Statistical Depth for Big Functional Data with Application to Neuroimaging

TL;DR

This paper introduces a new statistical depth measure for large-scale functional data, including images, satisfying key properties and applicable to neuroimaging data analysis.

Contribution

The paper proposes a novel functional depth that handles large and multi-dimensional data, with theoretical validation and practical application to PET neuroimaging.

Findings

Effective in ordering large functional datasets

Enables non-parametric deconvolution of PET data

Identifies representative subjects in neuroimaging studies

Abstract

Functional depth is the functional data analysis technique that orders a functional data set. Unlike the case of data on the real line, defining this order is non-trivial, and particularly, with functional data, there are a number of properties that any depth should satisfy. We propose a new depth which both satisfies the properties required of a functional depth but also one which can be used in the case where there are a very large number of functional observations or in the case where the observations are functions of several continuous variables (such as images, for example). We give theoretical justification for our choice, and evaluate our proposed depth through simulation. We finally apply the proposed depth to the problem of yielding a completely non-parametric deconvolution of Positron Emission Tomography (PET) data for a very large number of curves across the image, as well as to the problem of finding a representative subject from a set of PET scans.