Spatial Autocorrelation
D. A. Griffith, University of Texas at Dallas, Richardson, TX, USA
& 2009 Elsevier Inc. All rights reserved.
Glossary
Auto- A prefix literally meaning self; spatial
autocorrelation means self-correlation, or values within a
given variable are correlated, resulting in the variable
being correlated with itself.
Correlation A description of the nature and degree of
a relationship between a pair of quantitative variables.
Geary Ratio An index of spatial autocorrelation,
involving the computation of squared differences of
values that are geographic neighbors (i.e., paired
comparisons), that ranges from 0 to 1 for negative, and 1
to approximately 2 for positive, spatial autocorrelation,
with an expected value of 1 for zero spatial
autocorrelation.
Geographic Connectivity/Weights Matrix An n-by-n
matrix with the same sequence of row and column
location labels, whose entries indicate which pairs of
locations are neighbors.
Map Pattern The systematic organization of values
for some variable across a map resulting in visually
conspicuous texture that consists of global, regional,
and local trends, gradients, swaths, or mosaics.
Moran Coefficient An index of spatial autocorrelation,
involving the computation of cross-products of meanadjusted
values that are geographic neighbors (i.e.,
covariations), that ranges from roughly (–1, –0.5) to
nearly 0 for negative, and nearly 0 to approximately 1 for
positive, spatial autocorrelation, with an expected value
of –1/(n – 1) for zero spatial autocorrelation, where n
denotes the number of areal units.
Moran Scatterplot A scatterplot of standardized
versus summed nearby standardized values whose
associated bivariate regression slope coefficient is the
unstandardized Moran coefficient.
Negative Spatial Autocorrelation For the geographic
distribution of some variable across a map, high values
tend to be geographic neighbors of low values,
intermediate values tend to be geographic neighbors of
intermediate values, and low values tend to be
geographic neighbors of high values.
Positive Spatial Autocorrelation For the geographic
distribution of some variable across a map, high values
tend to be geographic neighbors of high values,
intermediate values tend to be geographic neighbors of
intermediate values, and low values tend to be
geographic neighbors of low values.
Redundant Information Information in data that is
duplicated, and hence unneeded for spatial statistical
analyses; for georeferenced data, this duplication arises
from locational closeness that results in mutually shared
information, allowing an awareness of nearby values
once the value for a given location is known.
Introduction
‘Spatial autocorrelation’ is the correlation among values
of a single variable strictly attributable to their relatively
close locational positions on a two-dimensional (2-D)
surface, introducing a deviation from the independent
observations assumption of classical statistics. Spatial
autocorrelation exists because real-world phenomena are
typified by orderliness, (map) pattern, and systematic
concentration, rather than randomness. Tobler’s first law
of geography encapsulates this situation: ‘‘everything is
related to everything else, but near things are more related
than distant things.’’ To this maximum should be
added the qualifier: ‘‘but not necessarily through the same
mechanisms.’’ In other words, spatial autocorrelation
means a dependency exists between values of a variable
in neighboring or proximal locations, or a systematic
pattern in values of a variable across the locations on a
map due to underlying common factors.
Selected physical models portray the existence of
spatial autocorrelation. First, and foremost, pictures
(analogous to map pattern) would not be discernible on a
television/computer screen without spatial autocorrelation.
Nor could sliding tile or jigsaw puzzles be solved.
Magnetic sculptures constructed from separate pieces of
metal piled upon a magnetized base more completely
illustrate spatial autocorrelation: when the pieces of
metal are placed on their magnetic base, they can be
sculpted into a 3-D figure; when the pieces of metal are
moved far from their magnetic base (as well as any other
magnetic source), they simply are independent pieces of
metal that only can be brushed into a pile. The spatial
autocorrelation mechanism here is the magnetic field that
is transferred from piece to piece by metal pieces
1
touching. Meanwhile, a number of real-world examples
also portray the existence of spatial autocorrelation. Most
mineral deposits cluster at relatively few locations on the
Earth’s surface; they are not ubiquitous. House prices and
house value assessments are established in the real estate
market by comparisons between a house and similar
nearby houses. Zoning ordinances force similar land-use
types to group together in coterminous locations. Diseases,
such as West Nile virus, creep across a landscape
through contagion. Finally, unabated water (wind) polluters
generate negative consequences for those downstream
(downwind) of their locations; a similar but more
fully 2-D example is furnished by defunct smelter
superfund sites.
