Summary We propose a method to test the correlation of two random fields when they are both spatially auto-correlated. In this scenario, the assumption of independence for the pair of observations in the standard test does not hold, and as a result we reject in many cases where there is no effect (the precision of the null distribution is overestimated). Our method recovers the null distribution taking into account the autocorrelation. It uses Monte-Carlo methods, and focuses on permuting, and then smoothing and scaling one of the variables to destroy the correlation with the other, while maintaining at the same time the initial autocorrelation. With this simulation model, any test based on the independence of two (or more) random fields can be constructed. This research was motivated by a project in biodiversity and conservation in the Biology Department at Stanford University.

a b s t r a c tIn this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher's linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.

We propose a nonparametric cluster algorithm based on local medians. Each observation is substituted by its local median and this new observation moves toward the peaks and away from the valleys of the distribution. The process is repeated until each observation converges to a fixpoint. We obtain a partition of the sample based on the convergence points. Our algorithm determines the number of clusters and the partition of the observations given the proportion α of neighbors. A fast version of the algorithm where only a subset of the observations from the sample is processed is also proposed. A proof of the convergence from each point to its closest fixpoint and the existence and uniqueness of a fixpoint in a neighborhood of each mode is given for the univariate case. © 2012 Wiley Periodicals, Inc. Statistical Analysis and Data Mining, 2012