For multivariate data, Tukey's half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey's half-space depth and the related median for probability distributions on any Banach space (which may be finite-or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 4, 1420-1434. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2011 ISI/BSTukey's half-space depth 1421 infinity. Moreover, if the underlying population distribution F has a spherically symmetric density f , that is, f (x) = ψ( x 2 ) for some ψ : R + → R + , the half-space depth turns out to be a decreasing function of x 2 = (|x 1 | 2 + · · · + |x d | 2 ) 1/2 . Consequently, when ψ is monotonically decreasing (i.e., f is unimodal), the half-space depth becomes an increasing function of f and vice versa. Therefore, in such cases, the half-space depth contours coincide with the contours of the density function. Because of this property of the halfspace depth, classification rules based on the ordering of the half-space depth functions coincide with the optimal Bayes classifier for discriminating among spherically symmetric unimodal populations differing in their centers of symmetry (see, e.g., [8]). Similarly, the use of the half-space depth functions to order and trim multivariate data sets (see, e.g., [6,17]) leading to the determination of central and outlying observations has a natural justification when the density contours coincide with the half-space depth contours. Also, due to this relation between half-space depth and spherical symmetry, half-space depth has been used to construct diagnostic tools for checking spherical symmetry of a data cloud (see, e.g., [13], pages 809-811). Another well-known feature of half-space depth is its characterization property. Koshevoy [10] proved that if the half-space depth functions of two atomic measures with finite support are identical, then the measures are also identical. Cuesta-Albertosa and Nieto-Reyes [4] proved this characterization property of Tukey depth for discrete distributions. Under some regularity conditions, Koshevoy [11] proved this characterization property for absolutely continuous probability distributions with compact support in finite-dimensional spaces. Hassairi and Regaieg [9] ge...

This paper explores the use of visualization through animations, coined visuanimation, in the field of statistics. In particular, it illustrates the embedding of animations in the paper itself and the storage of larger movies in the online supplemental material. We present results from statistics research projects using a variety of visuanimations, ranging from exploratory data analysis of image data sets to spatio-temporal extreme event modeling; these include a multiscale analysis of classification methods, the study of the effects of a simulated explosive volcanic eruption, and an emulation of climate model output. This paper serves as an illustration of visuanimation for future publications in Stat.

Modern global industrialization along with ever-increasing growth of population have resulted in continuous enhancement in the discharge and accumulation of various toxic and hazardous chemicals in the environment. These harmful...

This article uses projection depth (PD) for robust classification of multivariate data. Here we consider two types of classifiers, namely, the maximum depth classifier and the modified depth-based classifier. The latter involves kernel density estimation, where one needs to choose the associated scale of smoothing. We consider both the single scale and the multi-scale versions of kernel density estimation, and investigate the large sample properties of the resulting classifiers under appropriate regularity conditions. Some simulated and real data sets are analyzed to evaluate the finite sample performance of these classification tools.

CitationIn vitro structure-toxicity relationship of chalcones in human hepatic stellate cells 2015 Toxicology This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. In vitro AbstractXanthohumol (XN), the major prenylated chalcone from hops (Humulus lupulus L.), has received much attention within the last years, due to its multiple pharmacological activities including anti-proliferative, anti-inflammatory, antioxidant, pro-apoptotic, anti-bacterial and anti-adhesive effects. However, there exists a huge number of metabolites and structurallyrelated chalcones, which can be expected, or are already known, to exhibit various effects on cells. We have therefore analyzed the effects of XN and 18 other chalcones in a panel, consisting of multiple cell-based assays. Readouts of these assays addressed distinct aspects of cell-toxicity, like proliferation, mitochondrial health, cell cycle and other cellular features. Besides known active structural elements of chalcones, like the Michael system, we have identified several moieties that seem to have an impact on specific effects and toxicity in human liver cells in vitro. Based on these observations, we present a structure-toxicity model, which will be crucial to understand the molecular mechanisms of wanted effects and unwanted side-effects of chalcones.

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For data with more variables than the sample size, phenomena like concentration of pairwise distances, violation of cluster assumptions and presence of hubness often have adverse effects on the performance of the classic nearest neighbor classifier. To cope with such problems, some dimension reduction techniques like those based on random linear projections and principal component directions have been proposed in the literature. In this article, we construct nonlinear transformations of the data based on inter-point distances, which also lead to reduction in data dimension. More importantly, for such high dimension low sample size data, they enhance separability among the competing classes in the transformed space. When the classic nearest neighbor classifier is used on the transformed data, it usually yields lower misclassification rates. Under appropriate regularity conditions, we derive asymptotic results on misclassification probabilities of nearest neighbor classifiers based on the l 2 norm and the l p norms (with p ∈ (0, 1]) in the transformed space, when the training sample size remains fixed and the dimension of the data grows to infinity. Strength of the proposed transformations in the classification context is demonstrated by analyzing several simulated and benchmark data sets.

Total mercury levels were quantified in Tilapia mossambicus, Cirrhinus mrigela and Labio rohita, captured from East Calcutta Wetlands and Titagarh sewage fed aquaculture ponds. The bioconcentration factor of collected fish was assessed. Total mercury level ranged from 0.073 to 0.94 microg/g in both pre and post monsoon season. T. mossambicus in both season and C. mrigela at pre monsoon, cross the Indian recommended maximum limit (0.50 microg/g wet weight) for food consumption and according to World Health Organization guidelines all fish were not recommended for pregnant women and individuals under 15 years ages. A significant correlation was observed between mercury content of aquaculture pond water and fish muscle tissue. Total mercury concentration in experimental sites were higher than the control area (Wilcoxon Ranked-Sum test p > 0.05), which suggested the connection between mercury bioaccumulation and sewage fed aquaculture.