A notion of depth for sparse functional data
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A notion of depth for sparse functional data Carlo Sguera1
· Sara López-Pintado2
Received: 16 March 2020 / Accepted: 8 September 2020 © Sociedad de Estadística e Investigación Operativa 2020
Abstract Data depth is a well-known and useful nonparametric tool for analyzing functional data. It provides a novel way of ranking a sample of curves from the center outwards and defining robust statistics, such as the median or trimmed means. It has also been used as a building block for functional outlier detection methods and classification. Several notions of depth for functional data were introduced in the literature in the last few decades. These functional depths can only be directly applied to samples of curves measured on a fine and common grid. In practice, this is not always the case, and curves are often observed at sparse and subject dependent grids. In these scenarios, the usual approach consists in estimating the trajectories on a common dense grid, and using the estimates in the depth analysis. This approach ignores the uncertainty associated with the curves estimation step. Our goal is to extend the notion of depth so that it takes into account this uncertainty. Using both functional estimates and their associated confidence intervals, we propose a new method that allows the curve estimation uncertainty to be incorporated into the depth analysis. We describe the new approach using the modified band depth although any other functional depth could be used. The performance of the proposed methodology is illustrated using simulated curves in different settings where we control the degree of sparsity. Also a real data set consisting of female medflies egg-laying trajectories is considered. The results show the benefits of using uncertainty when computing depth for sparse functional data. Keywords Sparse functional data · Data depth · Modified band depth · Functional principal component analysis
Partial support from the National Institute of Mental Health (Grant Number: 1R21MH120534-01) is acknowledged.
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Carlo Sguera [email protected]; [email protected] Sara López-Pintado [email protected]
1
UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe, Spain
2
Department of Health Sciences, Northeastern University, Boston, USA
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C. Sguera, S. López-Pintado
Mathematics Subject Classification 62R10 · 62F07 · 62G35
1 Introduction and motivation Functional data analysis is an exciting developing area in statistics where the basic unit of observation is a function/curve. Many different statistical methods, such as principal components, analysis of variance and linear regression, have been extended to functional data. In the last two decades, there has been an intensive development of different notions of data depth which have been proven to be a powerful nonparametric tool for analyzing functional data. In general, a data depth is a function that measures the centrality (or outlyingness) of an observation within a population or sample. It provides a novel way of ran
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