State estimation for random closed sets

State estimation entails the estimation of an unobserved random closed set from (partial) observation of an associated random set. Examples include edge effect correction, cluster detection, filtering and prediction. We focus on inference for random sets based on points sampled on its boundary. Such data are subject to mis-alignment and noise. First, we ignore mis-alignment and discuss maximum likelihood estimation of the model and noise parameters in the Fourier domain. We estimate the unknown curve by back-transformation and derive the expectation of the integrated squared error. Then, we model mis-alignment by means of a shifted parametric diffeomorphism and minimise a suitable objective function simultaneously over the unknown curve and the mis-alignment parameters

Nearest-Neighbour Markov Point Processes on Graph with Euclidean Edges

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles

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

On the moments of Cox rate-and-state models

Rate-and-state models are widely used physical models for the relation betweenchanges in pore pressure due to fluid injection or gas extraction and the induced seismic hazard in a field. We consider the modification where the pore pressure measurements are affected by noise and provide explicit expressions for the first and second moments of the state variable. We show that when the pressure increases, there is positive correlation. In the case of decreasing pressure, both positive and negative correlation is possible. Using the delta method, approximate first and second moments of the rate variable are derived and compared to empirical moments