🤖 AI Summary
This work proposes a Bayesian nonparametric clustering method for replicated marked Poisson point process data, jointly inferring the latent cluster structure, the number of clusters, and the intensity surface associated with continuous marks. Built upon a Dirichlet process mixture model, the approach employs a squared link function to model the intensity surface and leverages variational Bayesian inference for efficient learning. To address sign ambiguity and nodal line issues inherent in the squared link, the method introduces a constrained Laplace approximation that reformulates the non-conjugate basis coefficient updates as a constrained optimization problem, thereby providing theoretical guarantees for mode-finding. Experimental results demonstrate that the proposed model achieves superior performance in clustering accuracy, intensity estimation, and computational efficiency on both synthetic and real-world datasets.
📝 Abstract
Marked point process data arise when events occur in a space with event-level marks. We study clustering of replicated marked Poisson point processes and introduce Dirichlet process mixtures of marked Poisson point processes, a Bayesian nonparametric model that jointly infers latent cluster structure, the number of clusters, and continuous mark-specific intensity surfaces. We use a squared link intensity representation to obtain tractable continuous domain likelihood terms without gridding or thinning. For posterior inference, we develop an efficient variational Bayes algorithm with a constrained Laplace approximation for the nonconjugate basis-coefficient block. The resulting coefficient update is formulated as a constrained optimization problem, which avoids the sign ambiguity and nodal-line issue of squared-link models. We further establish theoretical guarantees for mode finding optimization. We demonstrate the performance of the proposed model and algorithm through synthetic experiments and real-data analysis.