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 offered 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 generalized 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 distinguished 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：本文提出并发展了一种适用于检测功能轮廓形变位移的建模框架，该框架融合了Fréchet均值概念与形变模型理论。利用Fréchet均值概念提供的广义均值属性捕获所研究轮廓的典型模式，同时借助形变模型（特别是形状不变模型）的概念，实现对轮廓偏离典型形态的可解释参数化描述。本研究构建并提出了与数据函数特性及所采用形变模型相匹配的EWMA型控制图，通过利用所研究轮廓相对于广义均值意义的特定形状特征，能够识别与形状和/或形变过程相关的潜在位移。形变过程中的潜在位移进一步区分为所研究轮廓在幅值和/或相位上的显著偏移。该建模与位移检测框架已应用于实际案例研究，对雅典某地区每日空气污染物浓度轮廓进行了建模，并成功识别了多数情况下表征危险浓度水平的轮廓。