Evaluation of creating scoring opportunities for
teammates in soccer via trajectory prediction
Masakiyo Teranishi1 , Kazushi Tsutsui1 , Kazuya Takeda1 , and Keisuke Fujii1,2,3
Graduate School of Informatics, Nagoya University, Nagoya, Japan.
Center for Advanced Intelligence Project, RIKEN, Fukuoka, Japan.
3
PRESTO, Japan Science and Technology Agency, Saitama, Japan.

12

Abstract. Evaluating the individual movements for teammates in soccer players is crucial for assessing teamwork, scouting, and fan engagement. It has been said that players in a 90-min game do not have theball for about 87 minutes on average. However, it has remained difficult
to evaluate an attacking player without receiving the ball, and to reveal
how movement contributes to the creation of scoring opportunities for
teammates. In this paper, we evaluate players who create off-ball scoring
opportunities by comparing actual movements with the reference movements generated via trajectory prediction. First, we predict the trajectories of players using a graph variational recurrent neural network that
can accurately model the relationship between players and predict thelong-term trajectory. Next, based on the difference in the modified offball evaluation index between the actual and the predicted trajectory asa reference, we evaluate how the actual movement contributes to scoring opportunity compared to the predicted movement. For verification,
we examined the relationship with the annual salary, the goals, and therating in the game by experts for all games of a team in a professionalsoccer league in a year. The results show that the annual salary and the
proposed indicator correlated significantly, which could not be explainedby the existing indicators and goals. Our results suggest the effectivenessof the proposed method as an indicator for a player without the ball tocreate a scoring chance for teammates4 .
Keywords: multi-agent · deep learning · trajectory · sports · football

1

Introduction

Assessing the movements of individual players for teammates in team sports is
an important aspect of building teamwork, assessment of players’ salaries, player
recruitment, and scouting. In soccer, most analytics has focused on the outcomesof discrete events near the ball (on-ball) [30, 10, 35, 36, 3, 9, 29, 28] whereas muchof the importance in player movements exist in the events without the ball (offball). For example, it is said that players in a 90-min game do not have the ballfor about 87 minutes on average [14]. However, continuous off-ball movements
are usually not discretized and difficult to understand except for core fans, experienced players, and coaches. Also for the media and building fan engagement,
quantitative evaluation of off-ball players is an issue in demand, which provides
4

Appendix: https://arxiv.org/abs/2206.01899, *email: fujii@i.nagoya-u.ac.jp

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a common reference for beginners and experts in the sport e.g., when arguing aplay of a favorite player.
Regarding the off-ball player evaluation methods, the positioning itself related to the goal was evaluated from the location data of all players and theball. For example, the method called off-ball scoring opportunity (OBSO) toevaluate the player who receives the ball [41] and the method to evaluate themovement to create space [14] have been proposed. However, it has been still
difficult to clarify how movements contribute to the creation of scoring opportunities for teammates, to evaluate other attacking players who do not receiveit (e.g., a player moving tactically for teammates), and often to evaluate a scoreprediction to reflect the position of the multiple defenders.
In this paper, we propose a new evaluation indicator, Creating Off-Ball Scoring Opportunity (C-OBSO in Fig. 1A), aiming for evaluating players who create
scoring opportunities when the attacking player is without the ball. The overview
of our method is as follows. (i) First, we modify the score model in the framework of OBSO [41] with the potential score model that reflects the positionsof multiple defenders with a mixed Gaussian distribution (Fig. 1B). (ii) Next,
we accurately model the relationship between athletes and perform long-term
trajectory predictions (Fig. 1A) using the graph variational recurrent neural network (GVRNN) [46]. (iii) Finally, based on the difference in the modified off-ballevaluation index between the actual and the predicted trajectory (Fig. 1A), we
evaluate how the actual movement contributes to scoring opportunity relative
to the predicted movement as a reference.
In summary, our main contributions were as follows. (1) We proposed anevaluation method of how movements contributed to the creation of scoring opportunities compared to the predicted movements of off-ball players in teamsports attacks. (2) As a score predictor, we proposed a potential score modelthat considers the positions of multiple defenders in a mixed Gaussian distribution. (3) In the experiment, we analyzed the relationship between the annual
salary, the goals, and the game rating by experts, and show the effectivenessof the proposed method as an indicator for an off-ball player to create scoring
opportunities for teammates. Our approach can evaluate continuous movementsof players by comparing with the reference (here predicted) movements, which
are difficult to be discretized or labeled but crucial for teamwork, scouting, andfan engagement. The structure of this paper is as follows. First, we overview the
related works in Section 4 and present experimental results in Section 3. Next,
we describe our methods in Section 2 and conclude this paper in Section 5.

