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 $p \ge 3$. 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 $O\left(n^{p-1}\log(n)\right)$, 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

Fast computation of Tukey trimmed regions and median in dimension p>2p>2

Given data in $\mathbb{R}^{p}$, a Tukey $\kappa$-trimmed region is the set of all points that have at least Tukey depth $\kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension $p > 2$. We construct two novel algorithms to compute a Tukey $\kappa$-trimmed region, a na\"{i}ve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the na\"{i}ve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median

Quantile tomography: using quantiles with multivariate data

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development

Weighted lens depth: Some applications to supervised classification

Starting with Tukey's pioneering work in the 1970's, the notion of depth in statistics has been widely extended especially in the last decade. These extensions include high dimensional data, functional data, and manifold-valued data. In particular, in the learning paradigm, the depth-depth method has become a useful technique. In this paper we extend the notion of lens depth to the case of data in metric spaces, and prove its main properties, with particular emphasis on the case of Riemannian manifolds, where we extend the concept of lens depth in such a way that it takes into account non-convex structures on the data distribution. Next we illustrate our results with some simulation results and also in some interesting real datasets, including pattern recognition in phylogenetic trees using the depth--depth approach.Comment: 1

Dynamic Boundary Element Analysis of Machine Foundations

The central theme of this thesis is the further development of boundary element methods for the analysis of three-dimensional machine foundations, pertaining to various (translational and rotational) modes of vibration and, in particular, to high frequency response. Surface and embedded rectangular foundations are considered. The soil is assumed to behave approximately as a linear elastic material for small amplitudes of strain. The problem is formulated and solved in the frequency domain. This work includes rigorous theoretical studies, effective numerical techniques for the solution of the boundary integral equations, and efficient computer implementation of the algorithm. The derivation of the boundary integral formulation is reviewed and the dynamic fundamental solutions are examined in detail. The particular fundamental solutions for incompressible media has been derived in order to deal more effectively with these materials. Advanced integration schemes for non-singular and singular integrals have been developed in order to improve the computational accuracy and efficiency of the boundary element analysis. A novel infinite boundary element for dynamic analyses has been developed, which provides an efficient means for including far-field effects, without the necessity of explicit discrete representation outside the near field. The implementation and vectorization of the computer program using the IBM 3090-150 Vector Facility is described. Various numerical results for rectangular foundations are presented in order to illustrate the potential of the infinite boundary element formulation. Included among these are new results pertaining to the high frequency response of machine foundations