Conceptual Meanings of Spatial
Autocorrelation
Spatial autocorrelation has many interpretations. The
most dismissive is as a ‘nuisance parameter’. Spatial
autocorrelation is captured by a model specification because
its presence is necessary for a good description, but
it is not truly of interest and only interferes with the
estimation of other model parameters that are of true
interest. This interference tends to be for parameters
such as variances, rather than means. Scientists increasingly
are deciding that spatial autocorrelation should not
be treated as a nuisance parameter.
Interpreting spatial autocorrelation as ‘self-correlation’
is literal. Correlation arises from the geographic
context within which attribute values occur, and as such
can be expressed in terms of the Pearson product–moment
correlation coefficient (r) formula, but with
neighboring values of variable y replacing those of variable
x. Here, the correlation being ascertained is an
average of that between location-specific time series of a
variable for all possible pairs of locations. But these time
series are not observable, and hence the assumption invoked
is exchangeability (i.e., the set of time series can be
permuted without affecting results – the order in which a
time series mechanism generates values across a map is
irrelevant).
Interpreting spatial autocorrelation as ‘map pattern’
emphasizes conspicuous trends, gradients, swaths, or
mosaics across a map. Consider a constant, which is the
degenerate case (i.e., a constant has no variance) of perfect
positive spatial autocorrelation: once the value of a
constant is known at a single location, it is known at all
locations. Next, consider a variable that portrays a north–
south (or east–west) linear trend across a map. If this
variable has a mean of 0, then it is geometrically independent
of any constant. These north–south and east–
west oriented linear trend variables also are independent.
A variable with mean 0 whose values’ magnitudes form a
3-D symmetric hill (or valley) in the center of a map
constitutes yet another mutually independent map pattern.
These three variables display maximum levels of
positive spatial autocorrelation when geographic variance
is present, and may be described as global geographic
patterns. Alternating sequences of moderately large hills
and valleys with either an east–west or a north–south
orientation portray moderate positive spatial autocorrelation,
and constitute regional map patterns. Alternating
sequences of small hills and valleys with either an
east–west or a north–south orientation portray weak
positive spatial autocorrelation, and constitute local map
patterns. This fragmentation continues through randomness
(zero spatial autocorrelation) to arrangements of
increasingly alternating values (i.e., single value hills and
valleys), which portray increasing negative spatial autocorrelation.
Most substantive variables have geographic
distributions that can be described by combinations
of some subset of these mutually independent varying
hill–valley cluster size map patterns.
As a ‘diagnostic tool’, spatial autocorrelation plays a
crucial role in model-based inference, whose foundation
is a set of valid assumptions rather than a scientific (i.e.,
random) sampling design. Detected spatial autocorrelation
can signify model misspecification, including treating
nonlinear relationships with a linear specification,
such as an exponential relationship between a predictor
variable, x, and a response variable, y, that is described
with a straight trend line. It also can signify ‘missing
variables’ for a regression equation, and as such serves as
a ‘surrogate’ for variation otherwise unaccounted for
because these variables are missing. This surrogate role
occurs when autocorrelation map patterns displayed by
predictor variables align with autocorrelation map patterns
displayed by y. Moreover, the same set of distinct
map patterns is common to both a set of missing variables
and y. Accounting for spatial autocorrelation in this
context can do a surprisingly good job of representing
missing variables in an equation specification. In contrast,
spatial autocorrelation that remains unaccounted for
tends to distort classical correlation coefficient inter-
pretations.
