Agriculture Reference
In-Depth Information
X
X
w
ij
y
i
y
ð
Þ y
j
y
n
i
j
X
X
X
I
¼
i
,
j
¼ 1,
...
,
n
;
ð
1
:
36
Þ
2
w
ij
ð
y
i
y
Þ
i
j
i
where
n
is the number of locations under investigation,
y
i
is the observed variable,
y
is the mean of
y
, and
w
ij
is the generic element of the weight matrix W. This matrix
describes the observations that are in the neighborhood set of each location, and its
dimension is equal to the number of observations (
n
). Each row and column of
W corresponds to an observation pair (
i
,
j
). The elements,
w
ij
, assume a non-zero
value (1 for a binary matrix, but any other positive value for general weights) when
locations
i
and
j
are neighbors, and a zero value otherwise. By convention, the
diagonal elements of the weights matrix
w
ii
are set to zero. Furthermore, for ease of
interpretation, the weights matrix is often standardized such that the elements of a
row sum to one. Further details can be found in LeSage and Pace (
2009
).
Global Moran
s
I
index can be interpreted as a measure of correlation between
y
and its lagged value (the mean of the values observed in the neighborhood). If there
is no spatial autocorrelation, the mean of
I
is given by
'
1
n
1
:
EðÞ
¼
ð
1
:
37
Þ
Values larger than
E
(
I
) indicate positive spatial autocorrelation, with the size of the
value indicating the strength. Conversely, if
I
<
E
(
I
) there is negative spatial
autocorrelation.
The most common assumption (when determining the distribution of a test for
spatial autocorrelation under the null hypothesis) is that the data follow an
uncorrelated Gaussian distribution. Based on the properties of this distribution,
the moments of the
I
statistic under the null hypothesis can be analytically derived
(Cliff and Ord
1981
). Moreover, by applying an appropriate central limit theorem,
the statistic itself can be shown to tend to a Normal distribution. Thus, the statistic is
standardized by subtracting its expected value and dividing the result by the
corresponding standard deviation. The resulting values can be compared with a
table of standard Normal variates to assess the significance.
The
R
package for the computing Moran
s
I
index is spdep. In the following
code, we have tested the global autocorrelation using the eire data set, which is
available in the spdep package. The eire data set is composed of 26 observations
and 9 variables. We consider the OWNCONS variable, which represents the percent-
age of a country
'
'
s gross agricultural output that is consumed by itself.
>
library(spdep)
>
data(eire)
>
attach(eire.df)
>
eire.listw
<
- nb2listw(eire.nb, style
¼
"W")
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