Table 2:
Number of actions and sequences per time horizon
| Time Horizon | Actions | Sequences |
|---|---|---|
| 16-16 | 1,332 | 112,325 |
| 32-32 | 1,093 | 90,033 |
| 64-64 | 795 | 57,629 |
| 128-128 | 303 | 23,771 |
Fig. 1:
Illustration of 4 randomly selected target series P over different time horizons. The vertical dashed gray line represents the separation between known sequence on the left and prediction sequence on the right, the horizontal dotted black line represents the Stand Still baseline and the colored dotted lines the Constant Velocity baseline associated with the series displayed with the same color.
Fig. 2:
MSE and best epoch of the 5 best experiments for each model over the validation set.
Fig. 3:
Distribution of FDE relative improvement (in %) vs best baseline (Linear Regression for all cases) for Multivariate (M) and Univariate (U) inputs models for each time horizon. Each graph represents the results of the test procedure over the corresponding time horizon. The number in parenthesis indicates the number of models tested (see 4.1) and the model names located below the boxplots indicate the models reaching the lowest minimum FDE over the test set. The dashed red line at y = 0 materializes the score of the best performing baseline. Results from Mann-Whitney U tests are reported above the boxplots. Significance code: * p<0.05, ** p< 0.01, *** p<0.001, **** p<0.0001.
Fig. 4:
Analysis of Permutation importance for time horizon 128-128. The left panel displays the importance per feature type (position over the x-axis and y-axis, velocity and goal angle). The dashed red line materializes the 0-impact level while the dotted green line the importance of the performance feature. The middle panel indicates the importance of features per player. The heatmap of the right panel indicates feature importance over the two combined dimensions. The black line separates the forwards (player 1 to 8) and backs (players 9 to 15).
Table 1:
Testing summary - Time horizon 16-16
| Model | Inputs | n | Min. FDE | μ | σ |
|---|---|---|---|---|---|
| Crossformer | Multivariate | 12 | 3.90 | 3.94 | ±0.03 |
| Crossformer | Univariate | 7 | 3.95 | 3.96 | ±0.01 |
| Pyraformer | Multivariate | 9 | 4.11 | 4.13 | ±0.01 |
| StandStill | Baseline | 1 | 5.23 | 5.23 | - |
| Constant Velocity | Baseline | 1 | 6.13 | 6.13 | - |
| Linear Regression | Baseline | 1 | 4.50 | 4.50 | - |
Table 2:
Testing summary - Time horizon 32-32
| Model | Inputs | n | Min. FDE | μ | σ |
|---|---|---|---|---|---|
| Crossformer | Multivariate | 57 | 6.60 | 6.71 | ±0.05 |
| Crossformer | Univariate | 6 | 6.86 | 6.88 | ±0.02 |
| Non Stationary Transformer | Univariate | 6 | 7.46 | 8.49 | ±0.61 |
| Pyraformer | Multivariate | 13 | 7.04 | 7.10 | ±0.03 |
| Pyraformer | Univariate | 3 | 6.91 | 6.93 | ±0.02 |
| iTransformer | Multivariate | 5 | 6.93 | 6.96 | ±0.02 |
| StandStill | Baseline | 1 | 8.44 | 8.44 | - |
| Constant Velocity | Baseline | 1 | 11.41 | 11.41 | - |
| Linear Regression | Baseline | 1 | 7.82 | 7.82 | - |
Table 3:
Testing summary - Time horizon 64-64
| Model | Inputs | n | Min. FDE | μ | σ |
|---|---|---|---|---|---|
| Crossformer | Multivariate | 40 | 9.83 | 10.13 | ±0.13 |
| Crossformer | Univariate | 1 | 10.48 | 10.48 | - |
| PatchTST | Multivariate | 15 | 10.01 | 10.13 | ±0.07 |
| Pyraformer | Multivariate | 2 | 10.20 | 10.22 | ±0.03 |
| Pyraformer | Univariate | 18 | 10.43 | 10.57 | ±0.07 |
| iTransformer | Multivariate | 14 | 9.85 | 9.95 | ±0.06 |
| StandStill | Baseline | 1 | 11.79 | 11.79 | - |
| Constant Velocity | Baseline | 1 | 17.56 | 17.56 | - |
| Lineal Regression | Baseline | 1 | 11.14 | 11.14 | - |
Table 4:
Testing summary - Time horizon 128-128
| Model | Inputs | n | Min. FDE | μ | σ |
|---|---|---|---|---|---|
| Crossformer | Multivariate | 18 | 13.48 | 13.68 | ±0.11 |
| Crossformer | Univariate | 3 | 13.68 | 13.78 | ±0.09 |
| DLinear | Multivariate | 2 | 13.65 | 13.72 | ±0.10 |
| LSTM | Multivariate | 7 | 13.93 | 14.35 | ±0.28 |
| PatchTST | Multivariate | 88 | 12.29 | 13.07 | ±0.44 |
| Pyraformer | Multivariate | 7 | 13.51 | 13.99 | ±0.31 |
| Pyraformer | Univariate | 30 | 13.59 | 13.94 | ±0.25 |
| Transformer | Multivariate | 5 | 13.70 | 13.71 | ±0.01 |
| Transformer | Univariate | 4 | 14.38 | 14.68 | ±0.27 |
| iTransformer | Multivariate | 71 | 12.74 | 13.50 | ±0.35 |
| StandStill | Baseline | 1 | 15.09 | 15.09 | - |
| Constant Velocity | Baseline | 1 | 25.17 | 25.17 | - |
| Linear Regression | Baseline | 1 | 13.46 | 13.46 | - |