Bayesian multilevel step-and-turn models for evaluating player movement in American football

Abstract: In sports analytics, player tracking data have driven significant advancements in the task of player evaluation. We present a novel generative framework for evaluating the observed frame-by-frame player positioning against a distribution of hypothetical alternatives. We illustrate our approach by modeling the within-play movement of an individual ball carrier in the National Football League (NFL). Specifically, we develop Bayesian multilevel models for frame-level player movement based on two components: step length (distance between successive locations) and turn angle (change in direction between successive steps). Using the step-and-turn models, we perform posterior predictive simulation to generate hypothetical ball carrier steps at each frame during a play. This enables comparison of the observed player movement with a distribution of simulated alternatives using common valuation measures in American football. We apply our framework to tracking data from the first nine weeks of the 2022 NFL season and derive novel player performance metrics based on hypothetical evaluation.

• The paper introduces a Bayesian multilevel framework that simulates frame-level player movements by decomposing them into step length and turn angle components.
• It leverages high-resolution tracking data using multilevel Gaussian and von Mises models to effectively capture temporal and directional uncertainties.
• Posterior predictive simulation generates hypothetical movement alternatives, enabling robust evaluation of player performance through expected yards metrics.

Bayesian Multilevel Step-and-Turn Models for Evaluating Player Movement in American Football

Overview and Motivation

This work presents a generative statistical framework for player movement analysis within American football using high-resolution player tracking data. The authors develop Bayesian multilevel models for characterizing and simulating frame-level movement of ball carriers during NFL run plays. Their approach decomposes player movement into step length and turn angle attributes, enabling hypothetical evaluation by comparing the observed trajectory at every frame against a distribution of alternative, simulated steps. The methodology leverages recent advances in sports analytics, notably in multi-agent tracking and probabilistic evaluation (ghosting), but makes a crucial contribution by explicitly modeling uncertainty in movement and aggregating over entire plays rather than single time points.

Data Processing and Feature Construction

Player tracking data from the 2022 NFL season were used, comprising 5400 run plays over nine weeks, with position, velocity, orientation, and event tags for all 22 players and the football at 10 Hz. Frames from handoff to the termination event (tackle, out-of-bounds, touchdown) for each ball carrier were extracted. Spatial features were constructed for three groups: the ball carrier, the offense (excluding the ball carrier), and the defense, including distances and angular relationships, anchored on the ball carrier. Covariates for modeling included ball carrier features, nearest defender features, and aggregate directional groupings.

Generative Movement Models

Step Length Modeling

Step length (Euclidean distance traveled in one frame) is first normalized, then transformed with a scaled arcsine function to improve fit, overcoming deficiencies of standard positive-valued distributions (Gamma, log-normal). A multilevel Gaussian model is adopted, including random intercepts for each player and defensive team, and conditioning on previous frame's step length to capture temporal dependence.

Formally:

$$\tilde s_{ijt} \sim \mathcal{N}(\mu_{ijt}, \sigma^2); \qquad \mu_{ijt} = \alpha_0 + \mathbf{X}_{ijt} \boldsymbol{\beta} + u_j + v_k$$

with appropriate weak priors and MCMC inference.

Turn Angle Modeling

Turn angle (the change in direction between successive steps) is modeled with a von Mises distribution, employing tan-half link for the mean and log-link for concentration, conditioned on step length and prior turn angle to capture coupled movement dynamics and directional persistence.

Formally:

$$\varphi_{ijt} \sim \mathrm{vonMises}(\mu_{ijt}, \kappa_{ijt})$$

with random effect $w_j$ for each player reflecting individual change-of-direction variability.

Model Fitting and Assessments

Bayesian inference is performed using Stan and brms with weakly informative priors and extensive MCMC diagnostics. Posterior predictive checks demonstrate the superiority of the proposed transformation for step length over conventional distributions.

Simulation and Hypothetical Evaluation

Posterior predictive simulation generates distributions of hypothetical steps at each frame, conditioned on the observed spatial environment and covariates. The simulation framework synthesizes concepts from step selection analysis, common in animal movement literature, and ghosting in sports analytics. For each time point, multiple hypothetical steps are sampled for a prototypical ball carrier, holding all other players' positions fixed.

These simulated steps are mapped to hypothetical future player positions, and spatial features are re-extracted.

Player performance is evaluated by comparing observed movement at each frame to the simulated alternatives using a CatBoost regression model for expected yards gained—trained with full spatial features and cross-validation to optimize predictive RMSE. Key evaluation metrics include:

• $$\delta_{ijt}^{(h)} = \hat \ell_{ijt} - \hat \ell_{ijt}^{(h)}$$: difference between observed and hypothetical expected yards per frame.
• $\bar{\delta}_{ijt}$: average difference over all hypothetical steps at a frame.
• Aggregates over frames provide per-play summaries.

Results: Player Ratings and Performance Metrics

Posterior distributions of player-specific concentration random effects reveal substantial heterogeneity in directional variability among NFL running backs, with the highest concentration observed for runners like Jonathan Taylor, consistent with scouting reports indicating linear, speed-oriented running styles. Joint posterior means for step length and turn angle concentration illuminate diverse movement profiles—some runners excel with longer strides and low-turn variability, others with shorter steps and greater angular flexibility.

Hypothetical evaluation at play and season level enables comparison between observed player performance and simulated baselines. For example, yards success rate (fraction of simulations where the observed movement yields higher expected yards than hypothetical alternatives) and explosiveness (probability observed yards exceed the 95th percentile of hypothetical distributions) are constructed. Notably, these metrics provide strong discriminative power that aligns with qualitative assessments of player style and impact; the authors identify Josh Jacobs and Travis Etienne as leaders in these measures, consistent with known running profiles.

Practical and Theoretical Implications

The step-and-turn Bayesian generative framework advances movement modeling in sports analytics by:

• Providing multilevel structure with explicit uncertainty propagation, crucial for robust hypothetical comparisons.
• Allowing microanalysis of in-play movement and macroaggregation into performance metrics.
• Enabling flexible player evaluation across positions, contexts, and time horizons.
• Facilitating potential downstream integration with expected points and win probability models for more granular game valuation.

The approach is directly extensible to other sports with high-frequency tracking data (e.g., soccer, basketball, hockey) through modular substitution of value functions and movement context. The framework’s decomposability paves the way for integrating discrete action selection models, iterative trajectory forward simulation, and fully multi-agent simulation (contingent on modeling all on-field player movements and stochastic play termination events).

Limitations and Future Directions

Current limitations include simplified fixed-effects structure for covariates, simulation of only the ball carrier (with other player trajectories held fixed), and evaluation restricted to one-step ahead rather than full-play trajectory simulation. Key challenges remain in forward simulation (requiring well-calibrated tackle models and multi-player dynamics) and richer feature extraction to encapsulate complex spatial relationships.

Future research should focus on:

• Expanding covariate modeling via flexible ML or hierarchical architectures.
• Implementing full trajectory simulation with probabilistic play termination.
• Incorporating multi-agent models for coordinated team movement.
• Extending to action selection and decision-making contexts in other sports.

Conclusion

This paper introduces a robust Bayesian multilevel generative approach for frame-level player movement evaluation in American football, synthesizing step-and-turn modeling, posterior predictive simulation, and hypothetical baseline comparison. Empirical evidence highlights substantial player heterogeneity and strong discriminative ability of the proposed metrics. The framework provides a template for future sports analytics research, enabling granular, uncertainty-aware evaluation of individual and team performance in spatially dynamic, multi-agent environments.