A Topologically Valid Definition of Depth for Functional Data

Abstract

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function

Halfspace depths for scatter, concentration and shape matrices

“…Some instances are the band depth (López-Pintado and Romo, 2009), the functional halfspace depth (Claeskens et al, 2014) and the functional spatial depth (Chakraborty and Chaudhuri, 2014). The large variety of available depths made it necessary to introduce an axiomatic approach identifying the most desirable properties of a depth function; see Zuo and Serfling (2000) in the multivariate case and Nieto- Reyes and Battey (2016) in the functional one.…”

Recognizing VSC DC Cable Fault Types Using Bayesian Functional Data Depth

“…Nagy and Ferraty [35] use functional data analysis to represent discontinuous data. It was also used to analyze functional data [36,37]. By analyzing function curves, it is much easier to detect distant data.…”

Classification of Alzheimer’s Patients through Ubiquitous Computing

“…A functional classifier differs from a multivariate classifier in that it considers the given data as functional. The classifier we apply here is proposed in [ 8 ] and is based on the functional statistical depth [ 9 ] and a multivariate non-parametric kernel classifier. This multivariate classifier uses the euclidean norm and performs a non-parametric estimation of the density function of each of the three stages of the disease through the use of the well-known Nadaraya-Watson estimator.…”

“…Other depth functions give a higher value to the median of the distribution. We use here the h -mode depth because of the nice properties it satisfices according to [ 9 ].…”