Elastic analysis of irregularly or sparsely sampled curves

TL;DR

This paper introduces new statistical methods for analyzing irregularly and sparsely sampled curves using elastic distances, spline models, and algorithms, with applications to GPS tracking and handwriting analysis.

Contribution

It develops algorithms to approximate elastic distances for sparse curves and proposes spline-based models for elastic Fréchet means, addressing limitations of existing methods.

Findings

Effective clustering of GPS tracks using elastic distance.

Accurate classification of handwritten spirals for Parkinson's diagnosis.

Implementation of methods in the R package 'elasdics'.

Abstract

We provide statistical analysis methods for samples of curves when the image but not the parametrisation of the curves is of interest. A parametrisation invariant analysis can be based on the elastic distance of the curves modulo warping, but existing methods have limitations in common realistic settings where curves are irregularly and potentially sparsely observed. We provide methods and algorithms to approximate the elastic distance for such curves via interpreting them as polygons. Moreover, we propose to use spline curves for modelling smooth or polygonal Fr\'echet means of open or closed curves with respect to the elastic distance and show identifiability of the spline model modulo warping. We illustrate the use of our methods for elastic mean and distance computation by application to two datasets. The first application clusters sparsely sampled GPS tracks based on the elastic distance and computes smooth means for each cluster to find new paths on Tempelhof field in Berlin. The second classifies irregularly sampled handwritten spirals of Parkinson's patients and controls based on the elastic distance to a mean spiral curve computed using our approach. All developed methods are implemented in the \texttt{R}-package \texttt{elasdics} and evaluated in simulations.