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Extrapolation is defined as making predictions beyond the range of the data used to estimate a statistical model. In ecological studies, it is not always obvious when and where extrapolation occurs because of the multivariate nature of the data. Previous work on identifying extrapolation has focused on univariate response data, but these methods are not directly applicable to multivariate response data, which are more and more common in ecological investigations. In this paper, we extend previous work that identified extrapolation by applying the predictive variance from the univariate setting to the multivariate case. We illustrate our approach through an analysis of jointly modeled lake nutrients and indicators of algal biomass and water clarity in over 7000 inland lakes from across the Northeast and Mid-west US. In addition, we illustrate novel exploratory approaches for identifying regions of covariate space where extrapolation is more likely to occur using classification and regression trees., Comment: 28 pages, 2 supplementary files, 6 main figures, 2 supplementary figures, 2 supplementary tables

Regionalization is the task of dividing up a landscape into homogeneous patches with similar properties. Although this task has a wide range of applications, it has two notable challenges. First, it is assumed that the resulting regions are both homogeneous and spatially contiguous. Second, it is well-recognized that landscapes are hierarchical such that fine-scale regions are nested wholly within broader-scale regions. To address these two challenges, first, we develop a spatially constrained spectral clustering framework for region delineation that incorporates the tradeoff between region homogeneity and spatial contiguity. The framework uses a flexible, truncated exponential kernel to represent the spatial contiguity constraints, which is integrated with the landscape feature similarity matrix for region delineation. To address the second challenge, we extend the framework to create fine-scale regions that are nested within broader-scaled regions using a greedy, recursive bisection approach. We present a case study of a terrestrial ecology data set in the United States that compares the proposed framework with several baseline methods for regionalization. Experimental results suggest that the proposed framework for regionalization outperforms the baseline methods, especially in terms of balancing region contiguity and homogeneity, as well as creating regions of more similar size, which is often a desired trait of regions.