Clustering methods based on variational analysis in the space of measures

We formulate clustering as a minimisation problem in the space of measures by modelling the cluster centres as a Poisson process with unknown intensity function.We derive a Ward-type clustering criterion which, under the Poisson assumption, can easily be evaluated explicitly in terms of the intensity function. We show that asymptotically, i.e. for increasing total intensity, the optimal intensity function is proportional to a dimension-dependent power of the density of the observations. For fixed finite total intensity, no explicit solution seems available. However, the Ward-type criterion to be minimised is convex in the intensity function, so that the steepest descent method of Molchanov and Zuyev (2001) can be used to approximate the global minimum. It turns out that the gradient is similar in form to the functional to be optimised. If we discretise over a grid, the steepest descent algorithm at each iteration step increases the current intensity function at those points where the gradient is minimal at the expense of regions with a large gradient value. The algorithm is applied to a toy one-dimensional example, a simulation from a popular spatial cluster model and a real-life dataset from Strauss (1975) concerning the positions of redwood seedlings. Finally, we discuss the relative merits of our approach compared to classical hierarchical and partition clustering techniques as well as to modern model based clustering methods using Markov point processes and mixture distributions

A limit theorem for solutions of inequalities

Let $H(p)$ be the set {xin X:; h(x)leq p, where $h$ is a real-valued lower semicontinuous function on a locally compact second countable metric space $X$. A limit theorem is proved for the empirical counterpart of $H(p)$ obtained by replacing of $h$ with its estimator

Statistical models of random polyhedra

The paper discusses problems which appear in statistics of random polyhedra. Several parametric models of random polyhedra are considered. Particular attention is paid to the model of convex-stable compact random sets, which appear as weak limits of normalized convex hulls of systems of random points. It depends on two parameters only, but nevertheless provides sufficient flexibility of shape, and allows computations of several motion-invariant characteristics. The application of the model is demonstrated for real samples of particles

Set-valued means of random particles

Planar images of powder particles or sand grains can be interpreted as ``figures'', i.e., equivalence classes of directly congruent compact sets. The paper introduces a concept of set-valued means and real-valued variances for samples of such figures. In obtaining these results, the images are registered to have similar locations and orientations. The method is applied to find a mean figure of a sample of polygonal particles

Central limit theorem for a class of random measures associated with germ-grain models

In this paper we derive a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most $q$. By using this rank condition we establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system. Our proofs also comes with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

Morphology on convolution lattices with applications to the slope transformand random set theory

This paper develops an abstract theory for mathematical morphology on complete lattices. The approach is based upon the idea that objects are only known through information provided by a given collection of measurements (called evaluations in this paper). This abstract approach leads in a natural way to the concept of convolution lattice (where `convolution' has to be understood in the sense of an abstract Minkowski addition), the morphological slope transform, and the notion of `random lattice element'