Abstract
Statistical models for areal data are primarily used for smoothing maps revealing spatial trends. Subsequent interest often resides in the formal identification of ‘boundaries’ on the map. Here boundaries refer to ‘difference boundaries’, representing significant differences between adjacent regions. Recently, Lu and Carlin (Geogr Anal 37:265–285, 2005) discussed a Bayesian framework to carry out edge detection employing a spatial hierarchical model that is estimated using Markov chain Monte Carlo (MCMC) methods. Here we offer an alternative that avoids MCMC and is easier to implement. Our approach resembles a model comparison problem where the models correspond to different underlying edge configurations across which we wish to smooth (or not). We incorporate these edge configurations in spatially autoregressive models and demonstrate how the Bayesian Information Criteria (BIC) can be used to detect difference boundaries in the map. We illustrate our methods with a Minnesota Pneumonia and Influenza Hospitalization dataset to elicit boundaries detected from the different models.
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References
Anselin L (1988) Spatial econometrics: methods and models. Kluwer, Boston
Anselin L (1990) Spatial dependence and spatial structural instability in applied regression analysis. J Reg Sci 30:185–207
Assuncao R, Krainski E (2009) Neighborhood dependence in Bayesian spatial models. Biom J 51:851–869
Banerjee S, Carlin BP, Gelfand AE (2004) Hierarchical modeling and analysis for spatial data. Chapman and Hall/CRC Press, Boca Raton
Banerjee S, Gelfand AE (2006) Bayesian wombling: curvilinear gradient assessment under spatial process models. J Am Stat Assoc 101:1487–1501
Barbujani G, Sokal RR (1990) Zones of sharp genetic change in Europe are also linguistic boundaries. Proc Natl Acad Sci USA 87:1816–1819
Barry R, Pace RK (1999) A Monte Carlo estimator of the log determinant of large sparse matrices. Linear Algebra Appl 289:41–54
Berger JO (1985) Statistical decision theory and Bayesian analysis. Springer-Verlag, Berlin. ISBN 0-387-96098-8
Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc, B 36(2):192–236
Bocquet-Appel JP, Bacro JN (1994) Generalized wombling. Syst Biol 43:442–448
Carlin BP, Louis TA (2000) Bayes and empirical bayes methods for data analysis, 2nd edn. Chapman and Hall/CRC Press, Boca Raton
Cleveland WS, Devlin SJ (1988) Locally-weighted regression: an approach to regression analysis by local fitting. J Am Stat Assoc 83:596–610
Cressie NAC (1993) Statistics for spatial data, 2nd edn. Wiley, New York
Cromley EK, McLafferty SL (2002) GIS and public health. Guilford Publications, New York
Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1:209–230
Fortin MJ (1994) Edge detection algorithms for two-dimensional ecological data. Ecology 75:956–965
Fortin MJ, Drapeau P (1995) Delineation of ecological boundaries: comparisons of approaches and significance tests. Oikos 72:323–332
Fortin MJ (1997) Effects of data types on vegetation boundary delineation. Can J For Res 27:1851–1858
Gelman A, Carlin JB, Stern HS, Rubin DB (2004) Bayesian data analysis, 2nd edn. Chapman and Hall/CRC Press, Boca Raton
Hoeting JA, Madigan D, Raftery AE, Volinsky C (1999) Bayesian model averaging: a tutorial. Stat Sci 14:382–417
Jacquez GM, Greiling DA (2003). Local clustering in breast, lung and colorectal cancer in Long Island, New York. Int J Health Geogr 2:3
Jacquez GM, Greiling DA (2003). Geographic boundaries in breast, lung and colorectal cancers in relation to exposure to air toxics in long Island, New York. Int J Health Geogr 2:4
LeSage JP (1997) Analysis of spatial contiguity influences on state price level formation. Int J Forecast 13:245–253
LeSage JP, Pace K (2009) Introduction to spatial econometrics. Chapman and Hall/CRC, Boca Raton, FL
LeSage JP, Parent O (2007) Bayesian model averaging for spatial econometric models. Geogr Anal 39:241–267
Lu H, Carlin BP (2005) Bayesian areal wombling for geographical boundary analysis. Geogr Anal 37:265–285
Lu H, Reilly C, Banerjee S, Carlin BP (2007) Bayesian areal wombling via adjacency modeling. Environ Ecol Stat 14:433–452
Ma H, Carlin BP, Banerjee S (2010) Hierarchical and joint site-edge methods for Medicare hospice service region boundary analysis. Biometrics 66(2):355–364
Pace RK, Barry R (1997) Sparse spatial autoregressions. Stat Probab Lett 33:291–297
Raftery AE (1995) Bayesian model selection in social research (with discussion). In: Marsden PV (ed) Sociological methodology 1995. Blackwell, Cambridge, MA, pp 111–195
Raftery AE, Madigan D, Hoeting J (1997) Bayesian model averaging for linear regression models. J Am Stat Assoc 92:179–191
Robert C (2001) The Bayesian choice, 2nd ed. Springer, New York
Wall MM (2004) A close look at the spatial correlation structure implied by the CAR and SAR models. J Stat Plan Inference 121(2):311–324
Waller, Gotway (2004) Applied spatial statistics for public health data. Wiley, New York
Wheeler D, Waller L (2008) Mountains, valleys, and rivers: the transmission of raccoon rabies over a heterogeneous landscape. J Agric Biol Environ Stat 13:388–406
Womble WH (1951) Differential systematics. Science 114:315–322
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This work was supported in part by NIH grants 1-R01-CA95995 and 1-RC1-GM092400-01.
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Li, P., Banerjee, S. & McBean, A.M. Mining boundary effects in areally referenced spatial data using the Bayesian information criterion . Geoinformatica 15, 435–454 (2011). https://doi.org/10.1007/s10707-010-0109-0
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DOI: https://doi.org/10.1007/s10707-010-0109-0