Geography Reference
In-Depth Information
Central cell Cells excluded
Other cells included in calculation
Queen's case
contiguity
Rook's case
contiguity
Figure 4.9 Queen's case contiguity and rook's case contiguity.
case on the right, the neighbours of most cells have the same value and, therefore, the
values are positively spatially autocorrelated. h e only exceptions are the cells in the
middle two columns of the grid, which have dif erent values.
Using a small grid of values, Moran's I is illustrated below. Note that the method is
equally applicable to zones with irregular forms. h e sample grid is:
78 1
11
9
10
11
12
9
Values that are next to one another along rows or columns (e.g. 7 and 8 or 9 and 10)
will be counted as neighbours as will those that are next to one another diagonally (e.g.
8 and 10). As noted above, this is called queen's case contiguity.
First we will calculate (
- - —that is, the dif erence of each value from the
mean multiplied by the dif erence between each neighbouring value and the mean.
For example, the value 7 (top let cell) minus the mean (9.778) is -2.778. One of the
neighbours of this cell has the value 8 and its dif erence from the mean is -1.778. We
then multiply the two dif erences together, giving 4.938 (see the last entry of the top
row in Table 4.3). h is is done for every cell and its neighbours, as shown in Table 4.3.
Note that Table 4.3 includes only cells that are neighbours (so the weight in each case
is 1). h e sum of the products,
yyy y
)(
)
i
j
 Â
n
n
wy yy y
(
-
)(
-
)
, is 3.975.
ij
i
j
i
=
1
j
=
1
2
i y - , the squared dif erence between each value and the
mean. h e result s a re shown in Table 4.4. h e sum of squared dif erences from the
mean (
Next we will calculate
(
)
n
= Â ) is 21.556.
h ere are nine observations,
2
1 (
yy
)
i
i
n
n
ÂÂ , the sum of
squared dif erences from the mean is 21.556 and there are 20 adjacencies (the number
of rows in Table 4.3 is twice the number of adjacencies). As an example, the cells with
values 7 and 8 are neighbours (i.e. they are adjacent to one another). Each adjacency
(like all the others) is counted twice as we have 7 paired with 8 and 8 paired with 7.
wy yy y
(
-
)(
-
)
=
3.975
ij
i
j
i
=
1
j
=
1
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