Topics for exam NMTP438

1. random fields on lattice, Markov random field, Hammersley-Clifford theorem

2. examples of Markov random fields

3. Gaussian models on lattice, Gaussian Markov random field

4. spatial autocorrelation

5. random fields on continuous domain, stationarity, mean square properties

6. variogram and autocovariance function, properties, examples, spectral decomposition

7. space of locally finite measures, σ-algebra, weak and vague convergence

8. random measure, finite dimensional projections uniquely determine distribution, existence theorem

9. simple point process as a random locally finite set, void probabilities

10. binomial, Poisson and Cox point processes, relations among them

11. moment measures, Campbell theorem, Laplace transform of random measure

12. desintegration theorem, Campbell measure, Palm distribution of random measure

13. reduced Palm distribution of a point process, Campbell-Mecke theorem, Slivnyak theorem

14. stationary spatial point processes, intensity, Palm distribution, Campbell-Mecke theorem

15. characteristics of stationary spatial point processes

16. cluster point processes, thinned point processes

17. hard-core point processes

18. finite point processes with density, Papangelou conditional intensity, pairwise interaction point processes

19. Markov point processes, Hammersley-Clifford-Ripley-Kelly theorem

20. inhomogeneous spatial point processes

21. marked point processes, stationary mark distribution, two-point mark distribution

22. marking models, Poisson marked point process, independent marking, geostatistical marking

23. multivariate point processes, processes with quantitative marks, characteristics