2

Proposed framework

Here, we propose C-OBSO based on the motivation to evaluate players whocreate off-ball scoring opportunities for teammates. To this end, in Section 2.1,
we first propose a potential score model that reflects the positions of multiple
defenders with a mixed Gaussian distribution. Next, in Section 2.2, we predict
multi-agent trajectory using GVRNN [46] and evaluate the difference between
the actual value of the modified OBSO and the predicted value (as a reference) to
evaluate how the movement contributed to the creation of scoring opportunities.

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Fig. 1. Our C-OBSO example and potential score model. (A) Example of C-OBSO
computation. A1 is the player to be finally evaluated, A2 is the shooting player, D1
and D2 are the defender of A1 and A2, respectively. V1 is the C-OBSO value of A1,
2
′2
VOBSO
is the actual OBSO, and VOBSO
is the reference OBSO value of A2 using
the predicted trajectory. (B) Potential score model. Left: shot-blocking distribution
formed by defenders. Right: shot probability corresponding to each shot vector. The
vertical axis is the goal position (m), and the horizontal axis is the shot probability
corresponding to each shot vector.

2.1

Potential score model in modified OBSOFirst, we describe the base model of our evaluation method called OBSO [41]and then propose the potential score model. OBSO evaluates off-ball players bycomputing the following joint probability
P (G|D) = Σr∈R×R P (Sr ∩ Cr ∩ Tr |D)= Σr P (Sr |Cr , Tr , D)P (Cr |Tr , D)P (Tr |D),

(1)
(2)

where D is the instantaneous state of the game (e.g., player positions and velocities). P (Sr ) is the probability of scoring from an arbitrary point r ∈ R × R on
the pitch, assuming the next on-ball event occurs there. P (Cr ) is the probability
that the passing team will control a ball at point r. P (Tr ) is the probability
that the next on-ball event occurs at point r. Here, for simplicity, we can assume
that P (Sr |D), P (Tr |D), P (Cr |D) are independent if the parameter α = 0 in theoriginal work implementation (Eq. (6) in [41]). Then, the joint probability canbe decomposed into a series of conditional probabilities as follows:
P (G|D) = Σr∈R×R P (Sr |D)P (Cr |D)P (Tr |D).

(3)

P (Cr |D) is the probability that the attacking team will control the ball at pointr assuming the next on-ball event occurs there, which is called the potential
pitch control field (PPCF). P (Tr |D) is defined as a two-dimensional Gaussian
distribution with the current ball coordinates as the mean. P (Sr |D) is simply calculated as a value that decreases with the distance from the goal. We
used the grid data and computed P (Cr |D) and P (Tr |D) based on the code athttps://github.com/Friends-of-Tracking-Data-FoTD/LaurieOnTracking.
In the original OBSO [41], the scoring probability was calculated as theoutput P (Sr |D) of the score model as a function of the distance from the goal.
However, the scoring probability may depend on the angle to the goal and the4

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defensive position of the opponent. Therefore, in this paper, we propose a scoremodel that reflects the angle to the goal and the position of multiple defenders.
Here, we consider the shot-blocking distribution of the defenders who can block
shots in the field, and propose a potential model where the scoring probability
decreases when defenders exist. The basic idea shown in Fig. 1B is to calculate
the scoring probability from the angle to the goal at which the shot tends to bescored, considering the mixed distribution of the positions of multiple defenders.
The proposed scoring probability P (Srp |D) at a certain point r is calculated as
the sum of the shot value Vshot as follows:nX
P (Srp |D) =Vshot (⃗
si ),
(4)
i=1

