Geography Reference
In-Depth Information
gives the predicted fraction of quadrats containing
k
events for a mean intensity
l
(O'Sullivan and Unwin (2002) provide another account). In Figures 7.5 and 7.6 there
are 70 quadrats and 55 events. In both cases, the mean quadrat count (the mean intensity)
is
55
/
70
= 0.786. As an example, the probability of there being three events in a quadrat
is given by:
3
-
0.786
0.786
e
0.485588
¥
0.455664
0.221265
P
(3)
=
=
=
=
0.036877
3!
3
¥¥
2
1
6
Events like the number of points in quadrats can be modelled using the Poisson
distribution and a point pattern with a Poisson distribution may be described as con-
i rming to complete spatial randomness (CSR)—that is, there is no apparent structure.
Table 7.1 gives quadrat counts (given by #) for PP1 and PP2, the counts expressed as
proportions and
P
(
k
). Visual inspection of the two sets of fractions suggests that the
fractions for PP2 are much closer to those predicted given the Poisson distribution
than are those for PP1.
Following on from the discussion above, the VMR is expected to have a value of 1
if the distribution is Poisson—a value greater than 1 indicates clustering while a value
of less than 1 indicates a dispersed point pattern. Recall that, for the examples in
Figures 7.5 and 7.6, there are 70 quadrats and 55 events, giving a mean quadrat count
of
55
/
70
= 0.786 in both cases. For the point pattern in Figure 7.5 (PP1), the variance
(i.e. the mean squared dif erence between each quadrat count and the mean quadrat
count) is
131.786
/
70
= 1.883. For the point pattern in Figure 7.6 (PP2), the variance is
29.786
/
70
= 0.426. h e variance/mean ratios are then given by:
1.883
0.786
= 2.396
0.426
0.786
=
0.542
Table 7.1
Quadrat counts for PP1 and PP2, fractions and
P
(
k
)
k
PP1#
PP2#
PP1 fraction
PP2 fraction
P
(
k
)
0
42
24
0.600
0.343
0.45566381
1
17
37
0.243
0.529
0.35815175
2
5
9
0.071
0.129
0.14075364
3
1
0
0.014
0.000
0.03687745
4
2
0
0.029
0.000
0.00724642
5
1
0
0.014
0.000
0.00113914
6
2
0
0.029
0.000
0.00014923
7
0
0
0.000
0.000
0.00001676
8
0
0
0.000
0.000
0.00000165
9
0
0
0.000
0.000
0.00000014
10
0
0
0.000
0.000
0.00000001
Search WWH ::
Custom Search