A framework suitable for monitoring functional profiles combining the notion of Fr\'echet mean and the concept of deformation models is developed and proposed. The generalized sense of mean that the notion of the Fr\'echet mean offers is employed to capture the typical functional shape of the data, while the concept of deformation models allows for interpretable parameterizations of profile's deviations from the typical shape. Functional EWMA-type control charts are built and proposed based on shape characteristics of the functional data, allowing for (a) identifying shifts from the in-control behaviour and (b) providing causal relationships of the potential shifts with significant deviances of certain qualitative characteristics (e.g amplitude or phase deformations). The functional monitoring scheme is implemented to assess ambient air pollution. In particular, the method is implemented to a synthetic data example to assess its performance under various conditions, and to a real-world example using sensor data from an area in the city of Athens, where air pollutants profiles and their characteristics are successfully analyzed and out-of-control behaviours are identified.
翻译:已经制定和提议了一个适合于监测功能简介的框架,其中结合Fr\'echet平均值概念和变形模型概念,并结合了变形模型概念; 一种普遍的含义,即Fr\'echet表示的报价概念被用来捕捉数据的典型功能形状,而变形模型概念则允许对剖析特征偏离典型形状的可解释参数化; 功能EWMA型控制图是根据功能数据形状特征建立和提议的,以便:(a) 查明从控制状态变化的转变;(b) 提供潜在变化与某些质量特征(如振荡或阶段变形)的重大变形之间的因果关系; 功能监测计划用于评估环境空气污染; 具体地说,采用该方法为综合数据范例,评估其在不同条件下的性能,并采用来自雅典市的一个地区的传感器数据实例,在那里对空气污染简介及其特征进行了成功分析,并查明了控制外行为。