The term correlation alludes to the notion of ‘redundant
information’. If x and y are perfectly correlated,
then knowing x means exactly knowing y. In other words,
the information content of y is perfectly duplicated in x;
this degree of duplication decreases as the correlation
coefficient moves toward 0. Spatial autocorrelation extends
this notion of redundant information to georeferenced
data. The value at a given location can be
predicted with some degree of accuracy from the values
at nearby locations; this spatial data feature constitutes
the foundation of cartographic interpolation. Recalling
the example of housing prices, such redundant information
results from ‘spatial spillovers’: house values at
2 Spatial Autocorrelation
one location spill over to impact upon house values at
nearby locations. Building an expensive house near an
inexpensive house tends to reduce the value of the expensive
house while increasing the value of the inexpensive
house, all other things being equal. This
mechanism is similar to a river that floods its banks into
its flood plain. Another example, at the micro-level of
geographic resolution, is second-hand smoke generated
by smokers. And yet another example is the noxious
smell generated by such facilities as sewage treatment or
rendering plants that permeates their surrounding
neighborhoods.
Spatial autocorrelation can materialize from some
course of action operating over a geographic landscape,
such as contagious diffusion of a disease, resulting in it
being interpreted as a ‘spatial process mechanism’. The
diffusion of West Nile virus across the coterminous
United States (US) illustrates this interpretation. This
disease quickly became a serious problem in the US,
extremely rapidly diffusing from Long Island in 1999
throughout the remainder of the country following
northeast-to-west and -south paths. A weather front
moving across a geographic landscape can be viewed in a
similar way because it results in highly spatially autocorrelated
local weather conditions.
Finally, interpreting spatial autocorrelation as an
‘outcome of areal unit demarcation’ relates it to the
modifiable areal unit problem (MAUP), whereby results
from statistical analyses of georeferenced data can be
varied at will simply by changing the surface partitioning
used to demarcate areal units. For example, a standard
eight-by-eight checkerboard pattern exhibits negative
spatial autocorrelation between the red and black colors
of its squares. But if these squares are aggregated into
compact clusters of four (i.e., two-by-two groupings), and
the red and black colors averaged (resulting in a constant
dark red color across the checkerboard), then the spatial
autocorrelation becomes maximally positive. Political
redistricting involving gerrymandering (i.e., electoral
district or constituency boundaries are manipulated in
order to achieve a prespecified geographic aggregation
result, such as some political advantage) exemplifies this
general situation in practice.
Illustrations of Spatial Autocorrelation
Most empirical spatial autocorrelation cases involve
moderate, positive relationship tendencies between
nearby values on a map. Remotely sensed satellite images
are one exception, almost always displaying a very strong
positive relationship. Most socioeconomic/demographic
data display a moderate positive relationship. And,
negative spatial autocorrelation rarely is encountered in
practice.
Strong Positive Spatial Autocorrelation
Remotely sensed images are one exception to the georeferenced
data norm of moderate positive spatial autocorrelation,
frequently displaying very strong positive
spatial autocorrelation. This feature partly results from
light reflectance scattering, rather than being neatly
contained in pixel boundaries, which are imaginary,
hence spilling over into nearby pixels measured by a
satellite’s sensors.
A 1000-by-1000 pixel subset was extracted from a
satellite image of the Florida Everglades for illustrative
purposes (Figure 1a). The amount of green vegetation in
this region can be quantified with a normalized difference
vegetation index (NDVI), which is calculated with the
spectral reflectance values for the near-infrared (B5) and
visible red (B3) bands as follows:
B5 À B3
B5 þ B3
Positive values beyond about 0.3 tend to represent
green vegetation, whereas negative values tend to represent
swamp areas; positive values closer to 0 tend to
represent soil. The 1 000 000 selected pixels yield only
3033 different NDVI values. This measurement was
modified mathematically (i.e., transformed) in order to
better align it with a bell-shaped curve (Figure 1b).
The image appearing in Figure 1a displays very
strong positive spatial autocorrelation. The outline of the
Everglades is apparent, as is the southeast coast of
Florida, south of Miami. The absence of autocorrelation
would result in this image appearing as a shuffling of the
set of colors, similar to the picture-distorting momentary
white specks that appear on a television screen when, for
example, atmospheric static is dominant when, say, a
cable connection is lost.
Moderate Positive Spatial Autocorrelation
Maps of population density tend to display moderate
positive spatial autocorrelation, in part due to urbanization
at a regional or national scale, and zoning at a
local scale. Data from the 1993 census of Peru furnish
population counts by districts across the Cusco department;
an ArcGIS shapefile furnishes area measures for
these 108 districts. Population density tends to be skewed,
with a natural lower bound of 0, and few areal units with
relatively sizeable concentrations.
