Closed‐Form Maximum Likelihood Estimates of Nearest Neighbor Spatial Dependence

Abstract

Models of n2 potential spatial dependencies among n observations spread irregularly over space seem unlikely to yield simple structure. However, the use of the nearest neighbor leads to a very parsimonious eigenstructure of the associated adjacency matrix which results in an extremely simple closed form for the log determinant. In turn, this leads to a closed-form solution for the maximum likelihood estimates of the spatially autoregressive and mired regressive spatially autoregressive models. With the closed-form solution, one can find the neighbors and compute maximum likelihood estimates for 100,000 observations in under one minute. The model has theoretical, pedagogical, diagnostic, modeling, and methodological applications. For example, the model could serve as a more enlightened null hypothesis for geographic data than spatial independence.The task of modeling the n2 potential dependencies among n spatially scattered observations has generated a substantial amount of intellectual activity. Naturally, many possible ways exist to specify the spatial interactions. Virtually all distance-based weighting methods such as variograms or spatial weight matrices assign some positive influence to the nearest observation, the nearest neighbor. In this sense, the nearest neighbor is common to most methods.Given the ubiquitous nature of nearest neighbors, it seems worthwhile to examine a limiting model of spatial dependence, a model where observations depend only upon their nearest neighbor. Such special cases often help illustrate the more general case.The nearest neighbor graph exhibits a number of important regularities. These regularities ensure the associated adjacency matrix (spatial weight matrix in this case) has a remarkable structure-the eigenvalues are all -l,O, or 1 and the number of 1, -1 pairs equals the number of pairs of symmetric elements in the adjacency matrix. This leads to a simple closed form for log11 -aDI, where c1 represents a spatial autoregressive parameter and D represents the n by n

Transdisciplinary Foundations of Geospatial Data Science

“…Exact solutions, even with various algorithmic refinements [46,56] and parallelization strategies [51,57], may still suffer from high computational complexity and high memory cost (e.g., due to eigenvalues computation). Approximate solutions [58] aim to reduce complexity by providing reasonably good solutions (e.g., within a certain bound of the exact solution) instead of exact solutions.…”

Detection of Gauss–Markov Random Fields With Nearest-Neighbor Dependency

“…Proof : Apply LLN to (24). See appendix C. In the theorem above, the law of large numbers yields the same limit 7 under the Poisson or the binomial process. Thus, we provide a single expression for the error exponent under both the processes.…”

Detection of Gauss-Markov Random Fields with Nearest-Neighbor Dependency

“…In [7], this formulation is considered with (directed) nearest-neighbor interaction and a closed-form ML estimator of the AR spatial parameter is characterized. We do not consider this formulation in this paper.…”

“…In this paper, we consider the nearest-neighbor graph (NNG), which is the simplest proximity graph. The nearest-neighbor relation has been used in several areas of applied science, including the social sciences, geography and ecology, where proximity data is often important [6], [7].…”