Functional data analysis with Möbius waves: Applications to biomedical oscillatory signals

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Abstract
The Frequency Modulated Möbius (FMM) approach is a contemporary and versatile tool for analyzing oscillatory signals. Within the functional data analysis framework, the FMM model offers an accurate alternative for comprehending multidimensional synchronized oscillatory signals, which are common in various fields including biology and medicine. This approach decomposes the signals into scaled Möbius waves formulated in terms of four parameters, interpreted as measures of location and shape variation. Among these parameters, two are specific to individual signals, while the remaining two are global and describe the interconnections between signals. In this paper we extend the utility of the FMM by developing inferential procedures based on likelihood estimations for nonlinear models. These procedures cover the core parameters, the signals themselves, and their derivatives. We explore various techniques, including the Gauss–Seidel approach and profile likelihood, for obtaining estimators and their asymptotic properties. Our methodology undergoes rigorous validation, drawing on a combination of theoretical insights and numerical experiments. Furthermore, as our research is primarily motivated by the analysis of biomedical oscillatory signals, we apply this methodology to address two pertinent real-world problems involving electrocardiogram and pattern-reversal visual evoked potential data. These applications illustrate the practical efficacy of the new paradigm.

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