Vshot (⃗s) = C(c − Vblock ),(5)

where n is determined by the angle from the shooting position to the goal, and ⃗s
is a shot vector per degree (n is larger when the shot from the center and smallerfrom the side). The shot value Vshot is calculated by subtracting the shot block
value Vblock from a certain constant c (c, C are parameters determined from data
to be adjusted so that Vshot ≥ 0 and P (Srp ) ∈ [0, 1]). Let Vblock be the sum of
the shot block distribution values along the shot vector ⃗s. The shot blocking
distribution is the sum of the normal distributions (variance σ 2 = 0.5 + ld )assigned to each defender on the goal side of the shooting position (shot blockableplayers using legs), where ld is the distance between the shooting position and thedefender. We consider that goalkeepers have a shot blocking distribution withtwice the value of normal defenders because of a higher shot-blocking ability.
Here, we assume that the block distribution is not changed with the distancefrom the ball. A defender near to the ball may affect the ball, but far players use
the flight time of the ball for their movement. This formulation is left for future
work.
2.2

C-OBSO with trajectory prediction

Here, we describe the base model of our trajectory prediction method called
GVRNN [46] and then describe our C-OBSO framework. Our contribution here
is to evaluate how the actual “off-ball” movement contributes to scoring opportunity compared to the predicted movement (or trajectory) as a reference. In
our method, we use GVRNN [46], which is a VRNN [8] combined with a graphneural network (GNN [22]). In GVRNN, the graph encoder-decoder networkmodels the relationship between players as a graph, which is one of the best
performing models for predicting player trajectories in team sports [46]. This is
a probabilistic model which can sample multiple possible trajectories.
Based on the trajectory prediction, we propose an evaluation index C-OBSO
of players who create scoring opportunities for teammates. The basic idea isto evaluate an off-ball player from the difference in the modified OBSO values
between the predicted and actual movements of the players. The C-OBSO valueof a player i without the ball can be expressed as follows.
k′kVi = VOBSO− VOBSO,(6)

Evaluation of creating soccer scoring opportunities

5

where the player k is the ball carrier who performs a final action (e.g., shot),
k
′k
VOBSO
is the modified OBSO in the actual game situation, and VOBSO
is themodified OBSO based on the predicted trajectory as a reference. For example,
in Fig. 1A, C-OBSO is positive and the player to be evaluated (A1) contributes
more to the shooter (A2) than the referenced (predicted) player. Specifically,
A1 has created a more advantageous situation for A2 by attracting D1 morethan expected. C-OBSO can evaluate a player in such situations with an interpretable value (i.e., the increase in scoring probability). Theoretically, if perfectlypredicted, C-OBSO is always zero, but actually, if we apply this to a test data,
the perfect prediction is impossible. In other words, we assume the imperfect
trajectory prediction in this framework.

3

Experiments

In this section, we validate the proposed method of the potential score model,
the trajectory prediction model (GVRNN), and the C-OBSO itself. For our
implementation, the code is available at https://github.com/keisuke198619/COBSO.
3.1 Dataset
In this study, we used all 34 games data of Yokohama F Marinos in the Meiji J1League 2019 season to perform specific player-level evaluations in limited data.
Note that the tracking data for all players and timesteps were not publicly sharedin such amounts. The dataset includes event data (i.e., labels of actions, e.g.,
passing and shooting, recorded at 30 Hz and the simultaneous xy coordinatesof the ball) and tracking data (i.e., xy coordinates of all players recorded at 25
Hz) provided by Data Stadium Inc. The company was licensed to acquire thisdata and sell it to third parties, and it was guaranteed that the use of the data
would not infringe on any rights of the players or teams. For annual salaries, weused the salaries of the same team (Yokohama) in 2019 [40] because they weredifferent valuation criteria for different teams and the transfer of the players took
place during the season. The goals for each player in each match were collectedfrom [20]. The rating by experts in each match [39] was also used for verification,
which was scored in 0.5 point increments with a maximum of 10 points.
3.2 Data processing for verification
We used the attacking data of Yokohama F Marinos for the test and those of
the opponent teams for training the model or parameter fitting. Again, since thedata was limited in this study, we split the data in such a way, and if we havemore data, we can analyze all teams with the training data with the same team.
Here we describe the processing of the potential score model, the trajectory
prediction model, the C-OBSO, and their statistical analyses.
Potential score model. To validate the potential score model, the opponent’s
shots (345 shots, 34 goals) were used for fitting the parameters c and C, andYokohama F Marinos’ shots (494 shots, 59 goals) were used for verification. The
parameters c, C of the potential score model were determined to be c = 1.1, C =1/150 using the data of the opponents. The potential score model was verified by

