Gaussian Processes for Analyzing Positioned Trajectories in Sports

Feng Yin; Fredrik Gunnarsson; Fredrik Hultkrantz; Yuxin Zhao

arxiv: 1907.03043 · v1 · pith:URS2CJQQnew · submitted 2019-07-05 · 💻 cs.LG · eess.SP· stat.ML

Gaussian Processes for Analyzing Positioned Trajectories in Sports

Yuxin Zhao , Feng Yin , Fredrik Gunnarsson , Fredrik Hultkrantz This is my paper

Pith reviewed 2026-05-25 02:01 UTC · model grok-4.3

classification 💻 cs.LG eess.SPstat.ML

keywords Gaussian processgrey-box modelingcross-country skiingkinetic motion modelspositioned trajectoriespredictive uncertaintytrajectory analysissports analytics

0 comments

The pith

A grey-box model merging kinetic equations with Gaussian process regression reduces uncertainty in modeling cross-country skiing forces and velocities by 30 to 40 percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes combining established kinetic motion models for skier forces with Gaussian process regression to handle unknown factors in the system. This grey-box method is used to analyze both individual skier performances and groups during races. Data from the 2015 Falun World Championships shows the hybrid approach outperforms pure black-box Gaussian processes in reducing prediction uncertainty. The method applies to any scenario with positioned trajectories and kinetic measurements.

Core claim

The authors establish a grey-box modeling approach where a disciplined set of kinetic motion model formulae is combined with Gaussian process regression to account for everything unknown in the skiing system. This enables analysis of forces in cross country skiing races and kinetic flow for individuals and clusters of skiers. Evaluation on Men's 4x10 km relay data from the 2015 championships demonstrates that the grey-box approach reduces predictive uncertainty by 30% to 40% compared to black-box methods.

What carries the argument

The grey-box modeling approach, which integrates kinetic motion model formulae with data-driven Gaussian process regression to model only the unknowns in skier motion.

If this is right

Velocity models for skiers at different competition stages can be evaluated using the approach.
Forces during the races can be analyzed and compared between skiers.
The modeling can handle both individual skiers and clusters.
The approaches can be applied to other use cases with positioned trajectories and kinetic measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hybrid physics-informed machine learning models like this may generalize better to new conditions than pure data-driven ones.
Similar grey-box methods could be tested in other trajectory-based sports such as cycling or rowing.
The uncertainty reduction suggests potential for more reliable real-time performance predictions during competitions.

Load-bearing premise

The kinetic motion model formulae accurately represent the base forces and motion of skiers so that the Gaussian process only needs to capture the remaining unknowns.

What would settle it

Applying the grey-box and black-box models to a separate set of skiing trajectory data and finding that the grey-box does not reduce predictive uncertainty by 30-40% or more.

read the original abstract

Kernel-based machine learning approaches are gaining increasing interest for exploring and modeling large dataset in recent years. Gaussian process (GP) is one example of such kernel-based approaches, which can provide very good performance for nonlinear modeling problems. In this work, we first propose a grey-box modeling approach to analyze the forces in cross country skiing races. To be more precise, a disciplined set of kinetic motion model formulae is combined with data-driven Gaussian process regression model, which accounts for everything unknown in the system. Then, a modeling approach is proposed to analyze the kinetic flow of both individual and clusters of skiers. The proposed approaches can be generally applied to use cases where positioned trajectories and kinetic measurements are available. The proposed approaches are evaluated using data collected from the Falun Nordic World Ski Championships 2015, in particular the Men's cross country $4\times10$ km relay. Forces during the cross country skiing races are analyzed and compared. Velocity models for skiers at different competition stages are also evaluated. Finally, the comparisons between the grey-box and black-box approach are carried out, where the grey-box approach can reduce the predictive uncertainty by $30\%$ to $40\%$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Grey-box GP for skiing trajectories is a straightforward hybrid application, but the 30-40% uncertainty claim depends on an untested kinetic model.

