A State-Space Perspective on Modelling and Inference for Online Skill Rating

Abstract: We summarise popular methods used for skill rating in competitive sports, along with their inferential paradigms and introduce new approaches based on sequential Monte Carlo and discrete hidden Markov models. We advocate for a state-space model perspective, wherein players' skills are represented as time-varying, and match results serve as observed quantities. We explore the steps to construct the model and the three stages of inference: filtering, smoothing and parameter estimation. We examine the challenges of scaling up to numerous players and matches, highlighting the main approximations and reductions which facilitate statistical and computational efficiency. We additionally compare approaches in a realistic experimental pipeline that can be easily reproduced and extended with our open-source Python package, https://github.com/SamDuffield/abile.

• The paper introduces a state-space model framework that treats player skills as latent variables evolving over time using Markovian dynamics.
• It leverages advanced inference techniques like extended Kalman filtering and pairwise updating to efficiently manage high-dimensional sports data.
• Experiments show the approach outperforms traditional models like Elo in predicting outcomes, especially in scenarios with draws and variable skill volatility.

A State-Space Perspective on Modelling and Inference for Online Skill Rating

Introduction to Skill Rating Models

The paper explores advanced methodologies for skill rating in competitive sports, emphasizing models that account for the time-varying nature of player abilities. This perspective is crucial for accurately capturing the dynamic process of skill evolution based on match outcomes. Traditional systems such as Elo and TrueSkill are referenced, with enhancements proposed through a state-space model approach. This framework allows for modular, scalable, and interpretable models, which can be tailored to the specific needs of various sports and competitive scenarios.

State-Space Model Framework

The key contribution is the introduction of a state-space model (SSM) perspective for skill rating. The SSM setup views player skills as latent variables that evolve over time according to Markovian dynamics. Match results serve as observable data. This structure is flexible, supporting variations in match outcome distributions and dynamic skill trajectories, which can be continuous or discrete. This methodology contrasts with traditional approaches by decoupling model design from inference, empowering users to experiment with different configurations.

Model Specification

The model specifies skills as ordinal quantities within a totally ordered set $\mathcal{X}$, evolving in continuous time across matches. Match outcomes are linked probabilistically to these skill levels using potentially complex likelihood functions. Various dynamics are explored, including the discrete-time Markov jump process with a generator matrix, highlighting different modeling strategies dependent on the sport context.

Figure 1: Visualization of the different skill representations for Argentina's 2023 FIFA World Cup triumph. Each y-axis represents the skill-rating scale for the different approaches.

Inference in State-Space Models

Three inference tasks are addressed: filtering, smoothing, and parameter estimation. Filtering provides real-time estimates, smoothing refines past estimates based on all observed data, and estimation tunes model parameters to optimize fit. Computational strategies are explored to make these operations efficient in high-dimension settings typical of sports datasets, leveraging factorial approximations to maintain tractability.

Algorithmic Techniques

For filtering, the paper discusses methods like extended Kalman filtering, which apply Gaussian approximations to complex likelihoods. Pairwise updating schemes harness sparsity in match data, minimizing the computational overhead by focusing updates only on involved players at each event. The smoothing process involves backward recursions to incorporate prediction dynamics, refining historical skill estimates.

Figure 2: Log-likelihood grid and parameter estimation for WTA tennis data.

Practical Implementation and Extensions

Several practical extensions of the state-space framework are suggested, encompassing non-Gaussian dynamics, multi-player comparisons, and specific settings like home advantage in football. Furthermore, the availability of an open-source Python package, abile, facilitates the reproduction of results and adaptation of models for diverse applications.

Experiments and Results

Experiments demonstrate the effectiveness of the proposed methodologies on sports datasets, showing superior predictive capabilities over traditional models like Elo in scenarios involving draws and varying skill volatility. Analysis on real-world datasets showcases the utility for retrospective evaluations and strategic decision-making, enhancing model interpretability and predictive accuracy.

Figure 3: Extended Kalman filtering and smoothing with for Tottenham's EPL matches from 2011-2023.

Conclusion

The paper presents a compelling case for adopting a state-space approach to skill rating, highlighting its robustness, flexibility, and interpretability compared to traditional methods. Future work may explore broader applications across competitive domains and refine inference techniques to further enhance model performance and scalability.