  The spread in the use of tracking systems in sport has made fine-grained
spatiotemporal analysis a primary focus of an emerging sports analytics
industry. Recently publicized tracking data for men's professional tennis
allows for the first detailed spatial analysis of return impact. Mixture models
are an appealing model-based framework for spatial analysis in sport, where
latent variable discovery is often of primary interest. Although finite mixture
models have the advantages of interpretability and scalability, most
implementations assume standard parametric distributions for outcomes
conditioned on latent variables. In this paper, we present a more flexible
alternative that allows the latent conditional distribution to be a mixed
member of finite Gaussian mixtures. Our model was motivated by our efforts to
describe common styles of return impact location of professional tennis players
and is the reason we name the approach a 'latent style allocation' model. In a
fully Bayesian implementation, we apply the model to 142,803 return points
played by 141 top players at Association of Tennis Professional events between
2018 and 2020 and show that the latent style allocation improves predictive
performance over a finite Gaussian mixture model and identifies six unique
impact styles on the first and second serve return.
