The problem
Many spatial datasets — disease counts by county, species counts by site, incidents by region — are not homogeneous. Different parts of the map can follow genuinely different processes, sometimes with abrupt boundaries between them. Standard spatial models tend to assume smoothness everywhere, or fix the number of regions in advance, which can blur real boundaries or impose structure that isn’t there.
This project treats the map as a graph and asks a sharper question: which connected groups of areas behave as if they share one data-generating process, and where do those regions actually break?
The approach
A partition of the spatial graph is represented as a spanning forest — a set of trees whose connected components define the regions. Cutting an edge splits a region; adding one merges two. This gives a clean, fully connected representation of partitions that is easy to move around in.
Principled priors via the Matrix Tree Theorem
The Matrix Tree Theorem provides a tractable way to count and weight spanning trees and forests. The model uses it to place a principled prior over partitions — controlling how many regions are favored and how they are shaped — without ever enumerating the astronomically large space of possibilities.
Exploring partitions with split–merge–refinement
Posterior inference uses local MCMC moves — splitting a region, merging two adjacent regions, and refining boundaries — so the sampler can explore very different partitions while keeping every region connected by construction. Count likelihoods such as Poisson or negative binomial model the data within each region.
Representing partitions as spanning forests keeps regions connected by design and turns an intractable combinatorial search into a sequence of simple, reversible moves.
Why it matters
The result is a method that recovers connected regions with distinct behavior and quantifies uncertainty about where the boundaries lie — rather than returning a single hard map. That is valuable anywhere abrupt spatial structure carries meaning: epidemiology, ecology, public-health surveillance, and the social sciences.