Exploring spatial point patterns - Spatial Data Analysis: An Introduction for GIS Users - page 92

h is supports the notion of PP1 exhibiting clustering (as the value is greater than 1),
while visual inspection of PP2 suggests dispersal (with a value of less than 1). h is
supports the impression gained from inspection of Table 7.1. In addition, the ana-
lysis using Moran's I coei cient, summarized above, reached similar conclusions. Yet
another way of assessing the degree to which a point pattern conforms to CSR is to
conduct a chi-square ( c 2 ) test (O'Sullivan and Unwin, 2002). h is is given by the
variance (without division by n ) of the quadrat counts divided by the mean:

= Â

2

( k - m )

2 k

c

(7.4)

m

k

where m k is the mean quadrat count (the estimate of l ). h e calculations for the
variances of PP1 and PP2 are given in Table 7.2.
Note that k and m are the same for PP1 and PP2. h e c 2 statistic for PP1 and PP2 is
then calculated:

131.786
0.786

2

PP1: c = =167.666

29.786
0.786

2

PP2: c = = 37.895

h e number of degrees of freedom (see Section 3.2) is given by the number of quad-
rats (70) minus 1, i.e. 69. Examination of a table of critical values of c 2 (e.g. see Ebdon
(1985), Appendix C) indicates that the value for PP1 is signii cant to greater than the
0.001 level, while the value for PP2 is not signii cant even at the 0.1 level. In other
words, in the case of PP1 we can reject the null hypothesis that the point pattern was
generated by CSR. In the case of PP2 we cannot reject the null hypothesis (see Section
3.4 for a discussion on a related topic).
h e following section builds on the account of geographical weighting schemes in
Section 4.7, but in a point pattern analysis context.

Table 7.2 Quadrat counts for PP1 and PP2: derivation of the variances

k PP1# PP2# PP1 and PP2

( k - m )

PP1 and PP2
( k - m ) 2

PP1
# ( k - m ) 2

PP2
# ( k - m ) 2

0 42 24 - 0.786 0.617796 25.947432 14.827104

1 17 37 0.214 0.045796 0.778532 1.694452

2 5 9 1.214 1.473796 7.368980 13.264164

3 1 0 2.214 4.901796 4.901796 0.000000

4 2 0 3.214 10.329796 20.659592 0.000000

5 1 0 4.214 17.757796 17.757796 0.000000

6 2 0 5.214 27.185796 54.371592 0.000000

Total 70 70 131.785720 29.785720