The spatial distribution in infinite dimensional spaces and related quantiles and depths

Abstract

This reprint differs from the original in pagination and typographic detail. 1 2 A. CHAKRABORTY AND P. CHAUDHURIare curves or functions and can be modeled as random observations from probability distributions in infinite dimensional spaces. The ECG curves of patients observed over a period of time, the spectrometry curves recorded for a range of wavelengths, the annual temperature curves of different places, etc., are examples of such data. Many of the function spaces, where such data lie, are infinite dimensional Banach spaces. However, many of the wellknown multivariate medians like the simplicial depth median (see [23]), and the simplicial volume median (see [28]) do not have meaningful extensions into such spaces. On the other hand, the spatial median as well as the spatial quantiles extend easily into infinite dimensional Banach spaces (see [13,19] and [34]). The author of [18] proposed functional principal components using the sample spatial median and used those to analyze a data involving the movements of the lips. The authors of [8] considered an updation based estimator of the spatial median, and used it to compute the profile of a typical television audience in France throughout a single day. The spatial median has also been used in [12] to calculate the median profile for the electricity load data in France. Recently, the authors of [17] studied some direction-based quantiles for probability distributions in infinite dimensional Hilbert spaces. These quantiles are defined for unit direction vectors in such spaces, and they extend the finite dimensional quantiles considered in [22]. The principle quantile directions derived from these quantiles were used in [17] to detect outliers in a dataset of annual age-specific mortality rates of French males between the years 1899 and 2005. The purpose of this article is to investigate the spatial distribution in infinite dimensional Banach spaces, and study their properties along with the spatial quantiles and the spatial depth. There are several mathematical difficulties in dealing with the probability distributions in such spaces. These are primarily due to the noncompactness of the closed unit ball in such spaces. In Section 2, we prove some Glivenko-Cantelli and Donsker-type results for the empirical spatial distribution process arising from data lying in infinite dimensional spaces. In Section 3, we investigate the spatial quantiles in infinite dimensional spaces. A Bahadur-type linear representation of the sample spatial quantiles and their asymptotic Gaussianity are derived. We also study the asymptotic efficiency of the sample spatial median relative to the sample mean for some well-known probability distributions in function spaces. In Section 4, we investigate the spatial depth and its asymptotic properties in infinite dimensional spaces. We also demonstrate how exploratory data analytic tools like the depth-depth plot (DD-plot) (see [24]) can be developed for data in infinite dimensional spaces using the spatial depth. The proofs of the theorems are g...

Halfspace depths for scatter, concentration and shape matrices

“…In the last decade, depth has also known much success in functional data analysis, where it measures the centrality of a function with respect to a sample of functional data. 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.…”

Nonparametric depth and quantile regression for functional data

“…The conditional spatial depth SD(y | x) of a point y ∈ H given X = x is defined as SD(y | x) = 1 − S(y | x) (cf. [8]). Defining the conditional depth in the above way ensures that a point near the center of the conditional distribution of the response has higher depth than a point at the peripheral regions of the conditional distribution, and also that the conditional depth lies between 0 and 1.…”

“…Spatial depth for multivariate data was developed in [42] and [37], based on the ideas of spatial quantiles in [12] and [28]. The concept of spatial depth is extended to infinite dimensional data in [8]. In [38], multivariate spatial depth was employed in a functional data context by first discretizing the sample curves.…”

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