We study time-to-arrival (TTA) in sports: the shortest plausible time a player needs to reach an arbitrary location on the pitch given their current kinematic state, estimated as the lower envelope of observed locomotion times. Rather than modeling space ownership through position density estimation or closed-form physics-based models, we cast TTA as a conditional quantile regression problem, estimating the lower quantile of locomotion time so the model learns a player's lower-envelope arrival time. We train the model on optical tracking data from all 64 matches of the PFF FC FIFA World Cup 2022 dataset, and evaluate a lightweight multilayer perceptron against six player-motion models from the literature, re-expressed as TTA predictors, and a learned reachable- hull baseline. Across our evaluation metrics, the MLP consistently outperforms every physics-based baseline. To probe the generality of the learned locomotion model, we further evaluate it on basketball tracking data and a single-game NFL tracking dataset, analyzing the transferability of player motion dynamics across sports. Our project is available at https://github.com/mguti97/TTA-Quantile.
Figures & tables
Figure 1: Predicted TTA surfaces for the attacking and defending teams in football (top left), basketball (top right), and American football (bottom). The color map represents the difference between the predicted arrival times of the attacking and defending teams at each location. The ball is shown in white, and player arrows are scaled proportionally to player speed.
Feature
Meaning
Normalizer
d/120
distance to target (m)
Dnorm=120
cosθ
cosine of angle to target
∈[−1,1]
sinθ
sine of angle to target
∈[−1,1]
v/12
current speed (m/s)
Vnorm=12
ax/10
longitudinal acceleration (m/s 2 )
Anorm=10
ay/10
lateral acceleration (m/s 2 )
Anorm=10
Table 1: The six input features and their normalization constants.
Figure 2: Predicted TTA for a player moving upward at speeds of 2 , 5 , and 8 m/s. White contours correspond to the proposed learned model, while red ones correspond to the two-segment physical model [ 35 ] .
Model
MAE (s)
RMSE (s)
pinball
area @ 99% cov (m 2 )
physics
Constant speed [ 35 ]
3.166
3.75
0.0334
119.3
Constant accel. [ 35 , 46 ]
3.043
3.63
0.0326
46.9
Const. accel. + vmax [ 35 ]
3.101
4.75
0.2370
48.0
Two-segment [ 35 ]
2.992
3.60
0.0581
34.1
Two-segment +vmax (profile) [ 35 ]
2.900
3.50
0.0421
40.4
Free-direction accel. [ 48 ]
2.883
3.48
0.0307
43.6
Table 2: Evaluation on PFF FC test-set. Area is mean reachable area (m 2 ) at Δt=1 s with every model tuned to the same 99% coverage. Learned-model point metrics are mean ± std over five training seeds.
Figure 3: MAE by distance band on the PFF FC held-out test set, for the best precision physics baseline, the learned reachable-hull baseline and both quantile MLPs, stratified into four distance-to-target bands ( 0 – 5 , 5 – 15 , 15 – 30 and 30 m + ).
Figure 4: Mean reachable area at matched coverage , swept over three coverage levels ( 90 / 95 / 99% ) and three horizons ( Δt∈{1,3,5} s), on PFF FC test-set.
Test
Train
MAE (s)
RMSE (s)
pinball
area @ 99% cov (m 2 )
Football
Football
2.602
3.20
0.0280
31.6
Basketball
2.631
3.36
0.0808
35.1
NFL
2.652
3.25
0.0302
46.8
Two-segment
2.996
3.61
0.0581
35.7
Basketball
Basketball
2.462
3.08
0.0265
47.7
Football
2.490
3.09
0.0276
73.0
Table 3: Cross-sport transfer for every sport-trained MLP checkpoint scored on each sport’s own held-out test split, alongside that sport’s own fitted two-segment baseline. NFL rows use within-game (player-based) splitting.