Modelling between- and within-season trajectories in elite athletic performance data

Abstract

Athletic performance follows a typical pattern of improvement and decline during a career. This pattern is also often observed within-seasons, as an athlete aims for their performance to peak at key events such as the Olympic Games or World Championships. A Bayesian hierarchical model is developed to analyse the evolution of athletic sporting performance throughout an athlete's career and separate these effects whilst allowing for confounding factors such as environmental conditions. Our model works in continuous time and estimates both $g(t)$, the average performance level of the population at age $t$, and $f_i(t)$, the difference of the $i$-th athlete from this average. We further decompose $f_i(t)$ into a season-to-season trajectory and a within-season trajectory, which is modelled by a restricted Bernstein polynomial. The model is fitted using an adaptive Metropolis-within-Gibbs algorithm with a carefully chosen blocking scheme. The model allows us to understand seasonal patterns in athlete performance, how these differ between athletes, and provides individual fitted and trend performance trajectories. The properties of the model are illustrated using a simulation study and an application to 100 metres and 200 metres freestyle swimming for both female and male athletes.

Figures (20)

• Figure 1: Analysis of a 100 metres swimmer. Panel (a) shows the observed performances with individual performance trajectory (shown as posterior median (black line) and 95% credible interval) with each calendar year indicated by alternating grey and white bands. Panel (b) shows the individual career trend trajectory (shown as posterior median (black line) and 95% credible interval). Panel (c) shows posterior median within-season performance trajectories (light grey lines) and the posterior median of the athlete's mean within-season performance trajectory (black line).
• Figure 2: Results for the simulated example. Results for Population performance trajectory, Population, Individual and Seasonal within-season performance trajectories are shown as true value (dotted line), posterior median (solid line) and 95% credible interval. Results for the error density are shown as true value (dotted line) and posterior mean (solid line). The results for $\eta$ represent $\eta_{i, 1}, \dots, \eta_{i, 6}$ for one athlete and are shown as true value (block dot), posterior median (cross) with 95% credible interval.
• Figure 3: Simulation Study: The RMISE for (a) the Population Performance Trajectory $g(c\dot)$, (b) Population Within-Season Performance Trajectories $h^{\star}(\cdot)$, (c) Individual Within-Season Performance Trajectories $h^{\star}(\cdot)_i$, and (d) Seasonal Within-Season Performance Trajectories $h^{\star}_{i, s}(t)$. For all plots, the x-axis labels are $(i, j)$ where $i$ is the level of $M$ and $j$ represents the level of $(\sigma^2_a, \sigma^2_b)$, cross represents $p_1 = 0.2$ and circle represents $p_1 = 0.5$
• Figure 4: Simulation Study: (a) The ARMSE of $\eta_{i, j}$, (b) Spearman's rank correlation coefficient between $a_i$ and $\tau_i^2$, and (c) Spearman's rank correlation coefficient between $\bar{b}_i = \frac{1}{S_i}\sum_{j=1}^{S_i} \vert b_{i, j}\vert$ and $\lambda_i^2$. For all plots, the x-axis labels are $(i, j)$ where $i$ is the level of $M$ and $j$ represents the level of $(\sigma^2_a, \sigma^2_b)$, cross represents $p_1 = 0.2$ and circle represents $p_1 = 0.5$.
• Figure 5: Estimated population performance trajectory $g(\cdot)$ for both females and males in the 100 metre and 200 metre freestyle. The trajectories are shown as posterior median (black line) and 95% credible interval (grey shading).