Population density by district in the Cusco department
ranges from 0.8 to 11 579.2 per unit area. Its inverse
square root, after adding 11 to each density, better
mimics a bell-shaped curve (see Figure 2). This transformed
population density displays moderate positive
spatial autocorrelation, forming an elongated mound map
pattern with a single peak. The highest density is in the
city of Cusco, with the next highest densities stretching
Spatial Autocorrelation 3
along an economic corridor formed by the Vilcanota
River valley; the lowest densities are in the most rural
areas of this department.
Moderate Negative Spatial Autocorrelation
Few empirical examples of negative spatial autocorrelation
are reported in the literature. Usually this phenomenon
is discussed conceptually in terms of
geographic competition. In other words, if a finite amount
of land is available, gains in land size of one territory can
occur only through the loss of land size in nearby territories.
The World Wars fought on the European continent
illustrate this situation.
Continental Europe principally in terms of the
European Union is partitioned into 22 countries. Treating
the capital of each country as its seat of power, and
hence its focal point, this portion of the continent also
can be partitioned into Thiessen polygons (Figure 3b).
The ratio of each country’s actual to corresponding
Thiessen polygon land size gives an index of local competition,
whose normal quantile plot (Figure 3a) implies
a frequency distribution that conforms well to a bellshaped
curve. This areas ratio displays negative spatial
autocorrelation: Switzerland, Luxembourg, Slovenia, and
the Czech Republic are islands of very low ratio values
that are completely surrounded by countries that have
the highest ratio values (Figure 3c).
Estimators of Spatial Autocorrelation
Historically, once the spatial autocorrelation concept was
established and widely recognized, spatial scientists became
interested in quantifying it, and then testing
hypotheses about it, with an ultimate goal of incorporating
it into models. The two most commonly used
99.9999
99.99
99
95
80
50
20
5
1
0.01
0.000 1
Percent
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
tr-NDVI(a)
(c) (d) (e)
(b)
Figure 1 A Landsat 7 Enhanced Thematic Mapper Plus (ETM þ ) image of the Florida Everglades for 1 January 2002. (a) A composite
image constructed using all spectral bands, with the subregion of analysis outlined with a red rectangle and the nonimage areas blacked
out; (b) a quantile plot of the 1 000 000 transformed NDVI values for the subregion; (c) visible red spectral band (wavelength 0.63–0.69
microns (i.e., nanometers/1000)) for the subregion, with data values in the range 13–175; (d) mid-infrared spectral band (wavelength
1.55–1.75 microns) for the subregion, with data values in the range 1–173; and (e) transformed NDVI, with values in the range –0.84 to
1.89. Note: the gray scale for the maps is directly proportional to the pixel values.
4 Spatial Autocorrelation
quantitative indices are the Moran coefficient (MC) and
the Geary ratio (GR).
From r to Moran Coefficient
Exploiting the interpretation of self-correlation, spatial
autocorrelation can be expressed in terms of the formula
for r, but with neighboring values of variable y replacing
those of x
Pn
i¼1ðxi À ¯xÞð yi À ¯yÞ=n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1ðxi À ¯xÞ2
=n
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1ð yi À ¯yÞ2
=n
q
becomes
Pn
i¼1
Pn
j¼1 cij ð yi À ¯yÞð yj À ¯yÞ=
Pn
i¼1
Pn
j¼1 cij
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1ð yi À ¯yÞ2
=n
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
i¼1 ð yi À ¯yÞ2
=n
q
The left-hand expression converts to the right-hand one
by replacing y’s with x’s in the right-hand side, by computing
the numerator term only when areal units i and j
are nearby (cij is an indicator variable – often called a
spatial weight – whose value is 1 for neighbors, and 0
otherwise), and by averaging the numerator crossproduct
terms over the total number of pairs denoted as
being nearby. The denominator of this revised expression
is the sample variance of Y, s2
Y. But unlike the values 71
for r, the extreme values of the MC are determined by
sophisticated mathematical quantities called eigenvalues
that are computed when the set of n indicator variables is
100
80
60
40
20
−2000 0 2000 4000 6000 8000 10 000 12 000
0
pd
20
15
10
5
0
−2000 −1000 1000 2000 30000
transformed pd(a) (b)
Frequency
Frequency
(c) (d)
Figure 2 The 1993 population density across the department of Cusco, Peru. (a) A histogram constructed with the 108 raw population
densities; (b) a histogram constructed with the 108 transformed population densities; (c) a quintile map of the geographic distribution of
transformed population density values; and (d) a quintile map of the composite spatial autocorrelation map pattern latent in the
geographic distribution of transformed population density values. Note: the gray scale for the maps is directly proportional to population
density.
