Mean curvature and mean shape for multivariate functional data under Frenet-Serret framework

TL;DR

This paper introduces a novel geometric framework for analyzing multivariate functional data using Frenet-Serret equations, enabling the extraction of mean curvature and shape from noisy multidimensional curves.

Contribution

It proposes a new method to define and estimate mean curvature and shape of multidimensional curves based on differential geometry, extending functional data analysis.

Findings

Effective in capturing geometric features from noisy data

Demonstrated success with simulated and real datasets

Provides an efficient algorithm for estimation

Abstract

The analysis of curves has been routinely dealt with using tools from functional data analysis. However its extension to multi-dimensional curves poses a new challenge due to its inherent geometric features that are difficult to capture with the classical approaches that rely on linear approximations. We propose a new framework for functional data as multidimensional curves that allows us to extract geometrical features from noisy data. We define a mean through measuring shape variation of the curves. The notion of shape has been used in functional data analysis somewhat intuitively to find a common pattern in one dimensional curves. As a generalization, we directly utilize a geometric representation of the curves through the Frenet-Serret ordinary differential equations and introduce a new definition of mean curvature and mean shape through the mean ordinary differential equation. We formulate the estimation problem in a penalized regression and develop an efficient algorithm. We demonstrate our approach with both simulated data and a real data example.