## 6 Concluding Remarks
\begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline \multicolumn{1}{c}{**Rank**} & \multicolumn{1}{c}{**Name**} & \multicolumn{1}{c}{**Position**} & \multicolumn{1}{c}{**Age**} & \multicolumn{1}{c}{**Current**} & \multicolumn{1}{c}{**Peak**} & \multicolumn{1}{c}{**Current**} & \multicolumn{1}{c}{**Peak**} & \multicolumn{1}{c}{**Current**} & \multicolumn{1}{c}{**Peak**} \\ \hline
1 & [PERSON] & F & 28 & 0.340 & 0.349 & 0.052 & 0.065 & 0.393 & 0.410 \\
2 & [PERSON] & M & 26 & 0.280 & 0.286 & 0.106 & 0.108 & 0.386 & 0.390 \\
3 & [PERSON] & D,M & 25 & 0.248 & 0.260 & 0.118 & 0.134 & 0.367 & 0.389 \\
4 & [PERSON] & F & 27 & 0.282 & 0.302 & 0.084 & 0.091 & 0.366 & 0.389 \\
5 & [PERSON] & D & 28 & 0.286 & 0.296 & 0.075 & 0.087 & 0.362 & 0.379 \\
6 & [PERSON] & D,M & 28 & 0.211 & 0.225 & 0.114 & 0.124 & 0.326 & 0.345 \\
7 & [PERSON] & M, F & 26 & 0.251 & 0.260 & 0.058 & 0.069 & 0.309 & 0.325 \\
8 & [PERSON] & D,M & 28 & 0.200 & 0.209 & 0.107 & 0.119 & 0.307 & 0.324 \\
9 & [PERSON] & F & 33 & 0.288 & 0.348 & 0.016 & 0.076 & 0.305 & 0.420 \\
10 & [PERSON] & G & 31 & NA & NA & 0.300 & 0.330 & 0.330 & 0.330 \\
11 & [PERSON] & D,M & 33 & 0.186 & 0.248 & 0.104 & 0.169 & 0.291 & 0.413 \\
12 & [PERSON] & M & 30 & 0.121 & 0.153 & 0.165 & 0.185 & 0.286 & 0.334 \\
13 & [PERSON] & D,M, F & 24 & 0.162 & 0.166 & 0.120 & 0.139 & 0.283 & 0.301 \\
14 & [PERSON] & D,M & 31 & 0.134 & 0.180 & 0.143 & 0.175 & 0.277 & 0.351 \\
15 & [PERSON] & F & 34 & 0.257 & 0.323 & 0.011 & 0.087 & 0.268 & 0.406 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Top 15 rated players of Bayern München in July 2017, estimated using the full data set from August 2008 to July 2017.  
defensive ratings that are in opposition to current perceptions of players can be useful triggers for discussions. However, it remains an open question as to whether the split in offensive and defensive ratings lead to a noisy and seemingly arbitrarily set defensive ratings, or whether the new ratings are able to detect patterns that are not common knowledge. The reported experiments have some limitations. For one, they are based on a single data set, which, although large, may still be of insufficient size to calculate the best version of the offensive-defensive plus-minus ratings. In particular, it may be that additional seasons of data would lead to more stable results, and facilitate more robust settings for the hyperparameters of the rating model. In addition, there are few benchmarks to which the ratings can be compared. While this is true in general for overall player ratings, it is perhaps even more pronounced for defensive ratings. It may be that access to more granular data, such as event data, could facilitate improved defensive ratings. In conclusion, the new ratings perform on par with previous ratings when it comes to predicting future match outcomes. The offensive-defensive ratings allow additional insights into the game, by assessing the importance of the home field advantage and red cards on the individual scoring rates of the teams. For future research, generalizations of the model may be used to analyse additional elements, such as the effects of the playing schedule and player fatigue, or the effects of defensively or offensively oriented head coaches. The new rating model exploits information about player positions. It may be interesting to investigate whether more specific player position data could be used to improve the split into defensive and offensive ratings, for example, by taking into account the difference between center backs and full backs or wing backs. This research received no external funding. The author wishes to thank the three reviewers for their concrete and useful suggestions that helped to improve the manuscript. The author declares no conflicts of interest.