Bayesian GARCH Modeling of Functional Sports Data

In this work, we exploit the motivating example of describing performances of shot put athletes throughout their careers to propose a Bayesian model for quantitative sports data analysis. Specifically, we adapted the Bayesian latent factor regression model for functional and longitudinal data by Montagna et al. (2012) to propose a methodological and computational tool to deal with individual specific, seasonally clustered time series. The main goal of this approach is to characterize the performance curve of each athlete as a linear combination of a high-dimensional set of basis functions. We perform basis selection via the shrinkage method proposed by Bhattacharya et al. (2011). Moreover, with the aim of capturing the heterogeneity in athletes' mean performances across seasons, the model is enriched with an additive seasonal effect. In particular, we present a mixed effect model with generalized autoregressive conditional heteroskedasticity (GARCH) errors to describe how individuals perform over seasons with respect to an overall mean. Finally, a functional regression component completes our construction allowing for athlete-specific and time-varying covariates to impact on the shape of the estimated trajectories. From a computational point of view, we present an original algorithm which relies on Gibbs sampling steps to generate posterior samples for the parameters of interest, coupled with an extensive use of adaptive Metropolis-Hastings updates for posteriors not available in closed form. An extensive discussion of posterior results, parameter interpretation, model evaluation and convergence diagnostics completes this work and suggests further and future developments. This work is motivated by a Sport Analytic project in collaboration with Prof. Hopker (University of Kent). ​