Direction-Projection-Permutation for High-Dimensional Hypothesis Tests

High-dimensional low sample size (HDLSS) data are becoming increasingly common in statistical applications. When the data can be partitioned into two classes, a basic task is to construct a classifier that can assign objects to the correct class. Binary linear classifiers have been shown to be especially useful in HDLSS settings and preferable to more complicated classifiers because of their ease of interpretability. We propose a computational tool called direction-projection-permutation (DiProPerm), which rigorously assesses whether a binary linear classifier is detecting statistically significant differences between two high-dimensional distributions. The basic idea behind DiProPerm involves working directly with the one-dimensional projections of the data induced by binary linear classifier. Theoretical properties of DiProPerm are studied under the HDLSS asymptotic regime whereby dimension diverges to infinity while sample size remains fixed. We show that certain variations of DiProPerm are consistent and that consistency is a nontrivial property of tests in the HDLSS asymptotic regime. The practical utility of DiProPerm is demonstrated on HDLSS gene expression microarray datasets. Finally, an empirical power study is conducted comparing DiProPerm to several alternative two-sample HDLSS tests to understand the advantages and disadvantages of each method.     

Non parametric estimation of smooth stationary covariance functions by interpolation methods

In this paper we introduce a nonparametric approach for the estimation of the covariance function of a stationary stochastic process X _t indexed by The data consist of a finite number of observations of the process at irregularly spaced time points and the aim is to estimate the covariance at any lag point without parametric assumptions and in such a way that it is a positive definite function. After interpolating the process, we use the estimator designed by Parzen (Technometrics 3:167–190,1961) for continuous-time data. Our estimator is shown to be consistent under smoothness assumptions on the covariance. Its performance is evaluated by simulations.     

Stochastic structural analysis and lifetime oriented optimization

A stochastic structural anal sis for lifetime oriented ontimization is presented. The covariance analysis and a shaping filter-realized with singular value decomposition -enables a simultaneous formulation of load process and structure in an overall state space equation. A numerical example - industrial structure and wind load-illustrates the efficiency of the algorithm to estimate the probability density of stress ranges. The algorithm is used in lifetime oriented structural optimization     

The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism where U⊆X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators Lρ which may occur for given by α(f)=f○σ. We obtain examples of partial dynamical systems (XA,σA) such that the construction of the covariance algebra C^∗(XA,σA), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O(XA,α,L), recently introduced by the author and R. Exel, associated to this system are not equivalent, in the sense that there does not exist an invertible function ρ∈C(U) such that O(XA,α,Lρ)≅C^∗(XA,σA).