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Modelling of Functional Profiles and Explainable Shape Shifts Detection: An Approach Combining the Notion of the Fréchet Mean with the Shape-Invariant Model

Author

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  • Georgios I. Papayiannis

    (Section of Mathematics, Department of Naval Sciences, Hellenic Naval Academy, 18539 Piraeus, Greece
    Stochastic Modelling and Applications Laboratory, Athens University of Economics and Business, 10434 Athens, Greece)

  • Stelios Psarakis

    (Department of Statistics and Laboratory of Statistical Methodology, Athens University of Economics and Business, 10434 Athens, Greece)

  • Athanasios N. Yannacopoulos

    (Department of Statistics and Stochastic Modelling and Applications Laboratory, Athens University of Economics and Business, 10434 Athens, Greece)

Abstract

A modelling framework suitable for detecting shape shifts in functional profiles combining the notion of the Fréchet mean and the concept of deformation models is developed and proposed. The generalized mean sense offered by the Fréchet 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 the profile’s deviations from the typical shape. The 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 into significant shifts with respect to amplitude and/or the phase of the profile under study. The proposed modeling 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 modeled, while profiles indicating hazardous concentration levels are successfully identified in most cases.

Suggested Citation

  • Georgios I. Papayiannis & Stelios Psarakis & Athanasios N. Yannacopoulos, 2023. "Modelling of Functional Profiles and Explainable Shape Shifts Detection: An Approach Combining the Notion of the Fréchet Mean with the Shape-Invariant Model," Mathematics, MDPI, vol. 11(21), pages 1-24, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4466-:d:1269324
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    References listed on IDEAS

    as
    1. Sungkyu Jung & Ian L. Dryden & J. S. Marron, 2012. "Analysis of principal nested spheres," Biometrika, Biometrika Trust, vol. 99(3), pages 551-568.
    2. Javier Cano & Javier M. Moguerza & Stelios Psarakis & Athanasios N. Yannacopoulos, 2015. "Using statistical shape theory for the monitoring of nonlinear profiles," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 31(2), pages 160-177, March.
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