Spatial Autocorrelation 5
organized into a table, much like a Microsoft Excel
spreadsheet, called a matrix. These values rarely are 71;
more often, the lower bound is between –1 and –0.5,
and the upper bound is between 1 and 1.1. Nevertheless,
like the relative values for r, MC values greater than
–1/(n – 1) indicate positive spatial autocorrelation
(e.g., using the ratio MC/MCmax, where MCmax denotes
the maximum possible MC value, 0.25 to 0.50 denotes a
weak, 0.50 to 0.70 denotes a moderate, 0.70 to 0.90 denotes
a strong, and 0.90 to 1.00 denotes a marked degree),
whereas values less than –1/(n – 1) indicate negative
spatial autocorrelation.
The ‘MC’ is a covariation (i.e., pairwise products of
deviations from the mean) index, and is defined by the
preceding right-hand expression. Its sampling distribution
can be constructed in one of two ways: randomly
sampling from a hypothetical probability distribution or
taking the set of observed values as given and constructing
all possible permutations of their values across a
given surface partitioning. In the former case, present
distributional results assume that a normal probability
model underlies variable y. In either case, the expected
value of the sampling distribution is –1/(n – 1), the
boundary separating positive and negative spatial
autocorrelation.
The standard error formulae for the MC are rather
complicated, but can be well approximated by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Pn
i¼1
Pn
j¼1 cij
s
when n is at least 20; the sum
Pn
i¼1
Pn
j ¼1 cij counts the
number of 1’s in the n indicator variables, which equals
twice the number of neighbors. This standard error result
is not surprising, recalling that the standard error of a
sample correlation coefficient relates to n, and that all
neighbors are double counted (both i to j and j to i).
The MC values for the preceding empirical examples
are 0.78 for the Everglades transformed NDVI, 0.51 for
the Cusco transformed population density, and –0.33 for
the European areas ratio. Their respective standard deviations
are: 0.058 78, 0.000 71, and 0.171 18.
The ‘GR’, the other popular spatial autocorrelation
index, is based upon paired comparisons (i.e., pairwise
squared differences), and may be defined as follows:
99
95
90
80
70
60
50
40
30
20
10
5
1
Percent
−0.5 0.0 0.5 1.0 1.5 2.0 2.5
ratio(a) (b)
(c) (d)
Figure 3 The geographic distribution of hypothetical land size competition across continental Europe. (a) A quantile plot of the ratio
between actual and Thiessen polygon areas, by country; (b) the Thiessen polygon partitioning of the continent (gray lines), with country
capitals as focal points; (c) a quintile map of the geographic distribution of the areas ratio values; and (d) a quintile map of the composite
spatial autocorrelation map pattern latent in the geographic distribution of the areas ratio values. Note: the gray scale for the maps is
directly proportional to the size of the values mapped.
6 Spatial Autocorrelation
Pn
i¼1
Pn
j¼1 cij ðyi À yj Þ2
=
Pn
i¼1
Pn
j¼1 cij
2
Pn
i¼1ðyi À ¯yÞ2
=ðn À 1Þ
This expression, which involves an average squared differences
term, also involves the unbiased variance estimate
for variable y. A ‘2’ appears in the denominator
because of double counting
ðyi À yj Þ2
¼ ðyi À yj þ ¯y À ¯yÞ2
¼ ðyi À ¯yÞ2
þ ðyj À ¯yÞ2
À 2ðyi À ¯yÞðyj À ¯yÞ
The expected value of GR is 1. It ranges between
roughly 0 and 2, with 0–1 signifying positive spatial
autocorrelation (i.e., ðyi À yj Þ2
goes to 0), and with 1–2
signifying negative spatial autocorrelation (i.e., ðyi À yj Þ2
is increasing in magnitude). Again, the actual extreme
values are a function of eigenvalues affiliated with the
matrix constructed with their spatial weight indicator
variables. In addition, 0 can never be reached, because it
is associated with the degenerate case of a constant across
a geographic landscape, and hence yields a division by 0.
