Geography Reference
In-Depth Information
h e right-hand side (RHS) of Equation 7.8 is (1 -
d
i
2
/
t
2
)
2
. Taking one distance value
as an example (0.927) this leads to:
2
2
Ê
0.927
2
ˆ
0.85933
Ê
ˆ
2
1
-
=
1
-
=
0.99863
=
0.99726
Á
˜
Á
˜
2
Ë
¯
Ë
25
¯
625
Tables 7.3 (for point pattern A) and 7.4 (for point pattern B) show the
x
and
y
coordinates of each event, the distance of the event from the window centre, and com-
puted values of the RHS of Equation 7.8 followed by the LHS of the equation (which
has a value of 0.00152 for all cases given a bandwidth of 25 units) multiplied by the
RHS. h e sum of products in this last column is also given.
h e dif erence in the estimated intensities for the two methods will decrease as the
size of the bandwidth increases. h is is indicated by Figure 7.11, which shows the KE
values for bandwidths of 5, 10, 15, 20, 25, and 30 units. Note how for point pattern A,
which is visibly more clustered around the centre of the window, the KE value is much
larger for a bandwidth of i ve units than is the value for point pattern B, but this
dif erence decreases as the size of the kernel bandwidth is increased.
Table 7.3
Point pattern A: intensity estimate using the quartic kernel
with a bandwidth of 25 units (distance is from the centre of the kernel)
x
y
Distance
RHS
LHS
¥
RHS
-16.670
-2.156
0.927
0.997
0.00152
-19.794
-1.079
2.378
0.982
0.00150
-15.377
-5.496
4.235
0.943
0.00144
-14.084
1.507
4.832
0.927
0.00142
-21.087
1.615
4.960
0.923
0.00141
-18.717
-8.082
6.330
0.876
0.00134
-25.613
-5.389
8.802
0.767
0.00117
-8.912
-6.035
9.592
0.727
0.00111
Sum
0.01091
Table 7.4
Point pattern B: intensity estimate using the quartic kernel
with a bandwidth of 25 units (distance is from the centre of the kernel)
x
y
Distance
RHS
LHS
¥
RHS
-19.687
-1.079
2.276
0.983
0.00150
-4.603
2.153
13.554
0.499
0.00076
-5.572
-12.392
15.948
0.352
0.00054
-23.565
-19.180
18.334
0.214
0.00033
-35.309
-8.513
18.966
0.180
0.00028
-14.731
17.776
19.838
0.137
0.00021
-29.815
14.220
20.225
0.119
0.00018
3.801
-3.449
21.409
0.071
0.00011
Sum
0.00390
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