Inference for variograms

The empirical variogram is a standard tool in the investigation and modelling of spatial covariance. However, its properties can be difficult to identify and exploit in the context of exploring the characteristics of individual datasets. This is particularly true when seeking to move beyond description towards inferential statements about the structure of the spatial covariance which may be present. A robust form of empirical variogram based on a fourth-root transformation is used. This takes advantage of the normal approximation which gives an excellent description of the variation exhibited on this scale. Calculations of mean, variance and covariance of the binned empirical variogram then allow useful computations such as confidence intervals to be added to the underlying estimator. The comparison of variograms for different datasets provides an illustration of this. The suitability of simplifying assumptions such as isotropy and stationarity can then also be investigated through the construction of appropriate test statistics and the distributional calculations required in the associated p-values can be performed through quadratic form methods. Examples of the use of these methods in assessing the form of spatial covariance present in datasets are shown, both through hypothesis tests and in graphical form. A simulation study explores the properties of the tests while pollution data on mosses in Galicia (North-West Spain) are used to provide a real data illustration

On high-dimensional sign tests

Sign tests are among the most successful procedures in multivariate nonparametric statistics. In this paper, we consider several testing problems in multivariate analysis, directional statistics and multivariate time series analysis, and we show that, under appropriate symmetry assumptions, the fixed-$p$ multivariate sign tests remain valid in the high-dimensional case. Remarkably, our asymptotic results are universal, in the sense that, unlike in most previous works in high-dimensional statistics, $p$ may go to infinity in an arbitrary way as $n$ does. We conduct simulations that (i) confirm our asymptotic results, (ii) reveal that, even for relatively large $p$, chi-square critical values are to be favoured over the (asymptotically equivalent) Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial dependence in the high-dimensional case, Portmanteau sign tests outperform their competitors in terms of validity-robustness.Comment: Published at http://dx.doi.org/10.3150/15-BEJ710 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Gaussian process single-index models as emulators for computer experiments

A single-index model (SIM) provides for parsimonious multi-dimensional nonlinear regression by combining parametric (linear) projection with univariate nonparametric (non-linear) regression models. We show that a particular Gaussian process (GP) formulation is simple to work with and ideal as an emulator for some types of computer experiment as it can outperform the canonical separable GP regression model commonly used in this setting. Our contribution focuses on drastically simplifying, re-interpreting, and then generalizing a recently proposed fully Bayesian GP-SIM combination, and then illustrating its favorable performance on synthetic data and a real-data computer experiment. Two R packages, both released on CRAN, have been augmented to facilitate inference under our proposed model(s).Comment: 23 pages, 9 figures, 1 tabl

SAS/IML Macros for a Multivariate Analysis of Variance Based on Spatial Signs

Recently, new nonparametric multivariate extensions of the univariate sign methods have been proposed. Randles (2000) introduced an affine invariant multivariate sign test for the multivariate location problem. Later on, Hettmansperger and Randles (2002) considered an affine equivariant multivariate median corresponding to this test. The new methods have promising efficiency and robustness properties. In this paper, we review these developments and compare them with the classical multivariate analysis of variance model. A new SAS/IML tool for performing a spatial sign based multivariate analysis of variance is introduced.