Geography Reference
In-Depth Information
the nearest-neighbour distances are computed for randomly selected locations and
not for the point event locations. h e
F
function is given by:
#(
d
(
x
,
X
)
<
d
)
(7.11)
min
i
F d
()
=
m
where
d
min
(
x
i
,
X
) is the minimum distance from the location
x
i
in the randomly selected
set of locations to the nearest event in the point pattern,
X
, and
m
is the number of
randomly selected locations. Both
G
and
F
can be plotted against
d
to allow explora-
tion of changes in clustering with distance. h e
G
and
F
functions may vary, for exam-
ple, for a clustered pattern, values of
G
may be larger than values of
F
for small distances
as in the latter case only a random sample is used and the impact of clusters may be
diminished (Bailey and Gatrell, 1995; O'Sullivan and Unwin, 2002). One benei t of the
F
function is that the sample size
m
can be varied, with larger values of
m
giving a
smoother curve. Comparison of the
G
and
F
functions for the same point pattern may
be informative since the two functions bring out dif erent characteristics of the point
pattern (O'Sullivan and Unwin, 2002).
h e most widely used measure of spatial dependence in point patterns is the
K
function, which is the subject of the following section.
7.4.2
K
function
Whereas the
G
and
F
functions are based on nearest neighbours, the
K
function
is based on distances between all events in the region of study. h e
K
function for
distance
d
can be given by:
A
n
ˆ
()
Â
Kd
=
#( ( , ))
C
x
d
(7.12)
i
n
2
i
=
1
As dei ned previously, #(
C
(
x
i
,
d
)) indicates the number (#) of events in the circle
C
(
x
i
,
d
), which has as its centre the location
x
i
and radius
d
, and |
A
| indicates the area
of the study region. Recall that the hat (^) indicates that it is an estimate.
h e
K
function can be computed by following several steps:
1.
Go to an event and count the number of other events within a set radius of the
event.
Do the same for all other events, adding the number of points within that set
2.
radius to the number of points in that radius for all events visited.
Once all events have been visited, the total counts within the distance band,
3.
d
,
is scaled (multiplied) by
A
n
.
Increase the radius a i xed amount and go back to step 1, repeating the process
4.
to the maximum desired radius.
As an example, for PP1 (Figure 7.1) there were i ve events within 5 units of the i rst
event visited, so the unscaled version of
K
(5) = 5. h ere were four events within 5 units
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