An official website of the United States government
The existing quantitative geography literature contains a dearth of articles that span spatial autocorrelation (SA), a fundamental property of georeferenced data, and spatial optimization, a popular form of geographic analysis. The well-known location–allocation problem illustrates this state of affairs, although its empirical geographic distribution of demand virtually always exhibits positive SA. This latent redundant attribute information alludes to other tools that may well help to solve such spatial optimization problems in an improved, if not better than, heuristic way. Within a proof-of-concept perspective, this paper articulates connections between extensions of the renowned Majority Theorem of the minisum problem and especially the local indices of SA (LISA). The relationship articulation outlined here extends to the p = 2 setting linkages already established for the p = 1 spatial median problem. In addition, this paper presents the foundation for a novel extremely efficient p = 2 algorithm whose formulation demonstratively exploits spatial autocorrelation.
more »
« less
The Majority Theorem for the Single ( p = 1) Median Problem and Local Spatial Autocorrelation
Except for about a half dozen papers, virtually all (co)authored by Griffith, the existing literature lacks much content about the interface between spatial optimization, a popular form of geographic analysis, and spatial autocorrelation, a fundamental property of georeferenced data. The popularp‐median location‐allocation problem highlights this situation: the empirical geographic distribution of demand virtually always exhibits positive spatial autocorrelation. This property of geospatial data offers additional overlooked information for solving such spatial optimization problems when it actually relates to their solutions. With a proof‐of‐concept outlook, this paper articulates connections between the well‐known Majority Theorem of the 1‐median minisum problem and local indices of spatial autocorrelation; the LISA statistics appear to be the more useful of these later statistics because they better embrace negative spatial autocorrelation. The relationship articulation outlined here results in the positing of a new proposition labeled the egalitarian theorem.
more »
« less
- Award ID(s):
- 1951344
- PAR ID:
- 10363767
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Geographical Analysis
- Volume:
- 55
- Issue:
- 1
- ISSN:
- 0016-7363
- Page Range / eLocation ID:
- p. 107-124
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Both historically and in terms of practiced academic organization, the anticipation should be that a flourishing synergistic interface exists between statistics and operations research in general, and between spatial statistics/econometrics and spatial optimization in particular. Unfortunately, for the most part, this expectation is false. The purpose of this paper is to address this existential missing link by focusing on the beneficial contributions of spatial statistics to spatial optimization, via spatial autocorrelation (i.e., dis/similar attribute values tend to cluster together on a map), in order to encourage considerably more future collaboration and interaction between contributors to their two parent bodies of knowledge. The key basic statistical concept in this pursuit is the median in its bivariate form, with special reference to the global and to sets of regional spatial medians. One-dimensional examples illustrate situations that the narrative then extends to two-dimensional illustrations, which, in turn, connects these treatments to the spatial statistics centrography theme. Because of computational time constraints (reported results include some for timing experiments), the summarized analysis restricts attention to problems involving one global and two or three regional spatial medians. The fundamental and foundational spatial, statistical, conceptual tool employed here is spatial autocorrelation: geographically informed sampling designs—which acknowledge a non-random mixture of geographic demand weight values that manifests itself as local, homogeneous, spatial clusters of these values—can help spatial optimization techniques determine the spatial optima, at least for location-allocation problems. A valuable discovery by this study is that existing but ignored spatial autocorrelation latent in georeferenced demand point weights undermines spatial optimization algorithms. All in all, this paper should help initiate a dissipation of the existing isolation between statistics and operations research, hopefully inspiring substantially more collaborative work by their professionals in the future.more » « less
-
