Spatial Autocorrelation
Nilupa Gunaratna, Yali Liu, Junyong Park
1.
Definition
Observations made at different locations may not be independent.
For example,
measurements made at nearby locations may be closer in value than measurements made
at locations farther apart.
This phenomenon is called spatial autocorrelation.
Spatial autocorrelation measures the correlation of a variable with itself through space.
Spatial autocorrelation can be positive or negative.
Positive spatial autocorrelation
occurs when similar values occur near one another.
Negative spatial autocorrelation
occurs when dissimilar values occur near one another.
2.
Weight Matrix
To assess spatial autocorrelation, one first needs to define what is meant by two
observations being close together, i.e., a distance measure must be determined.
These
distances are presented in weight matrix, which defines the relationships between
locations where measurements were made.
If data are collected at
n
locations, then the
weight matrix will be
n n
×
with zeroes on the diagonal.
The weight matrix can be specified in many ways:
The weight for any two different locations is a constant.
All observations within a specified distance have a fixed weight.
K nearest neighbors have a fixed weight, and all others are zero.
Weight is proportional to inverse distance, inverse distance squared, or
inverse distance up to a specified distance.
Other weight matrices are possible.
The weight matrix is often rowstandardized, i.e., all
the weights in a row sum to one.
Note that the actual values in the weight matrix are up
to the researcher.
3. Measures of Spatial Autocorrelation
A. Moran's I
Moran's I (Moran 1950) tests for global spatial autocorrelation for continuous data.
It is based on crossproducts of the deviations from the mean and is calculated for
n
observations on a variable
x
at locations
i
,
j
as:
1
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0
(
)(
)
(
)
ij
i
j
i
j
i
i
w x
x x
x
n
I
S
x
x


=

∑∑
∑
,
where
x
is the mean of the
x
variable,
ij
w
are the elements of the weight matrix, and
0
S
is the sum of the elements of the weight matrix:
0
ij
i
j
S
w
=
∑∑
.
Moran’s I is similar but not equivalent to a correlation coefficient.
It varies from 1 to
+1.
In the absence of autocorrelation and regardless of the specified weight matrix, the
expectation of Moran’s I statistic is
1/(
1)
n


, which tends to zero as the sample size
increases.
For a rowstandardized spatial weight matrix, the normalizing factor
0
S
equals
n
(since each row sums to 1), and the statistic simplifies to a ratio of a spatial
cross product to a variance.
A Moran’s I coefficient larger than
1/(
1)
n


indicates
positive spatial autocorrelation, and a Moran’s I less than
1/(
1)
n


indicates negative
spatial autocorrelation. The variance is:
where
B. Geary's C
Geary’s C statistic (Geary 1954) is based on the deviations in responses of each
observation with one another:
2
2
0
(
)
1
2
(
)
ij
i
j
i
j
i
i
w x
x
n
C
S
x
x


=

∑∑
∑
.
Geary’s C ranges from 0 (maximal positive autocorrelation) to a positive value for high
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 Summer '08
 Staff
 Normal Distribution, Variance, Autocorrelation, Variogram, Geostatistics, A. Moran

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