von Westenholz, Mandala

Poisson Point Processes in application of data analysis

Poisson point processes (PPPs) are used to model and analyze random event distributions in time or space, under the assumption of independence. In data analysis, PPPs support tasks such as intensity estimation, clustering detection, and spatial simulation—critical in fields like epidemiology, logistics, and network analysis.

When analyzing the topological structure of point clouds arising from PPPs, simplicial complexes offer a powerful tool. They allow the construction of higher-order relationships between points—going beyond pairwise distances—to capture the shape or connectivity of the data. Techniques such as Vietoris–Rips or Čech complexes can be built on PPP-generated data to extract topological features like connected components, loops, or voids.

Together, Poisson point processes and simplicial complexes enable a deeper understanding of both the statistical and topological structure of complex datasets.

 Project team: Mandala von Westenholz (Ph.D. Student), Prof. Dr. Tim Römer (UOS)

Publications

von Westenholz, M. & Atzmueller, M. & Römer, T.: Simplicial complexes in network intrusion profiling. arXiv: 2408.09788 (2024)

von Westenholz, M. & Patzelt, M. & Römer, T. & Atzmueller, M.: Generalizing Local Pattern Mining on Attributed Graphs Using Simplicial Complex Abstraction. Complex Netzworks & Their Applications XIII. (pp. 298 -311) (2025)

Reitzner, M. & Römer, T. & von Westenholz, M.: Covariance matrices of length power functionals of random geometric graphs – an asymptotic analysis. Linear Algebra and Its Applications, 691 (2024).