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


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.


Moran coefficient Geary ratio Efficiency Power Geographic configuration Join count statistics 

JEL Classification

C12 C46 C55 



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


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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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