In the context of comparing two categorical maps, addressing the challenges posed by the dependence on marginal frequencies and spatial configuration is a formidable task. Map comparison typically relies on a numerical similarity index S, such as the spatial fuzzy kappa index, which has a probability distribution that varies according to the marginal frequencies of the classes and their spatial configuration in the maps. This variability makes it difficult to mathematically derive and assess the uncertainty associated with any empirical index. In this paper, we introduce two novel methods: the spatially thinned bootstrap and the random toroidal shift. These methods provide a robust framework for comparing two maps using an arbitrary similarity index while accounting for these marginal noise factors. Our approach is characterized by its generality and abstraction, enabling application across different indices of map similarity. To illustrate the effectiveness of our methods, we conduct an extensive study employing the spatial fuzzy kappa index, revealing valuable insights into the behavior and underlying distribution of similarity indices.

Multi-scale geographically weighted regression (MGWR) is among the most popular methods to analyze non-stationary spatial relationships. However, the current model calibration algorithm is computationally intensive: its runtime has a cubic growth with the sample size, while its memory use grows quadratically. We propose calibrating MGWR with gradient-based optimization. This is obtained by analytically deriving the gradient vector and the Hessian matrix of the corrected Akaike information criterion (AICc) and wrapping them with a trust-region optimization algorithm. We evaluate the model quality empirically. Our method converges to the same coefficients and produces the same inference as the current method but it has a substantial computational gain when the sample size is large. It reduces the runtime to quadratic convergence and makes the memory use linear with respect to sample size. Our new algorithm outperforms the existing alternatives and makes MGWR feasible for large spatial datasets.

Regionalization is the task of partitioning a set of contiguous areas into spatial clusters or regions. The theoretical and empirical literature focusing on regionalization is extensive, yet few quantitative comparisons have been conducted. We present a simulation study and explore the quality of frequently used and state-of-the-art regionalization algorithms, namely AZP, AZP-SA, AZPTabu, ARISEL, REDCAP, and SKATER, where the number of regions is an exogenous variable. The simulated benchmark data set consists of model realizations that represent various complexities in spatial data. Model families are defined with respect to regions’ shapes, value-mixing between regions, and the number of underlying spatial clusters. We evaluate the performance of different regionalization methods for realizations families using internal and external measures of regionalization quality. A large number of regionalization quality metrics expose a detailed profile of the analyzed methods’ strengths and weaknesses. We investigate the computational efficiency of every method as a function of the number of spatial units studied. We summarize results for different region families and discuss circumstances that make a certain method more desirable. We illustrate different regionalization algorithms’ implications on defining ecological regions for the conterminous US and compare them against expert-defined ecoregions.