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.

翻译：运动员的职业生涯中，其表现通常遵循着提升与衰退的典型模式。这种模式在赛季内也常被观察到，因为运动员的目标是在奥运会或世锦赛等关键赛事中达到表现峰值。本文开发了一个贝叶斯分层模型，用以分析运动员职业生涯中运动表现的演变过程，并在考虑环境条件等混杂因素的同时分离这些效应。我们的模型在连续时间上工作，同时估计 $g(t)$（即年龄 $t$ 时群体的平均表现水平）和 $f_i(t)$（即第 $i$ 位运动员与该平均值的差异）。我们进一步将 $f_i(t)$ 分解为赛季间轨迹和赛季内轨迹，后者通过受限伯恩斯坦多项式进行建模。模型采用自适应 Metropolis-within-Gibbs 算法进行拟合，并辅以精心设计的分块策略。该模型使我们能够理解运动员表现的季节性模式、这些模式在运动员之间的差异，并提供个体拟合及趋势表现轨迹。通过模拟研究以及对男女运动员 100 米和 200 米自由泳游泳成绩的应用，我们展示了该模型的性质。