Source-linked AI summary
Learning Human Sports Motion Feasibility via Quantile Regression on Tracking Data
Marc Gutiérrez-Pérez, Antonio Agudo
TL;DR
Sports space models need to estimate where a player can reach and how quickly, while existing motion models impose fixed forms and some data-driven alternatives omit arrival-time feasibility. This paper learns TTA as a lower-quantile regression problem from tracking data and evaluates the model against physics-based and learned baselines. The MLP outperforms every baseline on point error, RMSE, and pinball loss throughout every distance band.
Problem
Player-motion models underlie sports space measures, but existing approaches either commit to fixed functional forms or model space without an arrival-time feasibility bound.
Source: §1 · pp. 1–2Method
The paper learns the 1st-percentile arrival time from a player’s kinematic state and target using quantile regression, with a differentiable reachable-area term in the objective.
Source: §1 · p. 2, §3.1 · p. 6Results
The MLP beats every baseline on point error, RMSE, and pinball loss throughout every distance band; at matched coverage, a quantile MLP is tightest by reachable area in eight of nine coverage–horizon cells.
Source: §5 · p. 14Takeaways & Limitations
The same representation and objective are tested on basketball and American-football data to examine cross-sport transferability.
Source: §1 · p. 2Intent, fatigue, tactical constraints, and obstacles act conservatively, so the lower-envelope target reflects plausible arrival speed rather than unconstrained physical capability.
Source: §3.1 · p. 6
Abstract
from arXiv · showhide
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.
1 Introduction
Sports space models depend on predicting where players can reach and how quickly, yet existing approaches either impose fixed motion laws or estimate occupancy without arrival-time semantics. This work learns a player’s lower-envelope arrival time through quantile regression and evaluates the approach across sports.
1 Introduction
- Player-motion models underpin space measures such as pitch control, off-ball value, and pass-probability surfaces, which depend on where players can get and how fast.The paper notes related applications across football, basketball, and American football.Source: §1 · pp. 1–2
- The method predicts a player’s minimum plausible time-to-arrival from their kinematic state to any target, learning the lower envelope of observed locomotion rather than typical destination or mean arrival time.The authors use the 1st-percentile target because the goal is a feasibility bound.Source: §1 · p. 2
- The paper casts TTA as pinball quantile regression over physics-normalized, heading-invariant geometry.It adds a differentiable reachable-area term and jointly evaluates coverage and area against six literature motion models.Source: §1 · p. 2
- The work evaluates the same representation and objective on basketball and American-football tracking data.This probes transferability of the learned locomotion model across sports.Source: §1 · p. 2
2 Related work
Prior work either models reachability with interpretable but fixed-form physics, or uses data-driven space estimates that often omit arrival-time feasibility. This paper positions learned TTA as a way to retain an interpretable time output while drawing on data-driven regression.
Physical motion models
- Six physical motion models are re-expressed as TTA predictors and tuned with the reachable-area objective used by Renkin et al..The literature includes constant-speed, acceleration, capped-acceleration, and two-segment models, among others.Source: §2 · p. 3
Probabilistic and data-driven movement models
- Data-driven movement methods trade closed-form interpretability for flexibility, while many space models output occupancy or behavioral probabilities rather than physical reachability.Anzer and Bauer are an exception, estimating reachable regions as per-speed convex hulls of observed displacements.Source: §2 · p. 3
Applications resting on arrival times
- Many downstream applications consume arrival times, but their motion layer often uses fixed global parameters and is validated indirectly through downstream outcomes.The cited applications include pass targeting, pass quality, offside control, pressing, and pitch control.Source: §2 · p. 4
- Prior findings motivate per-player speed and horizon-sensitive motion: pitch-control surfaces are sensitive to top speed, while horizon-dependent parameters challenge a single fixed form.The passage connects these observations to a calibrated learned arrival time as a drop-in replacement.Source: §2 · p. 5
2 Related work
- The paper places learned TTA between interpretable motion models and flexible data-driven space models, retaining arrival time as an output that can be inverted to reachable area.It evaluates the approach against both literatures on their own terms.Source: §2 · p. 5
3 Method
The method estimates time-to-arrival as a conditional lower quantile of observed locomotion, so it predicts a player’s fastest reliably observed arrival at a target. It transforms tracking data into heading-frame inputs, benchmarks learned models against motion-model baselines, and uses reachable area as a dual view of the predicted TTA surface.
