Generalizing the Convex Hull of a Sample: The R Package alphahull

This paper presents the R package alphahull which implements the ñ-convex hull and the ñ-shape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the ñ-convex hull and the ñ-shape are able to reconstruct non-convex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations. The usefulness of the package is illustrated with two small simulation studies on boundary length estimation.

Connected Spatial Networks over Random Points and a Route-Length Statistic

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Probabilistic Bounds on the Length of a Longest Edge in Delaunay Graphs of Random Points in d-Dimensions

Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphsComment: 10 pages. 2 figures. In Proceedings of the 23rd Canadian Conference on Computational Geometry (CCCG 2011). Replacement of version 1106.4927, reference [5] adde

Extrema of locally stationary Gaussian fields on growing manifolds

We consider a class of non-homogeneous, continuous, centered Gaussian random fields $\{X_h(t), t \in {\cal M}_h;\,0 < h \le 1\}$ where ${\cal M}_h$ denotes a rescaled smooth manifold, i.e. ${\cal M}_h = \frac{1}{h} {\cal M},$ and study the limit behavior of the extreme values of these Gaussian random fields when $h$ tends to zero, which means that the manifold is growing. Our main result can be thought of as a generalization of a classical result of Bickel and Rosenblatt (1973a), and also of results by Mikhaleva and Piterbarg (1997).Comment: 28 pages, 1 figur

Relaxed Disk Packing

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted to CCCG 201

Advancing In Situ Modeling of ICMEs: New Techniques for New Observations

It is generally known that multi-spacecraft observations of interplanetary coronal mass ejections (ICMEs) more clearly reveal their three-dimensional structure than do observations made by a single spacecraft. The launch of the STEREO twin observatories in October 2006 has greatly increased the number of multipoint studies of ICMEs in the literature, but this field is still in its infancy. To date, most studies continue to use on flux rope models that rely on single track observations through a vast, multi-faceted structure, which oversimplifies the problem and often hinders interpretation of the large-scale geometry, especially for cases in which one spacecraft observes a flux rope, while another does not. In order to tackle these complex problems, new modeling techniques are required. We describe these new techniques and analyze two ICMEs observed at the twin STEREO spacecraft on 22-23 May 2007, when the spacecraft were separated by ~8 degrees. We find a combination of non-force-free flux rope multi-spacecraft modeling, together with a new non-flux rope ICME plasma flow deflection model, better constrains the large-scale structure of these ICMEs. We also introduce a new spatial mapping technique that allows us to put multispacecraft observations and the new ICME model results in context with the convecting solar wind. What is distinctly different about this analysis is that it reveals aspects of ICME geometry and dynamics in a far more visually intuitive way than previously accomplished. In the case of the 22-23 May ICMEs, the analysis facilitates a more physical understanding of ICME large-scale structure, the location and geometry of flux rope sub-structures within these ICMEs, and their dynamic interaction with the ambient solar wind