Geography Reference
In-Depth Information
Negative spatial autocorrelation
I
= 1.000
Positive spatial autocorrelation
I
= 0.865
Figure 4.8
Spatial autocorrelation: rook's case contiguity.
Black cells have a value of 1, white cells have a value of 0.
the calculations are computed for every combination of
i
and
j
and the results of each
calculation are added together, with the product being multiplied by the weight
w
ij
.
h e procedure to calculate the Moran's
I
value is demonstrated below. Negative values
of
I
indicate negative spatial autocorrelation—neighbouring values tend to be dif erent.
Positive values of
I
indicate positive spatial autocorrelation—neighbouring values
tend to be similar. Values of
I
close to zero indicate that there is no structure. h is section
i rst outlines the basic concepts, then gives a small, fully worked numerical example,
and i nally gives an example using a larger grid.
Values of
I
for two dif erent grids are given in Figure 4.8. h ese examples were
computed using the package GeoDa (Anselin
et al.
, 2006) and it should be noted that
the results are slightly dif erent to those that would be calculated using Equation 4.5
as GeoDa modii es the form of the weights (in that package the weights are what
are termed 'row standardized, i.e. they sum to 1), but that isn't a concern here. In this
example, black cells have a value of 1 while white cells have a value of 0. For this
example, cells which share an edge with another cell are compared and not cells which
share only corners. h rough the analogy with movement of pieces in chess, this is
called rook's case contiguity. Where cells which share corners (i.e. cells connected
diagonally) are also included, this is called queen's case contiguity. Rook's case and
queen's case contiguity are illustrated in Figure 4.9. In the case of irregularly shaped
zones, rook's case contiguity and queen's case contiguity can also be used, with the lat-
ter including zones that are connected only by vertices as well as by edges, while the
former includes only zones joined by edges. In packages such as GeoDa, dif erent
weighting functions (e.g. rook's case and queen's case) can be used and it is necessary
to consider which is used and how this may impact on the results. As well as rook and
queen contiguity, other weighting schemes can be used (see Section 4.7).
h e case on the let of Figure 4.8 indicates negative spatial autocorrelation—all
neighbours of a given cell are, using rook's case contiguity, dif erent to that cell. In the
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