# WP5 - Statistics for Stochastic geometry models

The aim is to develop new statistical inference procedures for stochastic geometry models. The main model examples are **spatial point processes** and **random fields**.

Typically, it is not possible to use classical statistical methods, based e.g. on likelihood functions, for the analysis of stochastic geometry models.

### WP5 - Subprojects

**WP5.1:** Asymptotics for excursion sets – extremal properties**WP5.2:** Monte Carlo envelope tests for random closed sets**WP5.3:** Estimation of sample spacing for stochastic processes

## Research questions

In the second funding period of CSGB, we have taken up two new projects concerning (a) excursion sets of random fields and (b) Monte Carlo envelope tests.

- Recently, the extremal behaviour of a random field defined as a kernel smoothing of a Lévy basis has been studied. It is planned to extend these studies to
**asymptotic results**for excursion sets. We expect to obtain results for the asymptotic size of an excursion set, and will furthermore study the asymptotic behaviour of geometric characteristics such as the number of critical points and the Euler characteristic. - The validity of stochastic geometry models may be assessed by a statistical test of
**goodness of fit**. In the point process community, a simulation-based test has recently been introduced. The disadvantage of this test is that it does not return a single*p*-value, but an interval. We want to explore the possibility for alternative tests. One realistic possibility is a test that treats the involved functions as high-dimensional vectors of function values taken on discrete arguments. The test could be based on a fine ordering of the vectors, using element-wise ranks.

In the second funding period, we also study statistical procedures for estimating **sample spacing** in stationary, isotropic random fields. This project is motivated by a problem studied in the first funding period of CSGB, viz. the estimation of the thickness of ultra thin sections, obtained by transmission electron microscopy.

### Selected references

- Mrkvička, T., Myllymäki, M. & Hahn, U. (2017): Multiple Monte Carlo testing, with applications in spatial point processes.
*Stat. Comput.***27**, 1239-1255. - Myllymäki, M., Mrkvička, T., Grabarnik, P., Seijo, H. & Hahn, U. (2017): Global envelope tests for spatial processes.
*J. Roy. Stat. Soc. B***79**, 381-404. - Rønn-Nielsen, A. & Jensen, E.B.V. (2016): Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure.
*J. Appl. Prob.***53**, 244-261. - Rønn-Nielsen, A. & Jensen, E.B.V. (2017): Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure.
*J. Appl. Prob.***54**, 833-851. - Rønn-Nielsen, A., Sporring, J. & Jensen, E.B.V. (2017): Estimation of sample spacing in stochastic processes.
*Image Anal. Stereol.***36**, 43-49.