Global Spatial Autocorrelation Indices  Moran's I, Geary's C and the General
CrossProduct Statistic
By M.Sawada
Department of Geography
University of Ottawa
Ottawa ON K1N 6N5
Introduction
The analysis of spatially located data is one of the basic concerns of the geographer and is becoming increasingly important in other fields
(Cliff and Ord, 1973). The assessment of spatial autocorrelation is generally considered to be one of the primary tasks of geographical data
analysis (Hubert and Arabie, 1991). Moran (1950) notes that the study of stochastic processes has naturally led to the study of stochastic
phenomena distributed in space in two or more dimensions which is important in such investigations as the study of soil fertility distribution or
the relation between velocities at different points in a turbulent fluid.
I will define spatial autocorrelation. I will explain how spatial autocorrelation works through explaining the general crossproduct statistic. I will
show that the common spatial autocorrelation measures of Moran’s I and Geary’s C are simply variations of the general crossproduct
statistic. I will also provide worked examples of the common spatial autocorrelation measures for which there are welldefined tests of
significance  which do not require randomization. I will then present a program that can be used to measure spatial autocorrelation called
Rooks Case v0.9
.
Defining Spatial Autocorrelation
There are a number of different definitions of spatial autocorrelation. Upton and Fingleton (1985) define spatial autocorrelation as a property
that mapped data possess whenever it exhibits an organized pattern. However, this definition is subjective because the exhibition of an
'organized pattern' can mean many things. The authors say that spatial autocorrelation exists whenever there is systematic spatial variation
in values across a map, or patterns in the values recorded at locations with the locations given. If high values at one locality are associated
with high values at neighboring localities the spatial autocorrelation is positive and when high values and low values alternate between
adjacent localities the spatial autocorrelation is negative (e.g., a checkerboard). Thus, Upton and Fingleton (1985) say that it is more useful
to define spatial autocorrelation by means of understanding lack of spatial autocorrelation. That is, if there is no connection between the
variables (X
i
, X
j
) at any pair of regions (
i
,
j
) in the study area, then the data exhibits a lack of spatial autocorrelation. In other words, a lack of
spatial autocorrelation should be found in a mapped pattern that does significantly deviate from a map where each value X
i
was assigned
randomly with equal probability to each (
i
,
j
) location on the map. Thus, notes Griffith (1991) and Goodchild (1987), spatial autocorrelation
deals simultaneously with both location and attribute information. In order to determine if the values in a mapped pattern deviates
significantly from a pattern in which the values are assigned randomly requires some sort of index of comparison.
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 Summer '08
 Staff
 Variance, Geography, Moran, spatial autocorrelation, Spatial data analysis

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