Geography Reference
In-Depth Information
of the second event visited, so the unscaled version of K (5) becomes 9 and so on until
all events have been visited.
For PP2 (Figure 7.2), the total number of events within 5 units of all events was zero
(i.e. there were no other events within 5 units of any event) and for PP1 it was 122. As
noted above, for each distance, the count is multiplied by A
n . h e area of the study
region is 6665.744 square units and there are 55 events in both PP1 and PP2. For PP1
ˆ (5)
K
is therefore obtained from:
6665.755
¥=
122
268.833
2
55
and for PP2, ˆ (5)
K
is:
6665.755
¥=
00
2
55
h e expected value of K is given by:
2
lp
d
(7.13)
EK d
[( ]
=
=
p
d
2
l
where l is the average intensity of the point pattern (but is not needed to calculate the
expected value of K ). Equation 7.13 gives the expected value for a point pattern that is
said to be the outcome of a CSR process (see Section 7.3.1). A value of K ( d ) greater
than the expected value (i.e. pd 2 ) suggests a clustered point pattern, while a value of
K ( d ) less than the expected value suggests a regular point pattern. In practice, the L
function, a transformed version of the K function, is ot en computed. h e L function
for distance d is given by:
ˆ ()
Kd
ˆ ()
(7.14)
Ld
=
-
d
p
Values of the L function of less than 0 indicate regularity, while values greater than
0 suggest clustering. h e K function is illustrated in Figure 7.13 using the point pat-
terns illustrated in Figures 7.1 (PP1) and 7.2 (PP2), with the expected values ( E ( K ))
also plotted. In this example, the smallest distance band was 2.5 units and this was
increased in steps of 2.5 units to a maximum of 25 units. h e L function is illustrated
in Figure 7.14.
Note that the values of K for PP1 are clearly larger than those for PP2 and the
expected value for smaller (£ 17.5 units) distances, while the expected values are larger
than those for PP1 for distances of 20 and greater. In contrast, values for PP2 are
clearly smaller than the values for both PP1 and the expected values up to a distance
band of 20 units. In words, PP1, dei ned as a 'clustered' point pattern, is indeed clustered
at short distances, while PP2, dei ned as a 'regular' point pattern, is dispersed at small
 
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