A modelling framework suitable for detecting shape shifts in functional profiles combining the notion of Fr\'echet mean and the concept of deformation models is developed and proposed. The generalized mean sense offerred by the Fr\'echet mean notion is employed to capture the typical pattern of the profiles under study, while the concept of deformation models, and in particular of the shape invariant model, allows for interpretable parameterizations of profile's deviations from the typical shape. EWMA-type control charts compatible with the functional nature of data and the employed deformation model are built and proposed, exploiting certain shape characteristics of the profiles under study with respect to the generalised mean sense, allowing for the identification of potential shifts concerning the shape and/or the deformation process. Potential shifts in the shape deformation process, are further distingu\-ished to significant shifts with respect to amplitude and/or the phase of the profile under study. The proposed modelling and shift detection framework is implemented to a real world case study, where daily concentration profiles concerning air pollutants from an area in the city of Athens are modelled, while profiles indicating hazardous concentration levels are successfully identified in most of the cases.

翻译：本文开发并提出了一个适用于检测功能轮廓形状偏移的建模框架，该框架结合了弗雷歇均值（Fréchet mean）概念与变形模型理论。利用弗雷歇均值所提供的广义均值概念来捕捉所研究轮廓的典型模式，而变形模型（尤其是形状不变模型）则允许对轮廓偏离典型形状的情况进行可解释的参数化描述。结合数据的函数特性及所使用的变形模型，构建并提出了EWMA型控制图，通过利用所研究轮廓在广义均值意义下的某些形状特征，能够识别潜在的形状和/或变形过程偏移。进一步地，将形状变形过程中的潜在偏移区分为所研究轮廓的振幅和/或相位方面的显著偏移。该建模与偏移检测框架被应用于实际案例研究中，对雅典市某区域的每日空气污染物浓度轮廓进行建模，并在大多数情况下成功识别出指示危险浓度水平的轮廓。