read the letter

The paper combines a set of kinetic motion equations with Gaussian process regression to model skier forces and velocities from positioned trajectory data. It evaluates the hybrid on 2015 Falun championship relay data and reports lower predictive uncertainty than a pure black-box GP. That is the main new piece: a concrete grey-box setup tailored to cross-country skiing rather than a generic GP extension. The work also sketches how the same structure could apply to other positioned-trajectory settings where both physics and measurements exist. The comparison to black-box GP is a useful check and the real-data example is welcome. The central result is the reported 30-40% uncertainty reduction. The problem is that this number only means what the authors want if the kinetic formulae already account for the dominant forces and the GP is left with small unstructured residuals. The abstract gives no residual diagnostics, no hold-out test of the physics-only predictor against measured trajectories, and no detail on how the kinetic parameters were chosen or validated. If the motion model misses drag, variable friction, or pole-push effects, the GP will absorb those systematic errors and the uncertainty shrinkage becomes an artifact of the construction. The paper is therefore for readers already working on sports trajectory modeling or hybrid physical-ML methods who need an example rather than a methodological advance. It is coherent on its own terms and shows honest engagement with the literature on GPs and kinetic models. A serious editor could send it to review, but only with the expectation that the authors supply the missing checks on the kinetic component before acceptance.

Referee Report

2 major / 1 minor

Summary. The paper proposes a grey-box modeling framework that combines a set of kinetic motion model formulae with Gaussian process regression to account for unknowns when analyzing forces and velocities from positioned trajectories in cross-country skiing. It applies the method to data from the 2015 Falun Nordic World Ski Championships (Men's 4x10 km relay), compares individual and cluster skier kinetics, evaluates velocity models across competition stages, and reports that the grey-box approach reduces predictive uncertainty by 30-40% relative to a black-box GP baseline.

Significance. If the kinetic models are shown to be a faithful representation of dominant forces and the uncertainty reduction is robustly validated, the hybrid approach would offer a useful template for incorporating physical priors into trajectory modeling, improving interpretability and uncertainty quantification over pure data-driven methods. The use of real-world competition data strengthens the practical relevance for sports analytics.

major comments (2)

[Abstract] Abstract: the central claim of 30-40% predictive uncertainty reduction is presented without any description of the validation procedure, error bars, data exclusion criteria, hold-out strategy, or parameterization of the kinetic motion models, rendering the quantitative improvement impossible to assess from the provided information.
[Abstract] Abstract and modeling description: the grey-box uncertainty reduction is only meaningful if the kinetic formulae already capture the dominant forces (leaving the GP to model only small residuals); no residual diagnostics, variance decomposition, or direct comparison of the physics-only predictor against measured trajectories are supplied to support this premise.

minor comments (1)

[Abstract] Abstract contains minor grammatical issues (e.g., 'large dataset in recent years' should be 'large datasets'; 'a disciplined set of kinetic motion model formulae is combined' is awkward).

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive review. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of 30-40% predictive uncertainty reduction is presented without any description of the validation procedure, error bars, data exclusion criteria, hold-out strategy, or parameterization of the kinetic motion models, rendering the quantitative improvement impossible to assess from the provided information.

Authors: We agree that the abstract would benefit from additional context. In the revised manuscript we will expand the abstract to briefly describe the hold-out strategy (using competition-stage splits), the parameterization of the kinetic models, the cross-validation procedure employed, and the reporting of uncertainty reduction with error bars. revision: yes
Referee: [Abstract] Abstract and modeling description: the grey-box uncertainty reduction is only meaningful if the kinetic formulae already capture the dominant forces (leaving the GP to model only small residuals); no residual diagnostics, variance decomposition, or direct comparison of the physics-only predictor against measured trajectories are supplied to support this premise.

Authors: We acknowledge the value of explicit diagnostics. The revised manuscript will add residual plots, a variance decomposition separating the physics and GP contributions, and direct comparisons of the physics-only predictor against the measured trajectories to demonstrate that the kinetic formulae capture the dominant forces. revision: yes

Circularity Check

0 steps flagged

No significant circularity; grey-box combines external kinetic formulae with independent GP fit

full rationale

The paper's central construction combines a set of kinetic motion model formulae (standard physics, external to the GP) with a data-driven Gaussian process for residuals. The reported 30-40% uncertainty reduction is obtained by direct empirical comparison of the hybrid model against a pure black-box GP on the same championship trajectory data. No equation reduces to its own input by construction, no parameter is fitted on a subset and then relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of standard kinetic motion models for skiing and the GP's capacity to capture residuals from trajectory data; no new entities are postulated.

free parameters (1)

Gaussian process kernel hyperparameters
Fitted during regression to capture unknown dynamics in the skiing system.

axioms (1)

domain assumption The kinetic motion model formulae provide an accurate base representation of skier forces and motion.
Invoked as the known component in the grey-box model to be combined with GP.

pith-pipeline@v0.9.0 · 5745 in / 1233 out tokens · 37367 ms · 2026-05-25T02:01:03.110820+00:00 · methodology

Gaussian Processes for Analyzing Positioned Trajectories in Sports

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)