Depth for curve data and applications

TL;DR

This paper introduces a new data depth concept for curve or trajectory data that is parametrization-independent, extending the statistical tool to functional and trajectory data with applications in brain imaging and handwriting recognition.

Contribution

It proposes a novel notion of data depth for curves, satisfying theoretical depth function properties and applicable to real-world trajectory data.

Findings

The curve depth measure is theoretically sound for trajectory data.

Applications demonstrate effectiveness in brain imaging analysis.

Method shows promise for pattern recognition tasks.

Abstract

John W. Tukey (1975) defined statistical data depth as a function that determines centrality of an arbitrary point with respect to a data cloud or to a probability measure. During the last decades, this seminal idea of data depth evolved into a powerful tool proving to be useful in various fields of science. Recently, extending the notion of data depth to the functional setting attracted a lot of attention among theoretical and applied statisticians. We go further and suggest a notion of data depth suitable for data represented as curves, or trajectories, which is independent of the parametrization. We show that our curve depth satisfies theoretical requirements of general depth functions that are meaningful for trajectories. We apply our methodology to diffusion tensor brain images and also to pattern recognition of hand written digits and letters. Supplementary Materials are available online.