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the root mean square error (RMSE) between the actual score and the calculated
scoring probability. We compared the RMSE with that of a simple score model
as a function of distance from the goal for implementing the original OBSO [41](see also Section 2.1). Although there have been more holistic score models suchas [15, 1], to fairly compare with our potential model as a component of themodified OBSO, we consider the simple score model as an appropriate baseline.
Trajectory prediction model. For the test data of trajectory prediction and
C-OBSO, we used 412 shot scenes of Yokohama F Marinos (we selected thesequences of consecutive events and excluded too short events such as a freekick). The trajectory prediction model was trained using the opponents’ data to
generate “league average” trajectories. The tracking data were down-sampled to
10 Hz (after prediction, up-sampled at the original 25 Hz) based on [17]. To verify
the accuracy of the long-term trajectory prediction, we set various time lengths(6, 8, 10, and 12 s) using mean trajectories in 10 samples. We divided into the
opponent data for batch training (6 s: 94208 sequences, 8 s: 49152 sequences,
10 s: 33536 sequences, 12 s: 24320 sequences) and the validation (6 s: 10477
sequences, 8 s: 5479 sequences, 10 s: 3730 sequences, 12 s: 2721 sequences). Notethat the end of all sequences was the moment of a shot. The input feature has
92 dimensions (the xy coordinates and the velocity of 22 players and the ball).
During training, the model was trained based on the one-step prediction errorof all combinations of the two attackers who invaded the attacking third. We
simultaneously predicted the three players: one of the off-ball attackers and thedefenders closest to each attacker. Note that we only consider the three players’interactions and ignore others’ interactions, because the prediction error will
increase if the numbers increase, and the increase of the predicted players is left
for future work.
For the test data of the 412 sequences, the three relevant players and theattacker from the same criterion were predicted. At the inference, using 2 s sequences as burn-in period, we predicted the sequences for the subsequent time
lengths (i.e., 4, 6, 8, and 10 s) by updating the estimated position and velocity(i.e., performed long-term prediction). For the training of the proposed and baseline models, we used the Adam optimizer [21] with a learning rate of 0.001 and10 training epochs. We set the batchsize to 256. For the performance metrics, weused the endpoint error (mean absolute error: MAE) from the actual trajectory.
C-OBSO. To compute C-OBSO, predicted trajectories with 4 s (total 6 s) wereused. This is because a longer prediction time will result in a larger prediction
error, while a shorter prediction time will not make a difference in the evaluation of C-OBSO. Although the negative values of C-OBSO are also calculated bycomparison with the reference, the negative values were calculated as 0, assuming that they may not have a negative effect on the behavioral players. This isbecause there were many situations with negative values in which the shooter’s
defender did not take an appropriate defensive position in the predicted trajectory.
Statistical analysis. For the verification of C-OBSO, we examined the relationship with the annual salary, the goals, and the expert’s rating. Note that

Evaluation of creating soccer scoring opportunities

7

there is no ground truth available for the verification. We also compared themwith the existing OBSO [41]. Since some of the data often did not follow normal distributions, we used Spearman’s rank correlation coefficient ρ for these
relationships. Regarding the RMSE in the potential score model and MAE in
the trajectory prediction, for the same reason, we used nonparametric statisticaltests to compare with the baselines. Regarding the potential score model, we
used the Wilcoxon rank sum test. For all statistical calculations, p < 0.05 wasconsidered as significant.
3.3

