Title: The Frenet-Serret equation for the estimation of the mean shape of multidimensional functional data
Authors: Nicolas Brunel - ENSIIE (France) [presenting]
Juhyun Park - ENSIIE (France)
Abstract: Variations of univariate curves can be efficiently analysed with functional data analysis tools, such as functional PCA or square root velocity transform to name a few. When considering multidimensional curves, these statistical methods can be generalised in several directions, as the notion of multidimensional amplitude can be defined in several ways. We propose a geometrical framework for the analysis of multidimensional functional data, where we aim at estimating the variations of shapes. As a first step, we define a mean shape for a population of curves. For space curves, we need to estimate curvatures and torsions, which are known to be very challenging when using discrete and noisy observations, as their definition involves higher order derivatives. We use the Frenet-Serret formula that defines an Ordinary Differential Equation in the Lie group or rotations, and we show that estimating the geometry of curve is equivalent to estimate time-varying coefficients. We build a penalised criterion and an adaptive estimator of the curvature, torsion and shape and derive an iterative algorithm based on the Magnus expansion. Finally, different simulation settings and real data case are discussed.