Geography Reference
In-Depth Information
Spatial dependence and spatial autocorrelation
4.8
A key concern in spatial data analysis is to examine spatial patterning in the variable
or variables of interest. For example, are values of a particular variable large in some
areas and small in others? Also, do similar values tend to cluster or are values visually
erratic? h e term 'spatial dependence' refers to the dependence of neighbouring val-
ues on one another (Haining, 2003). As outlined at the start of this chapter, the basis
of spatial dependence is that values close together in space tend to be more similar
than those that are farther apart. h e 'i rst law of geography' (Tobler, 1970) is a key
concept in geography in general and spatial data analysis in particular. In the context
of statistical measurement, this idea is related to spatial autocorrelation—the degree
to which a variable is spatially correlated with itself. A measure of spatial autocorrela-
tion may suggest spatial dependence (i.e. neighbouring values are similar—positive
spatial autocorrelation) or spatial independence (neighbouring values are dissimilar—
negative spatial autocorrelation).
h ere is a range of measures of spatial autocorrelation. h e joins count approach is
one means of summarizing the tendency of neighbouring observations to be the same
(O'Sullivan and Unwin, 2002). h e measure of spatial autocorrelation encountered
most frequently in the spatial analysis literature is the
I
coei cient proposed by Moran
(Moran, 1950; Clif and Ord, 1973). It is given by
n
n
ÂÂ
Â
n
w
(
y
-
y
)(
y
-
y
)
ij
i
j
(4.5)
i
=
1
j
=
1
I
=
n
ÂÂ
n
n
2
(
yy
-
)
w
i
ij
i
=
1
i
=
1
j
=
1
where the values
y
i
(of which there are
n
) have the mean
y
and the proximity between
locations
i
and
j
is given by
w
ij
. As before, this is a geographical weight and is ot en set
to 1 when locations
i
and
j
are neighbours and 0 when they are not. Note that here
y
is
a data value and not a coordinate. Elsewhere in this topic,
z
is used to represent data
values but
y
is used here to distinguish the use of
z
as a deviation of
y
from its mean in
the local spatial autocorrelation measures detailed in Section 8.4.1. Equation 4.5
includes double summations (note that single summation was introduced with respect
to Equation 3.1), that is:
n
n
ÂÂ
i
=
1
j
=
1
h is means start with
i
= 1 and
j
= 1, next,
i
= 1 and
j
= 2 then
i
= 1 and
j
= 3, and so on
until
j
=
n
. At er that point,
i
= 2 and we work through all values of
j
until all combina-
tions of
i
and
j
have been accounted for. At each stage, the computed values are added
to the values obtained previously. In this way all combinations of
i
and
j
are included.
With the numerator of Equation 4.5
n
n
ÂÂ
n
w
(
y
-
y
)(
y
-
y
)
ij
i
j
i
=
1
j
=
1
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