Modelling of functional profiles and explainable shape shifts detection: An approach combining the notion of the Fr\'echet mean with the shape invariant model

TL;DR

This paper introduces a novel framework combining the Fréchet mean and shape invariant models to detect shape shifts in functional profiles, with applications to air quality data for identifying hazardous pollution levels.

Contribution

It develops a new modeling and shift detection approach that integrates shape invariant models with Fréchet mean, enabling interpretable detection of shape and deformation shifts in functional data.

Findings

Successfully identified hazardous air pollutant concentration profiles

Developed EWMA-type control charts for functional data

Distinguished shifts in shape, amplitude, and phase

Abstract

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.