Monitoring of functional profiles combining the notion of Fr\'echet mean and the framework of deformation models with application in ambient air pollution surveillance

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.