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Journal of Geographical Systems

, Volume 21, Issue 2, pp 237–269 | Cite as

Spatial autocorrelation for massive spatial data: verification of efficiency and statistical power asymptotics

  • Qing Luo
  • Daniel A. Griffith
  • Huayi WuEmail author
Original Article
  • 102 Downloads

Abstract

Being a hot topic in recent years, many studies have been conducted with spatial data containing massive numbers of observations. Because initial developments for classical spatial autocorrelation statistics are based on rather small sample sizes, in the context of massive spatial datasets, this paper presents extensions to efficiency and statistical power comparisons between the Moran coefficient and the Geary ratio for different variable distribution assumptions and selected geographic neighborhood definitions. The question addressed asks whether or not earlier results for small n extend to large and massively large n, especially for non-normal variables; implications established are relevant to big spatial data. To achieve these comparisons, this paper summarizes proofs of limiting variances, also called asymptotic variances, to do the efficiency analysis, and derives the relationship function between the two statistics to compare their statistical power at the same scale. Visualization of this statistical power analysis employs an alternative technique that already appears in the literature, furnishing additional understanding and clarity about these spatial autocorrelation statistics. Results include: the Moran coefficient is more efficient than the Geary ratio for most surface partitionings, because this index has a relatively smaller asymptotic as well as exact variance, and the superior power of the Moran coefficient vis-à-vis the Geary ratio for positive spatial autocorrelation depends upon the type of geographic configuration, with this power approaching one as sample sizes become increasingly large. Because spatial analysts usually calculate these two statistics for interval/ration data, this paper also includes comments about the join count statistics used for nominal data.

Keywords

Moran coefficient Geary ratio Efficiency Power Geographic configuration Join count statistics 

JEL Classification

C12 C46 C55 

Notes

Acknowledgements

Funding was provided by The National Key Research and Development Program of China (Grant No. 2017YFB0503802) and China Scholarship Council (Grant No. 201406270075).

