Seminar stochastic geometry and its applications
Dr. Kirsten Schorning
Dr. Vitalii Makogin
Date and Place
It will be a one-day block-seminar. It takes place on 5th of July 2017, at 14-18 in room N24-254.
The level of difficulty in this seminar is varying between the different topics. The audience is at least supposed to be familiar with basic probability. There are a few talks that require the lecture Stochastics II.
Bachelor and Master Students in any mathematical programme of studies. The "(B)" in the list of talks, means that this talk is supposed to be given by a Bachelor's Student.
Poisson point processes and recent applications: Poisson point processes (PPPs) are very useful theoretical models for diverse applications involving the geometrical distribution of random occurrences of points in a multidimensional space. Both the number of points and their locations are modeled as random variables.
In this seminar, we consider PPPs and their most important properties. Several useful operations that map PPPs into other PPPs will be discussed, these include superposition, thinning, nonlinear transformation, and stochastic transformation. PPPs are characterized by their intensity function and we consider estimation of it from data using R packages.
The number of new applications of PPPs models growths constantly. In some applications, PPPs are extremely well matched to the physics and the engineering system. Tomography, especially positron emission tomography, multitarget tracking are excellent examples of this kind.
To register for the seminar, please write an E-Mail to Vitalii Makogin until 2nd May 2017. In the e-mail please give your name, matriculation number, your programme of studies, the title of supposed talk, and subjects you have taken in the area of Probability or Statistics.
Criteria to pass the seminar
Those who give a (good) talk and attend the seminar will pass the seminar.
List of Talks
- Construction of Poisson point processes: event space, definition and basic properties (conditional uniformity property, stationarity, isotropy), the Poisson limit theorem, general Point processes, realizations. (B)
- Characterization of PPPs: superposition, independent (Bernoulli) thinning, independent scattering, expectations of PPPs, Campbell’s theorem, Slivnyak’s theorem, probability generating functional, simulations ("spatstat" package in R: rpoispp). (B)
- Intensity Estimation of PPPs: likelihood function, maximum likelihood algorithms, expectation-maximization algorithm for affine Gaussian sum with PPP sample data, superposed intensities with histogram data, regularization, model fitting in "spatstat" (performed mainly by the function "ppm"). (B)
- PPP methods for tomographic imaging I: Positron emission tomography (PET), time-of-flight data, Shepp-Vardi algorithm for PPP sample data, image reconstruction, histogram data.
- PPP methods for tomographic imaging II: Single-photon computed emission tomography (SPECT), image reconstruction, transmission tomography, Lange-Carson algorithm.
- Approximate tracking methods based on PPP models: Intensity filters, PPP model Interpretation, predicted target and measurement processes, the final filter, particle methods, mean shift algorithm, covariance matrices, regularization,
- PPPs in extreme value theory: Laplace functional, weak convergence of point processes, point process of exceedances, point process description of records.
- Streit, Roy L. Poisson point processes: imaging, tracking, and sensing. Springer Science & Business Media, 2010.
- Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. Modelling extremal events: for insurance and finance. Vol. 33. Springer Science & Business Media, 2013.