The GR values for the preceding empirical examples
are 0.22 for the Everglades transformed NDVI, 0.41 for
the Cusco transformed population density, and 1.67 for
the European areas ratio.
Relationships between the Moran Coefficient
and the Geary Ratio
The MC and GR values tend to give consistent implications
when y is a normal random variable. In this case, a
useful rule of thumb is that the two coefficients should
sum to approximately 1. Dramatic deviations from 1 by
this sum suggest that y may be non-normal. Meanwhile,
most – but not all – empirical cases yield roughly the
same MC and GR values for raw data as well as data
transformed to better mimic a bell-shaped curve. And,
the MC is the statistically most powerful of the two indices:
it does the better job, on average, of differentiating
between null and alternative hypotheses. The algebraic
relationship between the MC and the GR highlights this
feature
GR ¼
n À 1
n
Pn
i¼1
Pn
j ¼1 cij
 
ðyi À ¯yÞ2
h i
=
Pn
i¼1
Pn
j ¼1 cij
Pn
i¼1ðyi À ¯yÞ2
=n
À MC
2
4
3
5
This equation reveals that the GR incorporates locational
information in addition to that included in the MC. This
additional information is the ratio of squared deviations
times their number of neighbors, divided by the sample
variance. If outliers are present, then this numerator can
become excessively large; if an areal unit has a large
number of neighbors, then this numerator can be markedly
influenced by the corresponding deviation. If variable
y conforms to a bell-shaped curve, then this
additional information term is approximately 1; as n
becomes increasingly large, the ratio (n – 1)/n converges
on 1. Therefore, adding MC to both sides of this equation
would result in (MC þ GR)E1.
The sign for the MC term in this preceding equation
is negative, emphasizing the negative relationship between
the MC and the GR: MC values approaching 1
correspond to GR values approaching 0 (positive spatial
autocorrelation), and MC values approaching –1 correspond
to GR values approaching 2 (negative spatial
autocorrelation). Zero spatial autocorrelation corresponds
to an expected value for the MC of –1/(n – 1),
which asymptotically converges on 0, resulting in the
corresponding GR value converging on 1, its expected
value for zero spatial autocorrelation. This negative relationship
can be seen simply by constructing a scatterplot
of MC and GR values for the three preceding
empirical examples.
Graphical Portrayals of Spatial
Autocorrelation
Spatial autocorrelation is a concept that lends itself to
visualization, including scatterplots and mappings. Various
graphical portrayals of this concept can be constructed,
including a Moran scatterplot, a semivariogram
plot (based upon the GR and interareal unit distances),
and a spatial correlogram (based upon MC and GR
values for 1st, then 2nd, then 3rd, and so forth, nearest
neighbors). The first of these, a Moran scatterplot, can be
constructed by: converting georeferenced data values to
z-scores (i.e., subtract the mean and then divide by the
standard deviation), summing the surrounding z-scores
for each areal unit (i.e.,
Pn
j¼1 cij zj ), and then plotting
these pairs of z-scores (horizontal axis) versus sums of
surrounding z-scores (vertical axis). The slope of the
resulting trend line is proportional to the MC (i.e., it
needs to be divided by
Pn
i¼1
Pn
j¼1 cij ). Moran scatterplots
for each of the three preceding empirical examples
appear in Figure 4.
The Moran scatterplot portrays
Pn
j¼1 cij zj versus zi,
whose trend line highlights the global trend across a
given geographic landscape. These sums of neighboring
values’ quantities also can be visualized with a map.
Doing so produces local indices of spatial autocorrelation
(LISA) statistics, which enable clusterings on a map to
become more conspicuous. Again, hills (i.e., clusters of
surrounding values above a mean) and valleys (i.e., clusters
of surrounding values below a mean) constitute the patterns
of interest. LISA quantities highlight local trends
across a given geographic landscape, emphasizing any
clusterings in the deviations from the global trend line.