Our model verification

First, we validated the potential score model needed to calculate the C-OBSO.
The RMSE with the actual scores was 0.324 ± 0.014 for the conventional scoremodel [41] without considering the defenders and goal angles, and 0.309 ± 0.0014
for the potential score model (p < 10−10 ). This result suggests that the proposed
method models the scores more accurately.
Figure 2 shows an example of the two methods in two actual situations wherea shot is attempted from a similar distance. In the existing method, the probabilities were the same (both 0.1237) because the shots were taken from almost thesame distance. The proposed method had a lower scoring probability with moredefenders (upper: 0.0489, lower: 0.1202). We indicate that the proposed method
reflects the position of multiple defenders and can model the score accurately.
Next, we show the results of the trajectory prediction model for computing C-OBSO. Endpoint errors (MAE and standard error, [m]) in GVRNN were0.608 ± 0.014, 0.867 ± 0.022, 1.701 ± 0.045, 1.606 ± 0.042 in 4, 6, 8, 10 s prediction. In GVRNN, longer predictions show larger prediction errors except for thedifference between 8 s and 10 s. Since the 4 s prediction of GVRNN achieved a
low the MAE of less than 0.7 m, the GVRNN trajectory prediction of 4 s wasused in the next C-OBSO.
3.4

C-OBSO results

Verification of C-OBSO is challenging because of no ground truth values or player
ratings. Therefore, we analyzed the relationship with the annual salary, the goals,
and the game rating by experts, whereas we admit that these variables include
various confounding factors. The relationships between the average C-OBSO
and OBSO values of each player of Yokohama F Marinos in 2019 and the annual
salary of each player in 2019 are shown in Fig. 3 (note that the tracking data for
all players and timesteps were not publicly shared). Here we analyzed 15 playerswith more than 10 sequences under evaluation. As a result, there was a significant
positive correlation between annual salary and C-OBSO (ρ = 0.45, p = 0.046).
In addition, the two players with the higher evaluation values but lower salaries(in red in Fig. 3A) were highly evaluated players, who won the individual awards(the most valuable player and valuable player award). In fact, their annual salary
for the following year (2020) was also increased (valuable player: increased from11 million yen to 40 million yen; the most valuable player: increased from 20
million yen to 60 million yen).

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Fig. 2. Comparison between the score model of conventional and the proposed potential model. The scoring probability of our model is lower (A) when the defenders are
crowded than (B), whereas that of the conventional score model in (A) was the sameas (B).

We found that these tendencies were similar to the C-OBSO and OBSO
without the potential score model. On the other hand, there was no significant correlation (ρ = −0.28, p = 0.154) for OBSO, which evaluates a player’s
own scoring opportunities (Fig. 3B). We also examined the relationship between annual salary and goals (Fig. 3C), and found no significant correlation(ρ = −0.23, p = 0.208). Therefore, there was no relationship between annual
salary and goals. There were many players with zero goals, and it is difficult toevaluate them only with the goals.
Next, in order to examine the relationship with player performance in moredetail, we show the relationship between C-OBSO and the rating by experts ofthe top three scorers (Nakagawa with 15 goals, Marcos with 15 goals, and Edigar
with 11 goals in this season) in Fig. 4. We analyzed the games in which there were
two or more C-OBSO evaluations using the average of C-OBSO values on eachgame (17 games for Nakagawa, 14 games for Marcos, and 10 games for Edigar).
A strong positive correlation was found only for Nakagawa (ρ = 0.75, p = 0.0003)
but not for Marcos (ρ = 0.27, p = 0.174) and Edigar (ρ = −0.37, p = 0.145).
Similarly, there were no significant correlations between them for all players(ρs < 0.190, ps > 0.05). In addition to the number of scoring opportunities for
teammates (17 times), the results found that Nakagawa would be subjectively
and quantitatively an outstanding player.

Evaluation of creating soccer scoring opportunities

9

Fig. 3. Relationship between indicators, goal, and annual salary in a team. (A) Relationship between C-OBSO and the salary. (B) Relationship between OBSO [41] andthe salary. (C) Relationship between each player’s goals and annual salary. Red players received individual awards (Hatanaka: valuable player Award, Nakagawa: the mostvaluable player award).

Fig. 4. Relationship between C-OBSO and the rating by experts of the top three scorers(A: Nakagawa, B: Marcos, C: Edigar) for each game.