3.1 Problem formulation
- The 1st-percentile conditional arrival-time estimate targets a feasibility bound—the lower envelope of observed locomotion—not a typical-case estimate or per-observation minimum.Intent, fatigue, tactical constraints, and obstacles act conservatively; the fast tail across many observations is used to discard choice-driven slowness.Source: §3.1 · p. 6
- A TTA surface and a reachable-area model are dual: the reachable region at horizon Δt is the surface’s sublevel set.This identity connects TTA predictions to player-motion models that directly predict reachable regions and enables evaluation with a reachable-area metric.Source: §3.1 · p. 6
3.2 Track cleaning
- Tracks are cleaned independently within match halves by smoothing gap-free runs, differentiating position, and excluding states that violate hard kinematic bounds.The input features and their normalization constants are listed in Table 1.Source: §3.2 · p. 6, Table 1 · p. 7
3.3 Coordinate transform and input features
- The heading-frame representation uses six physically normalized inputs and removes absolute pitch position and orientation from each state–target pair.The player is translated to the origin and rotated to face +x; the target is encoded by distance and heading-relative angle, and acceleration by longitudinal and lateral components.Source: §3.3 · p. 7, Table 1 · p. 7
3.4 Sample extraction and label-quality filter
- Training tuples pair each usable state with five uniformly sampled future horizons, while a conservative SUVAT-based screen removes only 0.3% of samples deemed physically implausible.Turning off the filter or widening its caps leaves the fitted floor unchanged but loosens matched-coverage areas and increases monotonicity violations, supporting its role as a noise screen rather than a physical prior.Source: §3.4 · pp. 7–8
3.5 Literature motion models as arrival-time predictors
- Six literature motion models are inverted into TTA predictors and tuned on training data using the same reachable-area validation score used for evaluation.The model classes range from constant-speed and acceleration disks to speed-limited, two-segment, free-direction, and friction-damped motion.Source: §3.5 · p. 8
3.6 Learned models
- The learned comparisons include a 6 → 64 → 64 → 1 MLP and a player-profile MLP, alongside a learned reachable-hull baseline.The player-profile model appends a maximum-speed estimate; the reachable-hull baseline uses per-speed-bin convex hulls of training displacements.Source: §3.6 · pp. 8–9
3.7 Quantile objective
- Learned models combine pinball loss with an area-aware auxiliary term; at τ = 0.01, over-prediction is penalized 99× more heavily than under-prediction.The auxiliary term is a differentiable relaxation of reachable-area fraction, evaluated on a fixed heading-frame grid; the total objective is L = Lτ + λ Larea, with λ = 0.01.Source: §3.7 · p. 9
4 Experiments
Experiments use World Cup tracking data, matched-coverage evaluation, and cross-sport datasets to assess quantile MLPs against physics-based and learned baselines. The MLPs outperform the physics baselines on point error and pinball loss, while transfer results vary by sport and the models run at live-match speeds.
4.1 Data
- The PFF FC corpus covers all 64 FIFA World Cup 2022 matches at 30 fps, split by match into 45/10/9 training, validation, and test matches.The basketball corpus contains 631 NBA games; the NFL corpus is a single game, split by player rather than match.Source: §4.1 · p. 10
4.2 Evaluation methodology
- Evaluation combines MAE, RMSE, τ = 0.01 pinball loss, and reachable area compared at matched coverage, avoiding rankings based on coverage or area alone.The evaluation reports canonical horizons from 1 to 5 seconds at a 99% coverage target.Source: §4.2 · pp. 10–11
4.3 Main comparison
- Both quantile MLPs beat every physics baseline on MAE and pinball loss; across distance bands, they also outperform the reachable-hull and two-segment baselines.The profile MLP improves on the base MLP, and the learned reachable-hull model trails both MLPs on every reported metric.Source: Table 2 · p. 11, §4.3 · pp. 11–12, Figure 3 · p. 12
- With the same train-split per-player vmax table, the two-segment closed-form baseline benefits most, with small gains concentrated at long horizons; the best personalized baseline still trails the profile MLP by 0.30 s of MAE.The advantage stems from the learned representation, not from the per-player feature.Source: §4.3 · pp. 11–12
4.4 Ablation studies
- A quantile MLP has the smallest reachable area in eight of nine coverage–horizon cells; the reachable-hull baseline is looser than the base MLP in all nine.The three coverage levels are 90%, 95%, and 99%; the three horizons are 1, 3, and 5 seconds.Source: §4.4 · p. 12, Figure 4 · p. 13
- Cross-sport transfer is asymmetric: the NFL-trained checkpoint beats each destination’s two-segment baseline on point error and pinball, while out-of-domain models have larger area than the destination baseline.The authors attribute the mismatch to NFL pass plays’ faster arrivals at matched distance and speed than football or basketball play.Source: §4.4 · pp. 12–13, Table 3 · p. 14
- The predicted sublevel sets are connected and contain the player in all 25,000 audited state–horizon cases, and are star-shaped in all 80,000 audited rays.Radial monotonicity violations occurred on 1.4% of rays, only beyond 4 seconds and 20 m, with a median dip of 16 ms.Source: §4.4 · p. 13
4.5 Real-time inference
- Evaluating the 1 m grid for all 22 players takes 2.4 ms on an NVIDIA RTX A5000, approximately 420 frames per second.That rate is an order of magnitude above the 25–30 fps sampling rate of the tracking system.Source: §4.5 · p. 13
5 Conclusion
The learned MLP outperforms the evaluated baselines across point-error, RMSE, and pinball-loss metrics, while cross-sport tests reveal asymmetric transfer. The results support using learned models for calibrated reachable-area estimates without a fixed closed-form motion model.
5 Conclusion
- The learned MLP beats every baseline on point error, RMSE, and pinball loss throughout every distance band.A single per-player max-speed feature improves on the MLP.Source: §5 · p. 14
- At matched coverage, the quantile MLP has tighter reachable area in eight of nine coverage×horizon cells; the profile variant wins every 95% and 99% cell at 3- and 5-second horizons.Source: §5 · p. 14
- In-domain, each sport’s own model beats that sport’s two-segment baseline on every metric, while transfer from American football is essentially free on point error and pinball.Every transferred checkpoint’s reachable area nevertheless loses to the destination’s own physics baseline; the authors trace the mismatch to label-regime differences, not the representation.Source: §5 · p. 14
- A fixed closed-form motion model is not necessary to produce a physically plausible, well-calibrated reachable-area estimate.Source: §5 · p. 14
- The study trains and evaluates the approach on a 64-match World Cup tracking corpus using a strict cross-match split.It benchmarks six literature motion models re-expressed as arrival-time predictors, refit at a matching 0.99 correctness gate, and a learned reachable-hull baseline.Source: §5 · p. 14
- Future work includes richer conditioning for player personalization and testing transfer across leagues and tracking providers within soccer.Source: §5 · p. 14