Point processes

Lecturer

Dr. Orkun Furat

Teaching assistant

Mr. Duc Nguyen

Time and place

This course is organized and carried out via the Moodle. Here is the link to the Moodle page of the course.

Lecture
Wednesday, 12-14, N24 - 226

Exercise
Friday, 08-10, (biweekly), N24 - 226

Type

The lecture will be held in English.

2 hours lecture + 1 hours exercises

Credits: 4

Prerequisites

Elementary course in probability. Prior knowledge in measure theory is recommended.

Intended Audience

Master students in Mathematics, Mathematical Economics, Mathematical Biometrics and Mathematical Finance

Content

The focus of this course is the stochastic modeling, statistical analysis and simulation of point patterns in the d-dimensional Euclidean space. The presented techniques can be applied for a wide range of spatial data sets. In collaboration with partners from other scientific disciplines and industry, the statistical analysis of point patterns is applied in our institute in various fields such as battery, fuel and solar cell research, solids process engineering for porous and polycrystalline materials, biotechnology, and weather models.

The lecture gives an introduction to the theory of random point processes. Among other things, properties such as stationarity and isotropy are discussed. Furthermore, basic classes of point process models are introduced. Besides the famous Poisson process, the model of complete spatial randomness, we discuss Cox and models of point processes, where clustering between the points is reflected in the model.

This course provides the mathematical basis for point processes as well as techniques for their implementation in applications. In this way approaches for the statistical analysis of point patterns are provided, which are—due to the increasing availability of spatial data—of increasing practical relevance.

Exercise sheets

The exercise sheets and scores will be published on Moodle.

References

V. Benes and J. Rataj. Stochastic Geometry: Selected Topics. Kluwer, Dordrecht, 2004.
S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and its Applications. J. Wiley & Sons, Chichester, 3rd edition, 2013.
N. Cressie. Statistics for Spatial Data. John Wiley & Sons, Chichester, 2015.
D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods. Springer, New York, 2nd edition, 2005.
D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure. Springer, New York, 2nd edition, 2008.
J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan. Statistical Analysis and Modelling of Spatial Point Patterns. J. Wiley & Sons, Chichester, 2008.
J. F. C. Kingman. Poisson Processes. Oxford University Press, Oxford, 1992.
G. Last and M. Penrose. Lectures on the Poisson Process. Cambridge University Press, Berlin, 2018.
J. Møller and R. P. Waagepetersen. Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, 2004.
V. Schmidt (Ed.). Stochastic Geometry, Spatial Statistics and Random Fields Models and Algorithms. Springer, Cham, 2014.
E. Spodarev (Ed.). Stochastic Geometry, Spatial Statistics and Random Fields Asymptotic Methods. Springer, Cham, 2013.