These individual contributions to an MC reveal whether
spatial autocorrelation essentially is the same in all parts,
Spatial Autocorrelation 7
or differs from one part to another part, of a geographic
landscape. Local indices can be constructed with a global
GR, too, as well as with statistics computed from geostatistical
quantities (e.g., the pair of Getis–Ord statistics).
Inspection of a map often suggests the nature and
degree of the spatial autocorrelation it contains. This
composite map pattern can be viewed in terms of global,
regional, and local map pattern components. For example,
the composite map pattern associated with the
geographic distribution of population density across the
Cusco department appears in Figure 2d, and comprises
14 individual distinct map patterns, of which 11 are
global and regional (strong-to-marked positive spatial
autocorrelation), and three are local (weak-to-moderate
positive spatial autocorrelation). The most prominent
among this selected set of map patterns is a northeast–
southwest trend, followed by the hill-shaped map pattern;
the most fragmented of these map patterns (i.e., a local
pattern) includes nine visually detectable clusters scattered
across the department, many of which are partly
merged. This composite map pattern accounts for
roughly 62% of the geographic variation (i.e., redundant
information), and for all but a trace amount of the
positive spatial autocorrelation in population density
across the department. Similar descriptions can be furnished
for the Everglades NDVI and the European areas
ratio maps. These local map patterns relate to LISAs.
The totality of these map patterns relates to geographically
weighted regression (GWR).
Theoretical Statistical Properties of
Spatial Autocorrelation
Classical statistics furnishes criteria to distinguish useful
from useless spatial autocorrelation measures, including
unbiasedness, efficiency, sufficiency, and consistency.
The arithmetic mean of the sampling distribution for
an ‘unbiased estimator’ equals its corresponding population
parameter. This property has been used extensively
to evaluate the case of zero spatial autocorrelation
measures, and the impact of nonzero spatial autocorrelation
on conventional sample statistics. For many,
but not all, conventional statistical models, sample means,
and regression coefficients tend to be unbiased, whereas
sample variances and correlation coefficients tend to be
biased, by nonzero spatial autocorrelation. Thus, the
principal impact of spatial autocorrelation is on standard
30
20
10
−10
−20
0
Cz
Cz
−5.0 −2.5 0.0 2.5 5.0 7.5
Z Z
10
5
−5
−3 −2 −1 0 1 2 3
−10
0
4
3
2
1
0
−1
−2
−2 −1 0 1 2
z-scores
sumofsurroundingz-scores(a) (b)
(c)
Figure 4 Moran scatterplots for the three empirical examples. (a) Everglades transformed NDVI subregion; (b) Cusco transformed
population density; and (c) European areas ratio.
8 Spatial Autocorrelation
error calculations. Because a correlation coefficient is the
ratio of a covariance and two standard deviations, spatial
autocorrelation impacts in both of these calculations
largely cancel each other when the ratio is calculated.
An ‘efficiency estimator’ is both unbiased and has the
smallest possible standard error. Because spatial autocorrelation
mostly impacts upon variance calculations,
which in turn are used to compute standard errors, this
statistical property is the one most affected by nonzero
spatial autocorrelation. In the presence of positive spatial
autocorrelation, a variance tends to be inflated. The net
result is that the corresponding sampling distribution is
flatter than traditional statistical theory indicates. Redundant
information introduced by spatial autocorrelation
results in, for example, sample sizes being misleadingly
large. Consequently, more geographic samples are needed
to acquire a given margin of error.
A ‘sufficient estimator’ utilizes all information contained
in a sample that is relevant to a particular parameter.
Conventional statistics applied to georeferenced
data overlook locational information contained in nearby
values. Recognition of this information is required for
sufficiency to be preserved. For example, an arithmetic
mean not only needs
Pn
i¼1 yi, but also the cross-product
term
Pn
i¼1 yi
Pn
j ¼1 cij yj , which is used to measure spatial
autocorrelation.
