Arrival 09:45
10:00 - 12:30
10:00 C.A.N. Biscio
10:30 M. Otto
11:00 Discussions
11:30 - 12:30 Lunch
12:30 - 17:00
12:30 M. Kiderlen
13:00 R. Waagepetersen
13:30 Coffee break
14:00 A. M. Svane
14:30 C. Hirsch
15:00 Discussions
Christophe Biscio
Statistical learning for general point processes and applications to intensity and density estimation
Point processes generalise iid random samples by allowing a random sample size and/or the sample points to be dependent. In this talk, we present the first statistical learning framework for general point processes and show how to use it for intensity, and thereby density, estimation under these non-iid circumstances.
Our new approach is based on a subtle combination of two new concepts in point process theory: prediction errors and cross-validation. The general idea is to split a point process in two, through thinning, and estimate parameters by predicting one part using the other. This allows us to introduce a variety of loss functions not only suitable for standard spatial statistical problems but for general estimation settings, without imposing the iid assumptions.
We will introduce our framework for a general statistical audience and illustrate how our methods can be used for intensity and density estimation. In particular, we will show numerically that it substantially outperforms state of the art in bandwidth selection for kernel intensity estimators. If time permits, we will also indicate how our new methodology could be applied in other point process settings.
Christian Hirsch
Goodness-of-fit tests for spatial tessellations based on the persistence diagram
In applications in materials science, it is common to work with tessellation data where the cell centers are not scattered entirely at random but are subject to repulsive interactions. Therefore, Gibbs-Voronoi and Gibbs-Laguerre tessellations are important building blocks when constructing stochastic geometry models. Moreover, recently the persistence diagram has become a popular tool to detect subtle topological features in data. Based on the framework in (Schreiber & Yukich, 2013), I will present a functional CLT for the persistence diagram on Gibbsian tessellations, which can form the basis for goodness-of-fit tests. I will also elaborate on how persistence vineyards can be used to design rigorous statistical hypothesis tests for 3D microstructure models based on data from 2D slices.
This talk is based on joint work with A. Cipriani, J. Krebs, C. Redenbach, and M. Vittorietti.
Markus Kiderlen
The Radial Spanning Tree of a Poisson Point Process.
Joint with Katerina Konasova.
We consider properties and generalizations of the radial spanning tree (RST) constructed on a stationary Poisson point process. This random graph model was introduced in [1]. Motivated by applications in the theory of wireless networks, we consider a hierarchical RST based on a marked Poisson point process with independent two-color marks. We first discuss local properties such as the expected edge length or the expected degree of a vertex of the RST in the colored setting.
We then turn to the deeper (and more involved) problem to quantify properties of an ancestral path from a given point to the origin. We discuss and revise the strategies using Markov chain arguments and establishing a stationary regenerative structure outlined in [1] to obtain laws of large numbers for spatial averages in the monochrome case and point out an interesting research question.
[1]: Baccelli F, Bordenave C (2007) The Radial Spanning Tree of a Poisson Point Process, Annals of Applied Probability 17, 305--359.
Moritz Otto
Poisson process approximation of functionals of determinantal processes
We consider functionals of determinantal point processes with a fast decay of correlations. Under a stabilization and a monotonicity assumption, we show a new Poisson process approximation result for the functional. Our proof is based on Stein’s method and uses a coupling of the original process and its Palm version. We present an application of our findings in the theory of random geometric graphs.
Anne Marie Svane
A K-function for inhomogeneous random measures with geometric features
We introduce a generalization of the classical Ripley’s K-function for assessing second-order properties of inhomogeneous random measures generated by marked point processes. The marks can e.g. be geometric objects like fibers or sets of positive volume. The K-function allows geometric features of the marks, such as tangent directions of fibers, to be taken into account.
The K-function requires an estimate of the inhomogeneous density function of the random measure. For this, we introduce parametric estimates for the density based on parametric models that represent large scale features of the inhomogeneous random measure. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.
This is joint work with Rasmus Waagepetersen.
Rasmus Waagepetersen
Second-order semiparametric inference for multivariate log Gaussian Cox processes
This talk presents an approach to inferring the second-order properties of a multivariate log Gaussian Cox process (LGCP) with a complex intensity function. We assume a semi-parametric model for the multivariate intensity function containing an unspecified complex factor common to all types of points. Given this model, we construct a second-order conditional composite likelihood to infer the pair correlation and cross pair correlation functions of the LGCP. We also introduce a cross-validation method for model selection and an algorithm for regularized inference that can be used to obtain sparse models for cross pair correlation functions. The methodology is applied to simulated data as well as data examples from microscopy and criminology.