On data depth in infinite dimensional spaces

Chakraborty, Anirvan; Chaudhuri, Probal

doi:10.1007/s10463-013-0416-y

Cited by 51 publications

(54 citation statements)

References 31 publications

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“…There are a few other notions of depth function for data in infinite dimensional function spaces (see, e.g., [16,25,26] and [32]). However, as shown in [10], some of these depth functions exhibit degeneracy for certain types of functional data, and hence are not very useful.…”

Section: 2mentioning

confidence: 99%

“…The properties of the spatial depth discussed above imply that it induces a meaningful center-outward ordering of the points in these spaces, and can be used to develop depth-based statistical procedures for data from such distributions. On the other hand, many of the well-known depths for infinite dimensional data like the half-space depth, the band depth and the halfregion depth do not possess such regular behavior and exhibit degeneracy for many Gaussian distributions (see [10]).…”

Section: 2mentioning

confidence: 99%

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

Chakraborty

Chaudhuri²

2014

Ann. Statist.

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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...

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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

Chakraborty

Chaudhuri²

2014

Ann. Statist.

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“…The research has been focused on computational issues, and the only major theoretical results are the consistency theorems: López-Pintado and Romo (2009, Theorem 4) for band depth and López-Pintado and Romo (2011, Theorem 3) for half-region depth. Recently, Chakraborty and Chaudhuri (2014, Section 3) pointed out some degenerate behaviour of the band depth and the half-region depth for certain probability distributions. For Φ-depths of Mosler and Polyakova (2012) there are, as far as we know, no results, yet, on treating its theoretical statistical properties.…”

Section: Introduction: Depth For Functional Datamentioning

confidence: 99%

Consistency of non-integrated depths for functional data

Gijbels

Nagy

2015

Journal of Multivariate Analysis

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In the analysis of functional data, the concept of data depth is of importance. Strong consistency of a sample version of a data depth is among the basic statistical properties that need to hold. In this paper we discuss consistency properties of three popular types of functional depth: the band depth, the half-region depth and the infimal depth. The latter is a special case of the recently introduced general class of Φ-depths. All three considered depth functions are of a non-integrated type. Counterexamples illustrate some problems with consistency results for these data depths. The main contribution of this paper consists of providing sufficient conditions for consistency of these non-integrated data depths to hold.

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“…Moreover an interesting question in such a scenario is the identification of an outlier. A study on data depth by Chakraborty & Chaudhuri () showed that for a large class of infinite‐dimensional distributions, the notion of SPD transforms all values to the interval

false(0, 1 false)

; see their Theorems 6 and 7. SPD is therefore still meaningful for data arising from a quite large class of infinite‐dimensional distributions.…”

Section: Robust Multivariate Regression For Sparse Datamentioning

confidence: 99%

Depth‐weighted robust multivariate regression with application to sparse data

Dutta

Genton

2017

Can J Statistics

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A robust method for multivariate regression is developed based on robust estimators of the joint location and scatter matrix of the explanatory and response variables using the notion of data depth. The multivariate regression estimator possesses desirable affine equivariance properties, achieves the best breakdown point of any affine equivariant estimator, and has an influence function which is bounded in both the response as well as the predictor variable. To increase the efficiency of this estimator, a re‐weighted estimator based on robust Mahalanobis distances of the residual vectors is proposed. In practice, the method is more stable than existing methods that are constructed using subsamples of the data. The resulting multivariate regression technique is computationally feasible, and turns out to perform better than several popular robust multivariate regression methods when applied to various simulated data as well as a real benchmark data set. When the data dimension is quite high compared to the sample size it is still possible to use meaningful notions of data depth along with the corresponding depth values to construct a robust estimator in a sparse setting. The Canadian Journal of Statistics 45: 164–184; 2017 © 2017 Statistical Society of Canada

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On data depth in infinite dimensional spaces

Cited by 51 publications

References 31 publications

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

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

Consistency of non-integrated depths for functional data

Depth‐weighted robust multivariate regression with application to sparse data

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