References

  1. Anselin L (1995) Local indicators of spatial association—LISA. Geogr Anal 27(2):93–115.  https://doi.org/10.1111/j.1538-4632.1995.tb00338.x Google Scholar
  2. Anselin L (1996) The Moran scatterplot as an ESDA tool to assess local instability in spatial association. In: Fischer M, Scholten H, Unwin D (eds) Spatial analytical perspectives on GIS. Taylor and Francis, London, pp 111–125Google Scholar
  3. Anselin L (2018) A local indicator of multivariate spatial association: Extending Geary’s c. Geogr Anal.  https://doi.org/10.1111/gean.12164 Google Scholar
  4. Bartels CPA, Hordijk L (1977) On the power of the generalized Moran contiguity coefficient in testing for spatial autocorrelation among regression distributions. Reg Sci Urban Econ 7(1):83–101.  https://doi.org/10.1016/0166-0462(77)90019-9 Google Scholar
  5. Bavaud F (2013) Testing spatial autocorrelation in weighted networks: The modes permutation test. J Geogr Syst 15(3):233–247.  https://doi.org/10.1007/s10109-013-0179-2 Google Scholar
  6. Bivand R, Müller WG, Reder M (2009) Power calculations for global and local Moran’s I. Comput Stat Data Anal 53(8):2859–2872.  https://doi.org/10.1016/j.csda.2008.07.021 Google Scholar
  7. Boots B (2003) Developing local measure of spatial association for categorical data. J Geogr Syst 5(2):139–160.  https://doi.org/10.1007/s10109-003-0110-3 Google Scholar
  8. Boots B, Tiefelsdorf M (2000) Global and local spatial autocorrelation in bounded regular tessellations. J Geogr Syst 2(4):319–348.  https://doi.org/10.1007/PL00011461 Google Scholar
  9. Carrijo TB, da Silva AR (2017) Modified Moran’s I for small samples. Geogr Anal 49(4):451–467.  https://doi.org/10.1111/gean.12130 Google Scholar
  10. Cheng T, Haworth J, Wang J (2012) Spatio-temporal autocorrelation of road network data. J Geogr Syst 14(4):389–413.  https://doi.org/10.1007/s10109-011-0149-5 Google Scholar
  11. Chun Y (2008) Modeling network autocorrelation within migration flows by eigenvector spatial filtering. J Geogr Syst 10(4):317–344.  https://doi.org/10.1007/s10109-008-0068-2 Google Scholar
  12. Chun Y, Griffith DA (2013) Spatial statistics and geostatistics: theory and applications for geographic information science and technology. SAGE, Thousand OaksGoogle Scholar
  13. Cliff AD, Ord JK (1969) The problem of spatial autocorrelation. In: Scott AJ (ed) Studies in regional science. Pion Ltd, London, pp 25–55Google Scholar
  14. Cliff AD, Ord JK (1970) Spatial autocorrelation: A review of existing and new measures with applications. Econ Geogr 46:269–292.  https://doi.org/10.2307/143144 Google Scholar
  15. Cliff AD, Ord JK (1973) Spatial autocorrelation. Pion Ltd, LondonGoogle Scholar
  16. Cliff AD, Ord JK (1981) Spatial process. Pion Ltd, LondonGoogle Scholar
  17. de Jong P, Sprenger C, van Veen F (1984) On extreme values of Moran’s I and Geary’s c. Geogr Anal 16(1):17–24.  https://doi.org/10.1111/j.1538-4632.1984.tb00797.x Google Scholar
  18. de la Mata T, Llano C (2013) Social networks and trade of service: Modelling interregional flows with spatial and network autocorrelation. J Geogr Syst 15(3):319–367.  https://doi.org/10.1007/s10109-013-0183-6 Google Scholar
  19. Diggle P (2010) Nonparametric methods. In: Gelfand AE, Diggle PJ, Fuentes M, Guttorp P (eds) Handbook of spatial statistics. CRC Press, Baca Raton, pp 299–316Google Scholar
  20. Dray S (2011) A new perspective about Moran’s Coefficient: Spatial autocorrelation as a linear regression problem. Geogr Anal 43(2):127–141.  https://doi.org/10.1111/j.1538-4632.2011.00811.x Google Scholar
  21. Geary RC (1954) The contiguity ratio and statistical mapping. Inc Stat 5(3):115–146.  https://doi.org/10.2307/2986645 Google Scholar
  22. Griffith DA (1987) Spatial autocorrelation: a primer. AAG, PennsylvaniaGoogle Scholar
  23. Griffith DA (1996) Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data. Can Geogr 40(4):351–367.  https://doi.org/10.1111/j.1541-0064.1996.tb00462.x Google Scholar
  24. Griffith DA (2003) Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer, BerlinGoogle Scholar
  25. Griffith DA (2004) Extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses. Linear Algebra Appl 388:201–219.  https://doi.org/10.1016/S0024-3795(03)00368-9 Google Scholar
  26. Griffith DA (2010) The Moran coefficient for non-normal data. J Stat Plan Inference 140(11):2980–2990.  https://doi.org/10.1016/j.jspi.2010.03.045 Google Scholar
  27. Griffith DA (2015) On the eigenvalue distribution of adjacency matrices for connected planar graphs. Quaest Geogr.  https://doi.org/10.1515/quageo-2015-0035 Google Scholar
  28. Griffith D, Chun Y (2016) Spatial autocorrelation and uncertainty associated with remotely-sensed data. Remote Sens 8(7):535.  https://doi.org/10.3390/rs8070535 Google Scholar
  29. Griffith DA, Luhanga U (2011) Approximating the inertia of the adjacency matrix of a connected planar graph that is the dual of a geographic surface partitioning. Geogr Anal 43(4):383–402.  https://doi.org/10.1111/j.1538-4632.2011.00828.x Google Scholar
  30. Haining RP (1978) The moving average model for spatial interaction. Trans Inst Br Geogr 3(2):202–225.  https://doi.org/10.2307/622202 Google Scholar
  31. Haynes D, Jokela A, Manson S (2018) IPUMS-Terra: Integrated big heterogeneous spatiotemporal data analysis system. J Geogr Syst 20(4):343–361.  https://doi.org/10.1007/s10109-018-0277-2 Google Scholar
  32. Hope ACA (1968) A simplified Monte Carlo significance test procedure. J R Stat Soc B 30(3):582–598Google Scholar
  33. Jackson MC, Huang L, Xie Q, Tiwari RC (2010) A modified version of Moran’s I. Int J Health Geogr 9:33.  https://doi.org/10.1186/1476-072X-9-33 Google Scholar
  34. Lee SI (2001) Developing a bivariate spatial association measure: an integration of Pearson’s r and Moran’s I. J Geogr Syst 3(4):369–385.  https://doi.org/10.1007/s101090100064 Google Scholar
  35. Lee J, Kang M (2015) Geospatial big data: challenges and oppurtunities. Big Data Res 2(2):74–81.  https://doi.org/10.1016/j.bdr.2015.01.003 Google Scholar
  36. Legendre P, Fortin MJ (1989) Spatial pattern and ecological analysis. Vegetatio 80(2):107–138.  https://doi.org/10.1007/BF00048036 Google Scholar
  37. Li S, Dragicevic S, Castro AC et al (2016) Geospatial big data handling theory and methods: a review and research challenges. ISPRS J Photogramm Remote Sens 115:119–133.  https://doi.org/10.1016/j.isprsjprs.2015.10.012 Google Scholar
  38. Luo Q, Griffith DA, Wu H (2017) The Moran coefficient and Geary ratio: some mathematical and numerical comparisons. In: Griffith DA, Chun Y, Dean DJ (eds) Advances in geocomputation. Advances in geographic information science. Springer, Cham, pp 253–269Google Scholar
  39. Moran PAP (1950) Notes on continuous stochastic phenomena. Biometrika 37(1/2):17–23.  https://doi.org/10.2307/2332142 Google Scholar
  40. Oden D (1995) Adjusting Moran’s I for population density. Stat Med 14(1):17–26Google Scholar
  41. Tait M, Tobin J (2017) Three conjectures in extremal spectral graph theory. J Comb Theory Ser B 126:137–161.  https://doi.org/10.1016/j.jctb.2017.04.006 Google Scholar
  42. Tiefelsdorf M, Boots B (1995) The exact distribution of Moran’s I. Environ Plan A 27(6):985–999.  https://doi.org/10.1068/a270985 Google Scholar
  43. van Zyl T (2014) Algorithmic design considerations for geospatial and/or temporal big data. In: Karimi HA (ed) Big data: techniques and technologies in geoinformatics. CRC Press, Baca Raton, pp 117–132Google Scholar
  44. Waldhör T (1996) The spatial autocorrelation coefficient Moran’s I under heteroscedasticity. Stat Med 15(7–9):887–892Google Scholar
  45. Weiss NA (2017) Introductory statistics, 10th edn. Pearson Education Ltd, LondonGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Information Engineering in Surveying, Mapping and Remote SensingWuhan UniversityWuhanChina
  2. 2.School of Economic, Political, and Policy ScienceThe University of Texas at DallasRichardsonUSA
  3. 3.Collaborative Innovation Center of Geospatial TechnologyWuhan UniversityWuhanChina

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