# WP3 - Spatial and spatio-temporal point processes

Point processes are the fundamental building blocks of stochastic geometry models. An important example is particle models that can be represented as marked point processes.

Modelling and statistical analysis of point processes in Euclidean spaces is very well developed. However, there is need for transferring this methodology to **non-Euclidean** spaces such as **linear networks** and **directed graphs**.

### WP3 - Subprojects

**WP3.1:** Point processes on linear networks**WP3.2:** Point process models and inference for attributed directed graphs**WP3.3:** Determinantal point process modelling**WP3.4:** Sparse models for multivariate spatial point processes**WP3.5:** Inference from quadrat count data

## Research questions

In the second funding period of CSGB, we want to transfer statistical methods for spatial and spatio-temporal point processes to **non-Euclidean spaces**.

- The main problem in transferring the well-established theory of spatial point processes in Euclidean spaces to
**linear networks**is the geometry of the network itself. Recently, members of CSGB have succeeded in adapting Ripley’s K-function to the linear network setting. The network geometry is taken into account. Our aim is to adapt other summary statistics as well. We also want to develop models on networks that exhibit second-order pseudo-stationarity. - We plan to develop a theory for prior modelling of attributed
**directed graphs**. A flexible marked point process representation will be investigated, which is tractable for inference and simulation. A hierarchical construction for first vertices and secondly edges given vertices will be used.

In the second funding period of CSGB, we also take up the challenge of modelling **multivariate spatial point processes**. The problem is here the high dimensionality of the parameter space. Inference for quadrat count data with explanatory variables will be studied. The research on determinantal point processes will be continued.

### Selected references

Coeurjolly, J.-F. and Møller, J. (2014): Variational approach for spatial point process intensity estimation. *Bernoulli* **20**, 1097-1125.

Jalilian, A., Guan, Y. and Waagepetersen, R. (2013): Decomposition of variance for spatial Cox processes. *Scand. J. Stat.* **40**, 119-137.

Lavancier, F., Møller, J. and Rubak, E. (2015): Determinantal point process models and statistical inference. *J. Roy. Stat. Soc. B* **77**, 853-877.

Møller, J. and Rasmussen, J.G. (2012): A sequential point process model and Bayesian inference for spatial point patterns with linear structures. *Scand. J. Stat.* **39**, 618-634.

Waagepetersen, R., Guan, Y., Jalilian, A. and Mateu, J. (2016): Analysis of multi-species point patterns using multivariate log Gaussian Cox processes. *J. Roy. Stat. Soc. C* **65**, 77-96.