Factor Point Processes
Jointly with S. Mellick, A. Khezeli solved the problem of the existence of factor point processes in the paper
- A. Khezeli and S. Mellick. On the existence of balancing allocations and factor point processes. arXiv preprint arXiv:2303.05137. 2023. https://doi.org/10.48550/arXiv.2303.05137
For more on the matter, see here.
Unsupervised Classification of Data Represented as Point Processes
Clustering methods are crucial for detecting and interpreting patterns in large datasets. In this work, S. Khaniha and F. Baccelli propose three hierarchical clustering algorithms, Clustroid Hierarchical Nearest Neighbor (CHN^2), Single Linkage Hierarchical Nearest Neighbor (SHN^2), and Hausdorff (Complete Linkage) Hierarchical Nearest Neighbor (H^2N^2), all of which handle countably infinite sets of points. These algorithms iteratively build clusters by linking nearest neighbors or nearest clusters, each using a distinct distance metric (clustroid, single linkage, or Hausdorff).
This work is focused on the homogeneous Poisson point process in Euclidean space, showing that each hierarchical clustering creates a phylogenetic forest with notable probabilistic properties, such as almost-sure finiteness of clusters at each level of the algorithm. The work also studies the limit structures formed as the number of clustering levels tend to infinity, proving that for SHN^2 on the Poisson point process, this limit is a subgraph of the Minimal Spanning Forest. Finally, we extend CHN^2 to certain stationary Cox point processes, showing that it preserves its finite-cluster properties and effectively detects Cox-driven aggregation.
Limit theory for asymptotically de-correlated dynamic spatial random models
In an ongoing work with D. Yogeshwaran and J. E. Yukich, B .Blaszczyszyn considers statistics of spatial random models evolving over a fixed time domain and which are asymptotically de-correlated over spatial domains. They establish the limit theory (LLNm CLT) of these statistics as the spatial domain increases up to R^d. The models involve three sources of randomness, namely the random collection of particle locations (sites), the initial states (covariates), and the particle system evolution. They adopt the language of particle systems, but the set-up is general and yieldsthe limit theory for statistics of other stochastic geometric structures such as interacting diffusions, geostatistical models, empirical random fields and Gibbsian models.
Palm Theory on Unimodular Spaces
A. Khezeli proposed a general approach for the construction of point processes on unimodular spaces in
- A. Khezeli. ‘Unimodular Random Measured Metric Spaces and Palm Theory on Them.’ arXiv preprint arXiv:2304.02863. 2023. https://doi.org/10.48550/arXiv.2304.02863
For more on the matter, see here.
Book Preprint
This book preprint Random Measures, Point Processes, and Stochastic Geometry by François Baccelli, Bartlomiej Blaszczyszyn, and Mohamed Kadhem Karray (Orange Labs) was first posted in 2020. A second and revised version was posted in 2024 and a third version is under preparation. This book is centered on the mathematical analysis of random structures embedded in the Euclidean space or more general topological spaces, with a main focus on random measures, point processes, and stochastic geometry. Such random structures have been known to play a key role in several branches of natural sciences (cosmology, ecology, cell biology) and engineering (material sciences, networks) for several decades. Their use is currently expanding to new fields like data sciences. The book was designed to help researchers finding a direct path from the basic definitions and properties of these mathematical objects to their use in new and concrete stochastic models.
The material is organized as follows. Random measures and point processes are presented first, whereas stochastic geometry is discussed at the end of the book. For point processes and random measures, parametric models are discussed before non-parametric ones. For the stochastic geometry part, the objects of interest are often considered as point processes in the space of random sets of the Euclidean space. We discuss both general processes such as, e.g., particle processes, and parametric ones like, e.g., Poisson Boolean models of Poisson hyperplane processes.
