, i.e., non-stretched defenses that allow distant, low-probability, shots but try to close the spaces closer to the goal line by creating a denser defense in the high-percentage shot areas. Figure 8 depicts two representative cases of teams with small and high fractal dimensionality. As we can see while both teams take approximately the same number of shots, the Orlando City Soccer Club is much more efficient, since it distributes its shots over a smaller area, which is also the area of high quality shots (i.e., close distance to the goal).