Shape-constrained functional data encompass a wide array of application fields especially in the life sciences, such as activity profiling, growth curves, healthcare and mortality. Most existing methods for general functional data analysis often ignore that such data are subject to inherent shape constraints, while some specialized techniques rely on strict distributional assumptions. We propose an approach for modeling such data that harnesses the intrinsic geometry of functional trajectories by decomposing them into size and shape components. We focus on the two most prevalent shape constraints, positivity and monotonicity, and develop individual-level estimators for the size and shape components. Furthermore, we demonstrate the applicability of our approach by conducting subsequent analyses involving Fr\'{e}chet mean and Fr\'{e}chet regression and establish rates of convergence for the empirical estimators. Illustrative examples include simulations and data applications for activity profiles for Mediterranean fruit flies during their entire lifespan and for data from the Z\"{u}rich longitudinal growth study.

翻译：形状约束函数数据涵盖广泛的应用领域，尤其在生命科学中，如活动模式分析、生长曲线、医疗保健与死亡率研究。现有的大多数通用函数数据分析方法往往忽略此类数据受内在形状约束的特性，而一些专门技术则依赖于严格的分布假设。我们提出一种建模方法，通过将函数轨迹分解为大小与形状分量，从而利用其内在几何结构。我们聚焦于两种最普遍的形状约束——正值性与单调性，并开发了针对大小与形状分量的个体水平估计量。此外，我们通过后续涉及弗雷歇均值与弗雷歇回归的分析，展示了该方法的适用性，并为经验估计量建立了收敛速率。说明性案例包括对地中海果蝇整个生命周期活动模式数据的仿真与应用分析，以及苏黎世纵向生长研究数据的应用。