Detecting Clustering Scales with the Incremental K-Function: Comparison Tests on Actual and Simulated Geospatial Datasets

  • Ran Tao
  • Jean-Claude ThillEmail author
  • Ikuho Yamada
Part of the Lecture Notes in Geoinformation and Cartography book series (LNGC)


The detection of so-called hot-spots in point datasets is important to generalize the spatial structures and properties in geospatial datasets. This is all the more important when spatial big data analytics is concerned. The K-function is regarded as one of the most effective methods to detect departures from randomness, high concentrations of point events and to examine the scale properties of a spatial point pattern. However, when applied to a pattern exhibiting local clusters, it can hardly determine the true scales of an observed pattern. We use a variant of the K-function that examines the number of events within a particular distance increment rather than the total number of events within a distance range. We compare the Incremental K-function to the standard K-function in terms of its fundamental properties and demonstrate the differences using several simulated point processes, which allow us to explore the range of conditions under which differences are obtained, as well as on a real-world geospatial dataset.


K-function Point data Hot spots Spatial clustering Scale Spatial big data 


  1. 1.
    Fotheringham AS, Zhan FB (1996) A comparison of three exploratory methods for cluster detection in spatial point patterns. Geogr Anal 28(3):200–218CrossRefGoogle Scholar
  2. 2.
    Yamada I, Thill J-C (2007) Local indicators of network-constrained clusters in spatial point patterns. Geogr Anal 39(3):268–292CrossRefGoogle Scholar
  3. 3.
    Getis A, Boots BN (1978) Models of spatial processes: an approach to the study of point, line and area patterns, vol 198. Cambridge University Press, CambridgeGoogle Scholar
  4. 4.
    Laney D (2001) 3D data management: controlling data volume, velocity and variety. META Group Research Note, 6Google Scholar
  5. 5.
    Miller HJ (2010) The data avalanche is here. Shouldn’t we be digging? J Reg Sci 50(1):181–201CrossRefGoogle Scholar
  6. 6.
    Tobler WR (1970) A computer movie simulating urban growth in the Detroit region. Econ Geogr 46(2):234–240Google Scholar
  7. 7.
    Waller LA (2009) Detection of clustering in spatial data. The Sage Handbook of Spatial Analysis. London, pp 299–320Google Scholar
  8. 8.
    Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266Google Scholar
  9. 9.
    Getis A, Franklin J (1987) Second-order neighborhood analysis of mapped point patterns. Ecology 68:473–477Google Scholar
  10. 10.
    Fotheringham AS (1997) Trends in quantitative methods I: stressing the local. Prog Hum Geogr 21(1):88–96CrossRefGoogle Scholar
  11. 11.
    Openshaw S, Charlton M, Wymer C, Craft A (1987) A mark 1 geographical analysis machine for the automated analysis of point data sets. Int J Geogr Inf Syst 1(4):335–358CrossRefGoogle Scholar
  12. 12.
    Besag J, Newell J (1991) The detection of clusters in rare diseases. J Roy Stat Soc A (Stat Soc) 154:143–155Google Scholar
  13. 13.
    Anselin L (1995) Local indicators of spatial association—LISA. Geogr Anal 27(2):93–115CrossRefGoogle Scholar
  14. 14.
    Getis A, Ord JK (1992) The analysis of spatial association by use of distance statistics. Geogr Anal 24(3):189–206CrossRefGoogle Scholar
  15. 15.
    Ord JK, Getis A (1995) Local spatial autocorrelation statistics: distributional issues and an application. Geogr Anal 27(4):286–306CrossRefGoogle Scholar
  16. 16.
    Aldstadt J, Getis A (2006) Using AMOEBA to create a spatial weights matrix and identify spatial clusters. Geogr Anal 38(4):327–343CrossRefGoogle Scholar
  17. 17.
    Widener MJ, Crago NC, Aldstadt J (2012) Developing a parallel computational implementation of AMOEBA. Int J Geogr Inf Sci 26(9):1707–1723CrossRefGoogle Scholar
  18. 18.
    Tang W, Feng W, Jia M (2014) Massively parallel spatial point pattern analysis: Ripley’s K function accelerated using graphics processing units. Int J Geogr Inf Sci 28(5):1107–1127 (Accepted)Google Scholar
  19. 19.
    Okabe A, Boots B, Satoh T (2010) A class of local and global k functions and their exact statistical methods. In: Perspectives on spatial data analysis. Springer, Berlin, pp 101–112Google Scholar
  20. 20.
    Boots B, Okabe A (2007) Local statistical spatial analysis: inventory and prospect. Int J Geogr Inf Sci 21(4):355–375CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Geography and Earth Sciences and Project MosaicUniversity of North Carolina at CharlotteCharlotteUSA
  2. 2.Department of Integrated Science and Engineering for Sustainable SocietyChuo UniversityTokyoJapan

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