Thep‐dispersion problem is a spatial optimization problem that aims to maximize the minimum separation distance among all assigned nodes. This problem is characterized by an innate spatial structure based on distance attributes. This research proposes a novel approach, named thedistance‐based spatially informed property(D‐SIP) method to reduce the problem size of thep‐dispersion instances, facilitating a more efficient solution while maintaining optimality in nearly all cases. The D‐SIP is derived from investigating the underlying spatial characteristics from the behaviors of thep‐dispersion problem in determining the optimal location of nodes. To define the D‐SIP, this research applies Ripley'sK‐function to the different types of point patterns, given that the optimal solutions of thep‐dispersion problem are strongly associated with the spatial proximity among points discovered by Ripley'sK‐function. The results demonstrate that the D‐SIP identifies collective dominances of optimal solutions, leading to buildingthe spatially informed p‐dispersion model. The simulation‐based experiments show that the proposed method significantly diminishes the size of problems, improves computational performance, and secures optimal solutions for 99.9% of instances (999 out of 1,000 instances) under diverse conditions.more » « less
-
Abstract AimPatterns of genetic diversity within species’ ranges can reveal important insights into effects of past climate on species’ biogeography and current population dynamics. While numerous biogeographic hypotheses have been proposed to explain patterns of genetic diversity within species’ ranges, formal comparisons and rigorous statistical tests of these hypotheses remain rare. Here, we compared seven hypotheses for their abilities to describe the geographic pattern of two metrics of genetic diversity in balsam poplar (Populus balsamifera), a northern North American tree species. LocationNorth America. TaxonBalsam poplar (Populus balsamiferaL.). MethodsWe compared seven hypotheses, representing effects of past climate and current range position, for their ability to describe the geographic pattern of expected heterozygosity and per cent polymorphic loci across 85 populations of balsam poplar. We tested each hypothesis using spatial and non‐spatial least‐squares regression to assess the importance of spatial autocorrelation on model performance. ResultsWe found that both expected heterozygosity and per cent polymorphic loci could best be explained by the current range position and genetic structure of populations within the contemporary range. Genetic diversity showed a clear gradient of being highest near the geographic and climatic range centre and lowest near range edges. Hypotheses accounting for the effects of past climate (e.g. past climatic suitability, distance from the southern edge), in contrast, had comparatively little support. Model ranks were similar among spatial and non‐spatial models, but residuals of all non‐spatial models were significantly autocorrelated, violating the assumption of independence in least‐squares regression. Main conclusionsOur work adds strong support for the “Central‐Periphery Hypothesis” as providing a predictive framework for understanding the forces structuring genetic diversity across species’ ranges, and illustrates the value of applying a robust comparative model selection framework and accounting for spatial autocorrelation when comparing biogeographic models of genetic diversity.more » « less
-
ABSTRACT Fluctuations in Lyman-α (Ly α) forest transmission towards high-z quasars are partially sourced from spatial fluctuations in the ultraviolet background, the level of which are set by the mean free path of ionizing photons (λmfp). The autocorrelation function of Ly α forest flux characterizes the strength and scale of transmission fluctuations and, as we show, is thus sensitive to λmfp. Recent measurements at z ∼ 6 suggest a rapid evolution of λmfp at z > 5.0 which would leave a signature in the evolution of the autocorrelation function. For this forecast, we model mock Ly α forest data with properties similar to the XQR-30 extended data set at 5.4 ≤ z ≤ 6.0. At each z, we investigate 100 mock data sets and an ideal case where mock data matches model values of the autocorrelation function. For ideal data with λmfp = 9.0 cMpc at z = 6.0, we recover $$\lambda _{\text{mfp}}=12^{+6}_{-3}$$ cMpc. This precision is comparable to direct measurements of λmfp from the stacking of quasar spectra beyond the Lyman limit. Hypothetical high-resolution data leads to a $$\sim 40~{{\ \rm per\ cent}}$$ reduction in the error bars over all z. The distribution of mock values of the autocorrelation function in this work is highly non-Gaussian for high-z, which should caution work with other statistics of the high-z Ly α forest against making this assumption. We use a rigorous statistical method to pass an inference test, however future work on non-Gaussian methods will enable higher precision measurements.more » « less
