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Bayesian wombling for spatial point processes. (English) Zbl 1180.62175
Summary: In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point-referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point-process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult.
We extend existing point-level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point-referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point-referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient’s distance to facilities likely to offer helpful cancer screening options.

62P10 Applications of statistics to biology and medical sciences; meta analysis
62F15 Bayesian inference
62M99 Inference from stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
65C60 Computational problems in statistics (MSC2010)
92C50 Medical applications (general)
Full Text: DOI
[1] Banerjee, Bayesian wombling: Curvilinear gradient assessment under spatial process models, Journal of American Statistical Association 101 pp 1487– (2006) · Zbl 1171.62347 · doi:10.1198/016214506000000041
[2] Banerjee, Gaussian predictive process models for large spatial datasets, Journal of the Royal Statistical Society, Series B 70 pp 825– (2008) · Zbl 05563371 · doi:10.1111/j.1467-9868.2008.00663.x
[3] Berman, Approximating point process likelihoods with GLIM, Applied Statistics 41 pp 31– (1992) · Zbl 0825.62614 · doi:10.2307/2347614
[4] Besag, Spatial interaction and the statistical analysis of lattice systems (with discussion), Journal of the Royal Statistical Society, Series B 36 pp 192– (1974) · Zbl 0327.60067
[5] Diggle, Bayesian geostatistical design, Scandinavian Journal of Statistics 33 pp 53– (2006) · Zbl 1120.62112 · doi:10.1111/j.1467-9469.2005.00469.x
[6] Guan, A thinned block bootstrap variance estimation procedure for inhomogeneous spatial point patterns, Journal of the American Statistical Association 102 pp 1377– (2007) · Zbl 1332.62108 · doi:10.1198/016214507000000879
[7] Hossain, Approximate methods in Bayesian point process spatial models, Computational Statistics and Data Analysis (2009) · Zbl 05689051 · doi:10.1016/j.csda.2008.05.017
[8] Kaufman, Finding Groups in Data: An Introduction to Cluster Analysis (1990) · Zbl 1345.62009
[9] Spatial Cluster Modeling (2002)
[10] Liang , S. Carlin , B. P. Gelfand , A. E. 2007 Analysis of marked point patterns with spatial and non-spatial covariate information
[11] Lu, Bayesian areal wombling for geographical boundary analysis, Geographical Analysis 37 pp 265– (2005) · doi:10.1111/j.1538-4632.2005.00624.x
[12] Ma, Evaluation of Bayesian models for focused clustering in health data, Environmetrics 18 pp 871– (2007) · doi:10.1002/env.850
[13] Mardia, Multi-dimensional multivariate Gaussian Markov random fields with application to image processing, Journal of Multivariate Analysis 24 pp 265– (1988) · Zbl 0637.60065 · doi:10.1016/0047-259X(88)90040-1
[14] Møller, Statistical Inference and Simulation for Spatial Point Processes (2004) · Zbl 1044.62101
[15] Pascutto, Statistical issues in the analysis of disease mapping data, Statistics in Medicine 19 pp 2493– (2000) · doi:10.1002/1097-0258(20000915/30)19:17/18<2493::AID-SIM584>3.0.CO;2-D
[16] Royle, An algorithm for the construction of spatial coverage designs with implementation in S-PLUS, Computional Geoscience 24 pp 479– (1998) · doi:10.1016/S0098-3004(98)00020-X
[17] Waagepetersen, An estimating function approach to inference for inhomogeneous Neyman-Scott processes, Biometrics 63 pp 252– (2007) · Zbl 1122.62073 · doi:10.1111/j.1541-0420.2006.00667.x
[18] Waagepetersen, Two-step estimation for inhomogeneous spatial point processes, Journal of the Royal Statistical Society, Series B. (2009) · Zbl 1250.62047 · doi:10.1111/j.1467-9868.2008.00702.x
[19] Womble, Differential systematics, Science 114 pp 315– (1951) · doi:10.1126/science.114.2961.315
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