Finally, a ‘consistent estimator’s’ sampling distribution
concentrates at the corresponding parameter value as n
increases. The efficiency criterion implies that this will
occur for unbiased statistics, such as the arithmetic mean,
but at a slower rate when spatial autocorrelation is present
than is suggested by conventional statistics.
Asymptotic concentration will occur even with variance
calculations, although not necessarily at the correct value.
If geographic sampling intensifies in a given region (infill
sampling), then sample points increasingly become closer
as n increases, resulting in spatial autocorrelation increasing,
and hence concentration increasingly slowing
down. If geographic sampling involves expanding a region
while maintaining the same average spacing between
sample point (increasing domain sampling), then
the rate of convergence will be constant rather than decreasing,
and thus consistency relies only on the finiteness
of all or part of the globe.
Summary and Contemporary Issues
Spatial autocorrelation has many faces, with its most
common interpretations being expressed in terms of selfcorrelation,
map pattern, and redundant information. Its
preferred measure is the MC, and one of its two most
popular graphical portrayals is the associated Moran
scatterplot. Few empirical examples of negative autocorrelation
have been found, with most empirical
examples involving moderate positive spatial autocorrelation;
remotely sensed images tend to display
strong positive spatial autocorrelation. Impacts of spatial
autocorrelation on conventional statistics can be assessed
with standard mathematical statistics criteria, such as
unbiasedness, efficiency, sufficiency, and consistency.
Today, spatial scientists routinely compute measures
of spatial autocorrelation, and rather than test hypotheses
about its presence, automatically include it in their
model specifications. Doing so often costs only 1 degree
of freedom. Spatial autocorrelation with linear models is
well understood, and has yielded spatial autoregressive
tools used in spatial statistics and spatial econometrics.
Spatial autocorrelation with generalized linear (mixed)
models is not well understood, with only a few cumbersome
tools available to handle it. Spatial filtering,
which is in its infancy and exploits the map pattern interpretation,
offers an approach that spans both linear
and nonlinear statistical models with tools that account
for the presence of spatial autocorrelation. This methodology
decomposes an underlying composite map pattern
into global, regional, and local components of spatial
autocorrelation.
The frontiers of spatial autocorrelation research entail
fuller development of contemporary techniques such as
spatial filtering, and efficient extensions of existing
techniques to massively large datasets. For example, the
Everglades remotely sensed image contains 41 611 007
land coverage pixels, whereas the simple preceding analysis
was challenged by dealing with only 1 000 000 of
these pixels. Furthermore, a need still exists for development
of quality spatial autocorrelation measures, especially
robust ones, for non-normal data.
See also: Regression, linear and non-linear; Segregation
indices; Spatial clustering, detection and analysis of;
Spatially autoregressive models; Statistics, Spatial.
Further Reading
Anselin, L. (1995). Local indicators of spatial association – LISA.
Geographical Analysis 27, 93--115.
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice
systems. Journal of the Royal Statistical Society B 36, 192--225.
Getis, A. and Ord, J. K. (1992). The analysis of spatial association by
use of distance statistics. Geographical Analysis 24, 189--206.
Griffith, D. (1987). Spatial Autocorrelation: A Primer. Washington, DC:
Association of American Geographers Resource Publication.
Griffith, D. (1992). What is spatial autocorrelation? Reflections on the
past 25 years of spatial statistics. l’Espace Ge´ographique 21,
265--280.
Griffith, D. (1996). Spatial autocorrelation and eigenfunctions of the
geographic weights matrix accompanying geo-referenced data. The
Canadian Geographer 40, 351--367.
Mardia, K. and Marshall, R. (1984). Maximum likelihood estimation of
models for residual covariance in spatial regression. Biometrika 71,
135--146.
Spatial Autocorrelation 9
Richardson, S. and He´ mon, D. (1981). On the variance of the sample
correlation between two independent lattice processes. Journal of
Applied Probability 18, 943--948.
Tiefelsdorf, M. and Boots, B. (1995). The exact distribution of Moran’s I.
Environment and Planning A 27, 985--999.
Relevant Websites
http://www.ecoevol.ufg.br/sam/
http://www.geoda.uiuc.edu
http://ncg.nuim.ie/ncg/GWR/
http://spatialfiltering.com/
10 Spatial Autocorrelation