The Annals of Regional Science

, Volume 48, Issue 2, pp 529–539 | Cite as

Local spatial heteroscedasticity (LOSH)

  • J. Keith Ord
  • Arthur GetisEmail author
Special Issue Paper


In keeping with the Annals of Regional Science 50-year tradition of emphasizing spatial analytic contributions, a new statistic, H i , is introduced as an extension of the recent work on map pattern analysis using local spatial criteria. In conjunction with local statistics for the mean level of a spatial process, H i tests for local spatial heteroscedasticity. The statistic measures spatial variability while attempting to avoid the pitfalls of using the non-spatial F test in georeferenced situations. H i is defined and suitably scaled, and an inferential procedure is given. Three artificial cases are explored: a random pattern, a cluster pattern of high values, and a binary-based cluster. The statistic may be used to identify boundaries of clusters and to describe the nature of heteroscedasticity within clusters.

JEL Classification

C21 C46 C51 


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  1. Aldstadt J, Getis A (2006) Using AMOEBA to create a spatial weights matrix and identify spatial clusters. Geogr Anal 38: 327–343CrossRefGoogle Scholar
  2. Anselin L (1995) Local indicators of spatial association—LISA. Geogr Anal 27(2): 93–115CrossRefGoogle Scholar
  3. Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc B 57(1): 289–300Google Scholar
  4. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31: 307–327CrossRefGoogle Scholar
  5. Boos DD, Browne C (2004) Comparing variances and others measures of dispersion. Stat Sci 19(4): 571–578CrossRefGoogle Scholar
  6. Box GEP (1953) Non-normality and tests on variances. Biometrika 40: 318–335Google Scholar
  7. Castro MC, Singer BH (2006) Controlling the false discovery rate: a new application to account for multiple and dependent tests in local statistics of spatial association. Geogr Anal 38: 180–208CrossRefGoogle Scholar
  8. Cliff AD, Ord JK (1969) The problem of spatial autocorrelation. In: Scott AJ (ed) London papers in regional science. Pion, London, pp 25–55Google Scholar
  9. Cliff AD, Ord JK (1973) Spatial autocorrelation. Pion, LondonGoogle Scholar
  10. Cliff AD, Ord JK (1981) Spatial processes: models and applications. Pion, LondonGoogle Scholar
  11. Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation. Econometrica 50: 987–1008CrossRefGoogle Scholar
  12. Fischer, MM, Getis, A (eds) (2010) Handbook of applied spatial analysis: software tools, methods and applications. Springer, BerlinGoogle Scholar
  13. Getis A, Ord JK (1992) The analysis of spatial association by distance statistics. Geogr Anal 24:189–206. [Reprinted in: Anselin L, Rey S (eds) Perspectives on spatial data analysis. Springer: Berlin]Google Scholar
  14. Getis A, Ord JK (2000) Seemingly independent tests: addressing the problem of multiple simultaneous and dependent tests. 39th Annual meeting of the Western Regional Science Association, Kauai, HawaiiGoogle Scholar
  15. Levene H (1960) Robust tests for equality of variances. In: Olkin I (ed) Contributions to probability and statistics. Stanford University Press, Stanford, pp 278–292Google Scholar
  16. Ord JK, Getis A (1995) Local spatial autocorrelation statistics: distributional issues and an application. Geogr Anal 27: 286–306CrossRefGoogle Scholar
  17. Simes RJ (1986) An improved Bonferroni procedure for multiple tests of significance. Biometrika 73(3): 751–754CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.McDonough School of BusinessGeorgetown UniversityWashingtonUSA
  2. 2.Department of GeographySan Diego State UniversitySan DiegoUSA

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