第5回思考院セミナー / The 5th Seminar of the School of Statistical Thinking

Consider a discrete random field parameterized on the multi-dimensional integer lattice. For a fixed threshold u, the excursion set above u decomposes into connected components, or clusters. The size of a cluster is defined as the number of lattice points it contains, which is random. In this talk, we study the probability distribution of these cluster sizes. For stationary random fields, we derive an exact expression for the cluster size distribution. To accommodate nonstationary settings, we further develop a peak-based cluster size distribution, which characterizes the distribution of cluster sizes conditional on the presence of a local maximum above u. The framework applies broadly to Gaussian and non-Gaussian fields, relying only on their joint dependence structure. Our results provide a theoretical foundation for quantifying signal extent in discretely sampled spatiotemporal data, with applications to medical imaging, geoscience, environmental monitoring, and other scientific areas where thresholded random fields naturally arise (the preprint will be available soon).