For reference, we also show the relationship between the goals of the top threescorers and the ratings by experts in Fig. 5. We analyzed the games in which
each player played (33 games for Nakagawa, 33 for Marcos, and 16 for Edigar).
For each player, there were strong correlations between the goals and the rating (Nakagawa ρ = 0.63, p = 4.33 × 10−5 , Marcos ρ = 0.71, p = 1.98 × 10−6 ,Edigar ρ = 0.91, p = 4.40 × 10−7 ). We found that the rating of each game depends on a rare event (i.e., goals). Similarly, there were significant correlations
between them for all players (ρs > 0.516, ps < 0.018). Recall that there was astronger correlation between C-OBSO and Nakagawa’s rating than for the othertwo players. Nakagawa also had higher average ratings than the other players(6.26 for Nakagawa, 5.97 for Marcos, and 6.09 for Edigar), and he was the player
who won the most valuable player award. The game rating by experts would
depend on the goals, but it may also evaluate the creation of scoring opportunities only for Nakagawa. From these results, we speculate that Nakagawa washighly evaluated not only for his scoring but also for his contribution to otherattacking players. Our method can also evaluate players difficult to be evaluatedby conventional indicators, which is crucial for assessing teamwork and player
salary, player recruitment, and scouting.

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Fig. 5. Relationship between the goals and the rating by experts of the top three scorers(A: Nakagawa, B: Marcos, C: Edigar) for each game. The size of the circle represents
the frequency because there are many combinations of the goals and the rating withthe same value.

4

Related work

In the tactical behaviors of team sports, agents select an action that follows apolicy (or strategy) in a state, receives a reward from the environment and others,
and updates the state [16]. This is similar to a reinforcement learning framework(e.g., [2]). Due to the difficulty in modeling the entire framework from data forvarious reasons [44] (e.g., a sparse reward and difficulty in estimating intents),
we can adopt two approaches: to estimate the related variables and functions
from data (i.e., inverse approach) as a sub-problem, and to build a model (e.g.,
reinforcement learning model) to generate data in virtual space (i.e., forward
approach, e.g., [23, 37]). Here, we focus on the former approach and introduce
the research from the view of inverse approaches.
There have been many approaches to quantitatively evaluate the actions ofattacking players about the scoring, such as based on the expected scores using tracking data [30, 10, 35, 36, 3], action data such as dribbling and passing [9,13], and estimating state-action value function (Q-function) [45, 29, 28]. Some
researchers have evaluated passes [32, 4, 12], and others evaluated actions to receive a ball by assigning a value to the location with the highest expected score[41, 27] and a rule-based manner [19]. In particular, Spearman [41] proposedan evaluation metric called OBSO to evaluate behavior based on location data
and rule-based modeling. Defensive behaviors have also been evaluated basedon data-driven [34, 43] and rule-based manners (e.g., [42]). However, these score
evaluations do not often reflect the position of multiple defenders and goal angles
in rule-based manner.
From the perspective of reinforcement learning, there have been many studies on inverse approaches. As for the study of state evaluation, there are several
studies based on score expectation (e.g., [6, 7, 15]) and based on the value of
space (e.g., [5, 14]). There is also research on estimating reward functions byinverse reinforcement learning [31, 33]. Researchers performed trajectory prediction sometimes in terms of the policy function estimation, as imitation learning[25, 24, 42, 17] and behavioral modeling [47, 46, 26, 18] to mimic (not optimize)
the policy using neural network approaches. In this paper, we first propose amethod to evaluate how the actual “off-ball” movement contributes to scoringEvaluation of creating soccer scoring opportunities

11

opportunity based on the difference between the state values generated from theactual and the reference policies.

5

Conclusion

In this paper, we evaluated players who create off-ball scoring opportunities bycomparing actual movements with the reference movements generated by trajectory prediction. Our results suggest the effectiveness of the proposed method
as an indicator for a player without the ball to create scoring opportunities for
teammates. For future work, although the number of players to be evaluated wasdetermined in the minimum setting, it is possible to evaluate the contribution
to the scoring opportunities for teammates in a less limited way by predicting
a larger number of players in both offense and defense. Furthermore, since ourmethod evaluates off-ball players by comparing them with the referenced trajectory, the value becomes too small. Computing the evaluation value in a moreinterpretable way (e.g., in a score scale) would be future work. Finally, computing our indicators from broadcast videos (e.g., [11]) or other videos (e.g., top- or
side-view [38]) would also be future work.

Acknowledgments
This work was supported by JSPS KAKENHI (Grant Numbers 20H04075 and21H05300) and JST Presto (Grant Number JPMJPR20CA).

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