89 research outputs found
Определение глубины данных на многомерных выборках
Concepts of depth and depth trimmed regions are considered, their main properties and classification are presented. 5 notions of data depth and results of investigating their correspondence with requirements, imposed on depth and regions, are discussed. One of the latest forms of depth zonoid depth and algorithms for depth computation and constructing regions are outlined. These mathematical tools for nonparametric data analysis can be applied in location and dispersion estimation, cluster analysis and risk estimation
Intrinsic data depth for Hermitian positive definite matrices
Nondegenerate covariance, correlation and spectral density matrices are
necessarily symmetric or Hermitian and positive definite. The main contribution
of this paper is the development of statistical data depths for collections of
Hermitian positive definite matrices by exploiting the geometric structure of
the space as a Riemannian manifold. The depth functions allow one to naturally
characterize most central or outlying matrices, but also provide a practical
framework for inference in the context of samples of positive definite
matrices. First, the desired properties of an intrinsic data depth function
acting on the space of Hermitian positive definite matrices are presented.
Second, we propose two computationally fast pointwise and integrated data depth
functions that satisfy each of these requirements and investigate several
robustness and efficiency aspects. As an application, we construct depth-based
confidence regions for the intrinsic mean of a sample of positive definite
matrices, which is applied to the exploratory analysis of a collection of
covariance matrices associated to a multicenter research trial
Depth and Depth-Based Classification with R Package ddalpha
Following the seminal idea of Tukey (1975), data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification. ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the DDα-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition
Nonparametrically consistent depth-based classifiers
We introduce a class of depth-based classification procedures that are of a
nearest-neighbor nature. Depth, after symmetrization, indeed provides the
center-outward ordering that is necessary and sufficient to define nearest
neighbors. Like all their depth-based competitors, the resulting classifiers
are affine-invariant, hence in particular are insensitive to unit changes.
Unlike the former, however, the latter achieve Bayes consistency under
virtually any absolutely continuous distributions - a concept we call
nonparametric consistency, to stress the difference with the stronger universal
consistency of the standard NN classifiers. We investigate the finite-sample
performances of the proposed classifiers through simulations and show that they
outperform affine-invariant nearest-neighbor classifiers obtained through an
obvious standardization construction. We illustrate the practical value of our
classifiers on two real data examples. Finally, we shortly discuss the possible
uses of our depth-based neighbors in other inference problems.Comment: Published at http://dx.doi.org/10.3150/13-BEJ561 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A Comparison of Depth Functions in Maximal Depth Classification Rules
Data depth has been described as alternative to some parametric approaches in analyzing many multivariate data. Many depth functions have emerged over two decades and studied in literature. In this study, a nonparametric approach to classification based on notions of different data depth functions is considered and some properties of these methods are studied. The performance of different depth functions in maximal depth classifiers is investigated using simulation and real data with application to agricultural industry
Fast implementation of the Tukey depth
Tukey depth function is one of the most famous multivariate tools serving
robust purposes. It is also very well known for its computability problems in
dimensions . In this paper, we address this computing issue by
presenting two combinatorial algorithms. The first is naive and calculates the
Tukey depth of a single point with complexity ,
while the second further utilizes the quasiconcave of the Tukey depth function
and hence is more efficient than the first. Both require very minimal memory
and run much faster than the existing ones. All experiments indicate that they
compute the exact Tukey depth.Comment: 16 pages, 13 figure
- …