The goal of this course is to discuss the mathematics involved in the analysis of spatial data.

Such data may come in various forms:

In classical geostatistics, some spatially varying variable is observed at a given number of fixed locations and one is interested in its value at locations where it was not observed. An example is the prediction of ore content in the soil based on measurements at some conveniently placed boreholes.

Alternatively, the data may be collected in aggregated forms as region counts or as a discrete image. Typical examples include tomographic scans, disease maps or yields in agricultural field trials. In this case, the objective is spatial smoothing rather than predicting.

Finally, if both the number of points and their locations are random and their arrangement is of prime importance, a spatial point process is called for. These arise for example in the study of earthquakes.

We will describe the mathematical foundations as well as useful models for each of the data forms mentioned above, discuss statistical inference and give pointers to open source software.

Part 1. Spatial interpolation

- Gaussian random fields

- Kriging

Part 2. Random fields on graphs

- Markov random fields

- CAR/SAR model

- Markov chain Monte Carlo

- Bayesian hierarchical modelling

Part 3. Spatio-temporal point processes

- Densities and moment measures

- Cox and cluster processes

- Gibbs processes

- Monte Carlo likelihood based inference

Part 4. Software

- R and its packages

**Prerequisites**

**Maturity in probability and statistics at the level expected of a master student in mathematics, engineering or statistics.**

- Docent: Marie-Colette van Lieshout