AI 中文总结
本研究开发两阶段框架RACE（含第二阶段模型STAR），用于建模运动员相对性能包络的衰老，通过模拟验证其参数可识别性，并应用于棒球数据分离运动员水平与衰老速率。
AI 中文摘要
运动员职业生涯产生稀疏、不规则的纵向序列：每名运动员仅有少数赛季数据、路径不完整，且存在持续参赛的选择效应。科学研究常关注与峰值可达表现（即上限）的接近程度，而非平均轨迹，以及这种接近程度如何与衰老速率共变。标准工具仅能解决该问题的部分方面：线性混合模型描述平均衰老；功能主成分描述主要变异模式；形状不变模型将曲线配准至平均模板；生长图表估计群体百分位数，但无法得到个体潜在轨迹。我们开发了RACE（Relative Aging Curves via Envelopes，基于包络的相对衰老曲线），这是一个两阶段框架，先估计作为年龄条件高百分位数的群体性能包络，再将每名运动员建模为该包络的低维几何变换。第二阶段混合模型为STAR（Shape Translation And Rotation，形状平移与旋转），包含水平、时机和速率参数。当职业生涯较短时，包络几何决定这些参数的可识别性：近线性包络仅可识别水平和速率，而曲线包络可识别全部三个参数。将STAR嵌入非线性混合效应层级结构，使水平-速率关联成为随机效应协方差的参数，而非单独拟合的事后关联。模拟研究考察稀疏数据下该关联是否可恢复、几何如何控制可识别性，以及结果对包络误设的敏感性。应用于美国职业棒球大联盟Statcast的冲刺速度和bolt率数据，分析表明该嵌入方法如何在功能衰老空间中分离运动员水平和衰老速率，并解释为何bolt率的时机参数可识别而冲刺速度的时机参数不可识别。
英文摘要
Athletic careers yield sparse, irregular longitudinal series: few seasons per athlete, incomplete paths, and selection into continued play. Scientific interest often centres on proximity to peak attainable performance-a ceiling-rather than on the mean trajectory, and on how that proximity co-varies with the rate of decline. Standard tools address only pieces of this problem. Linear mixed models describe average ageing; functional principal components describe dominant modes of variation; shape-invariant models register curves about a mean template; growth charts estimate population centiles but stop short of individual latent trajectories. We develop RACE (Relative Aging Curves via Envelopes), a two-stage framework that estimates a population performance envelope as an age-conditional high centile and then models each athlete as a low-dimensional geometric transformation of that envelope. STAR (Shape Translation And Rotation) is the Stage 2 mixed model, with parameters for level, timing, and tempo. Envelope geometry determines which of these parameters are identifiable when careers are short: near-linear envelopes identify level and tempo only, whereas curved envelopes identify all three. Embedding STAR in a nonlinear mixed-effects hierarchy makes the level-tempo association a parameter of the random-effect covariance rather than a post-hoc correlation of separate fits. Simulations ask whether that association is recoverable under sparsity, how geometry governs identifiability, and how sensitive results are to envelope misspecification. Applied to Major League Baseball Statcast sprint speed and bolt rate, the analysis demonstrates how the proposed embedding separates athletic level and ageing tempo in Functional Ageing Space, and shows why timing is identifiable for bolt rate but not for sprint speed.
