title: High dimensional data analysis using multivariate generalized spatial quantiles.
abstract: High dimensional data routinely arises in image analysis, genetic experiments, network analysis, and various other research areas. Many such datasets do not correspond to well-studied probability distributions, and in several applications the data-cloud prominently displays non-symmetric and non-convex shape features. We propose using spatial quantiles and their generalizations, in particular, the projection quantile, for describing, analyzing and conducting inference with multivariate data. Minimal assumptions are made about the nature and shape characteristics of the underlying probability distribution, and we do not require the sample size to be as high as the data-dimension. We present theoretical properties of the generalized spatial quantiles, and an algorithm to compute them quickly. Our quantiles may be used to obtain multidimensional confidence or credible regions that are not required to conform to a pre-determined shape. We also propose a new notion of multidimensional order statistics, which may be used to obtain multidimensional outliers. Many of the features revealed using a generalized spatial quantile-based analysis would be missed if the data was shoehorned into a well-known probabilistic configuration.
authors.affiliation: Virginia Commonwealth University, Department of Biostatistics, Richmond VA 23298, United States.
authors.forename: Nitai D
authors.initials: ND
authors.lastname: Mukhopadhyay
authors.affiliation: School of Statistics, University of Minnesota, Minneapolis MN 55455, United States.
authors.forename: Snigdhansu
authors.initials: S
authors.lastname: Chatterjee
keywords: Brain imaging; Generalized spatial quantile; High dimensional data visualization; Multidimensional coverage sets; Multivariate order statistics; Multivariate quantile; Projection quantile; Spatial quantile