Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Statistical modelling of spatial animal movements

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State of the Art in Patterns for Point Cluster Analysis

International audienceNowadays, an abundance of sensors are used to collect very large da-tasets containing spatial points which can be mined and analyzed to extract meaningful patterns. In this article, we focus on different techniques used to summarize and visualize 2D point clusters and discuss their relative strengths. This article focuses on patterns which describe the dispersion of data around a central tendency. These techniques are particularly beneficial for detecting out-liers and understanding the spatial density of point clusters

An Application of Advanced Spatio-Temporal Formalisms to Behavioural Ecology

Abstract. There is great potential for the development of many new applications using data on mobile objects and mobile regions. To promote these kinds of applications advanced data management techniques for the representation and analysis of mobility-related data are needed. Together with application experts (behavioural ecologists), we investigate how two novel data management approaches may help. We focus on a case study concerning the anal-ysis of fauna behaviour, in particular crested porcupines, which represents a typical example of mobile object monitoring. The first technique we experiment with is a recently developed conceptual spatio-temporal data modelling approach, MADS. This is used to model the schema of the database suited to our case study. Relying on this first outcome a subset of the problem is represented in the logical language MuTACLP. This allows us to formalise and solve the queries which enable the behavioural ecologists to derive crested porcupines behaviour from the raw data on animal movements. Finally, we investigate the support from a commercial Geographical Information System (GIS) for the analysis of spatio-temporal data. We present a way to integrate MuTACLP and a GIS, combining the advantages of GIS technology and the expressive power of MuTACLP.