Results 1  10
of
86
Quantile Functions for Multivariate Analysis: Approaches and Applications
, 2001
"... Despite the absence of a natural ordering of Euclidean space for dimension greater than one, effort to define vectorvalued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we in ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
Despite the absence of a natural ordering of Euclidean space for dimension greater than one, effort to define vectorvalued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we introduce relevant criteria, including a notion of "medianoriented quantile function". On this basis we compare recent quantile approaches and several multivariate versions of trimmed mean and interquartile range. We also discuss a univariate "generalized quantile" approach that enables particular features of multivariate distributions, for example scale and kurtosis, to be studied by twodimensional plots. Methods based on statistical depth functions are found to be especially attractive for quantilebased multivariate inference.
Structural Properties and Convergence Results for Contours of Sample Statistical Depth Functions
, 2000
"... Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness, and compactness, and almost ..."
Abstract

Cited by 37 (14 self)
 Add to MetaCart
Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness, and compactness, and almost sure convergence results for sample depth contours, are established. Also, specialized results are established for some popular depth functions, including halfspace depth, and for the case of elliptical distributions. Finally, some needed foundational results on almost sure convergence of sample depth functions are provided.
Outlier Detection with the Kernelized Spatial Depth Function
, 2008
"... Statistical depth functions provide from the “deepest ” point a “centeroutward ordering” of multidimensional data. In this sense, depth functions can measure the “extremeness” or “outlyingness” of a data point with respect to a given data set. Hence they can detect outliers – observations that appe ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
(Show Context)
Statistical depth functions provide from the “deepest ” point a “centeroutward ordering” of multidimensional data. In this sense, depth functions can measure the “extremeness” or “outlyingness” of a data point with respect to a given data set. Hence they can detect outliers – observations that appear extreme relative to the rest of the observations. Of the various statistical depths, the spatial depth is especially appealing because of its computational efficiency and mathematical tractability. In this article, we propose a novel statistical depth, the kernelized spatial depth (KSD), which generalizes the spatial depth via positive definite kernels. By choosing a proper kernel, the KSD can capture the local structure of a data set while the spatial depth fails. We demonstrate this by the halfmoon data and the ringshaped data. Based on the KSD, we propose a novel outlier detection algorithm, by which an observation with a depth value less than a threshold is declared as an outlier. The proposed algorithm is simple in structure: the threshold is the only one parameter for a given kernel. It applies to a oneclass learning setting, in which “normal ” observations are given as the training data, as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels. We give upper bounds on the false alarm probability of a depthbased detector. These upper bounds can be used to determine the threshold. We perform extensive experiments on synthetic data and data sets from real applications. The proposed outlier detector is compared with existing methods. The KSD outlier detector demonstrates competitive performance.
A depth function and a scale curve based on spatial quantiles
 In Statistical Data Analysis Based On the L1Norm and Related Methods
, 2002
"... Abstract. Spatial quantiles, based on the L1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume ” functional based on spatial “central regions ” of increasing size. A plot of this functional as a “spatial scale curve ” provides a conv ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
Abstract. Spatial quantiles, based on the L1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume ” functional based on spatial “central regions ” of increasing size. A plot of this functional as a “spatial scale curve ” provides a convenient twodimensional characterization of the spread of a multivariate distribution of any dimension. We discuss this curve and establish weak convergence of the empirical version. As a tool, we introduce and study a new statistical depth function which is naturally associated with spatial quantiles. Other depth functions that generate L1based multivariate quantiles are also noted. 1.
On the StahelDonoho Estimator and DepthWeighted Means of Multivariate Data, The Annals of Statistics 22
, 2004
"... The depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the Stahel–Donoho estimator as a special case. In contrast to maximal depth estimators, the depthweighted means are shown to be asymptotically normal under a ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
The depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the Stahel–Donoho estimator as a special case. In contrast to maximal depth estimators, the depthweighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature. We also confirm through a finitesample study that the Stahel–Donoho estimator achieves a desirable balance between robustness and efficiency at Gaussian models. 1. Introduction. Depth
Multivariate quantiles and multipleoutput regression quantiles: from L1 optimization to halfspace depth (with discussion
 Annals of Statistics
, 2010
"... A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multipleoutput regression problems. In their empirical version, those quantiles can be computed efficiently via linear progra ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
A new multivariate concept of quantile, based on a directional version of Koenker and Bassett’s traditional regression quantiles, is introduced for multivariate location and multipleoutput regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depthrelated multivariate rankbased inference.
Projection Based Depth Functions and Associated Medians
 The Annals of Statistics
, 2000
"... A class of projection based depth functions is introduced and studied. These projection based depth functions possess desirable properties of "statistical depth functions" and their sample versions possess the strong and order √n uniform consistency. Depth regions and contours in ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
A class of projection based depth functions is introduced and studied. These projection based depth functions possess desirable properties of "statistical depth functions" and their sample versions possess the strong and order &radic;n uniform consistency. Depth regions and contours induced from projection based depth functions are investigated. Structural properties of depth regions and depth contours and general continuity and convergence results of sample depth regions are obtained. Affine equivariant multivariate medians induced from projection based depth functions are studied. The limiting distributions as well as the strong and order &radic;n consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of the leading depth induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that multivariate medians induced from projection based depth functions possess much hig...
Equivariance and Invariance Properties of Multivariate Quantile and Related Functions, and the Role of Standardization
, 2009
"... Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invar ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invariance of procedures, three important types of matrixvalued functional are studied: “weak covariance ” (or “shape”), “transformationretransformation ” (TR), and “strong invariant coordinate system ” (SICS). Clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artifacts of SICSstandardized data are invariant under affine transformation of the original data followed by standardization using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardization are described.
On the Performance of Some Robust Nonparametric Location Measures Relative to a General Notion of Multivariate Symmetry
 J. STATIST. PLANN. INFERENCE
, 1999
"... Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, name ..."
Abstract

Cited by 16 (10 self)
 Add to MetaCart
Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, namely "halfspace symmetry", generalizing the wellknown notions of "central" and "angular" symmetry. Characterizations of these symmetry notions are established, permitting their properties and interrelations to be illuminated. The particular location measures considered consist of several nonparametric notions of multidimensional median: the "L²" (or "spatial"), "Tukey/Donoho halfspace", "projection", and "Liu simplicial" medians, all of which are robust in the sense of nonzero breakdown point. It is established that the rst three of these in general do identify the point of symmetry when it exists, whereas the latter, however, fails to do so in some circumstances. Combining this finding ...
Lower Bounds for Computing Statistical Depth
, 2001
"... Given a finite set of points S, two measures of the depth of a query point with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires n log n) time, which matches the upper bound complexities o ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Given a finite set of points S, two measures of the depth of a query point with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires n log n) time, which matches the upper bound complexities of the algorithms of Rousseeuw and Ruts. Our lower bound proofs may also be applied to two bivariate sign tests: that of Hodges, and that of Oja and Nyblom.