  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.
