Point Processes


Prof. Dr. Volker Schmidt

Teaching Assistant

Orkun Furat

Time and Place

Friday, 8-10, He18, E120

Exercise Session
Monday, 16-18 (biweekly), He18, E20


2 hours lecture + 1 hours exercises

Credits: 4


Probability and Calculus

Intended Audience

The lecture is for master students. Since the lecture is based directly on the basic course WR, bachelor students are also welcome and can get additional credits for the master's programme. The lecture deals with a branch of stochastics that goes beyond the mandatory courses, which makes a multitude of practical applications possible. The contents of the course are designed in such a way that it is easy to establish links to current research projects, which are carried out at our institute in cooperation with partners from science and industry. Attending the lecture is therefore a good preparation for the participation in these projects, be it in the form of internships or a thesis.


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, telecommunications networks and georisk research.

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, such as Poisson, Cox, and Gibbs processes are introduced. On this basis, a large number of structural properties of point patterns, such as spatial inhomogeneities or attraction or repulsion effects of the points, can be characterized.

Another aspect of the lecture is the study of the so-called Palm distribution of a random point process, with the help of which the probability of events can be calculated "from the point of view of a randomly picked point". For example, such an event could be that there is no other point in a sphere of radius 1 around a randomly picked point.

In addition, random point patterns are discussed which describe points in the Euclidean space with additional (random) marks. For example, these marks can reflect the size, shape and orientation of a particle at the respective particle's center of gravity. By means of marked point process models one can gain insights into the spatial correlation structure of the marks and their temporal changes. 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 spatial data are provided, which are of increasing practical relevance, in particular due to the rapid developments in the field of imaging techniques.


At the end of the semester there will be an oral exam.

Prerequisite: 50% of all credits from the exercise sheet.

Lecture notes

 lecture notes

Lecture notes on the generation of pseudo-random numbers can be found here.

Exercise sheets

 The exercise sheets and scores will be published on Moodle.


[1] Last, G., Penrose, M.
Lectures on the Poisson Process, Vol. 7. Cambridge University Press, Cambridge 2017.

[2] Baddeley, A., Bárány, I., Schneider, R., Weil, W. (Hrsg.)
Stochastic Geometry. Lecture Notes in Mathematics, Vol. 1892, Springer, Berlin 2007

[3] Benes, V., Rataj, J.
Stochastic Geometry. Kluwer, Boston 2004

[4] Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.
Stochastic Geometry and its Applications. J. Wiley & Sons, Chichester 2013

[5] Daley, D.J., Vere-Jones, D.
An Introduction to the Theory of Point Processes, Vol. I. Springer, New York 2005

[6] Daley, D.J., Vere-Jones, D.
An Introduction to the Theory of Point Processes, Vol. II. Springer, New York 2008

[7] Diggle, P.J.
Statistical Analysis of Spatial Point Patterns. Arnold, London 2003

[8] Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P. (Hrsg.) 
Handbook of Spatial Statistics. Chapman & Hall / CRC, Boca Raton 2010

[9] Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.
Statistical Analysis and Modelling of Spatial Point Patterns. J. Wiley & Sons, Chichester 2008

[10] Kallenberg, O.
Foundations of Modern Probability. Springer, New York 2001

[11] Kendall, W. S. and Molchanov, I. (Hrsg.)
New Perspectives in Stochastic Geometry. Springer, Berlin 2010

[12] Kingman, J.F.C.
Poisson Processes. Oxford University Press, Oxford 1993

[13] Matheron, G.
Random Sets and Integral Geometry. J. Wiley & Sons, New York 1975

[14] Møller, J., Waagepetersen, R.P.
Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall / CRC, Boca Raton 2004

[15] Ohser, J. and Schladitz, K.
3D Images of Materials Structures - Processing and Analysis. Wiley-VCH, Weinheim 2009

[16] Ripley, B.D.
Spatial Statistics. J. Wiley & Sons, New York 1981

[17] Schmidt, V. (Hrsg.)
Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol. 2120, Springer, Cham 2015.                                               

[18] Schneider, R., Weil, W.
Stochastic and Integral Geometry. Springer, Heidelberg 2008

[19] Spodarev, E. (Hrsg.)
Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol. 2068, Springer, Berlin 2013.

[20] Stoyan, D., Stoyan, H.
Fractals, Random Shapes and Point Fields. J. Wiley & Sons, Chichester 1994





Prof. Dr. Volker Schmidt
Office hours: on appointment
Phone: +49 (0)731/50-23532
E-Mail: volker.schmidt(at)uni-ulm.de


Exercise sessions

Orkun Furat
Office hours: on appointment
Phone: +49 (0)731/50-23555
E-Mail: orkun.furat(at)uni-ulm.de


  • The second exercise session will be on Monday, May 6th.