1 Spatial Modelling of the Terrestrial Environment - Georeferencial

Characterizing Land Use in Urban Systems via Built-Form Connectivity Models 209
relational graph, {N, r i } ≡ G (Barr and Barnsley, 1997). Non-relational properties of the
regions (e.g. their area and perimeter) are represented as attributes of the nodes, while
relational properties (e.g. distance and cardinal direction (orientation)) between any two
regions are represented as attributes of the edges in the set EP.
Measuring Constellation→Built-Form Spatial Structure. The overall objective of this
study is to quantitatively compare the spatial and morphological structure of different urban
constellations. In this context, it is hypothesized that constellations of the same land-use
category will tend to exhibit similar structural properties, while those of different land-use
categories will tend to exhibit dissimilar properties. As the built-form connectivity model
used here is limited to Constellation → Built-Form Unit relationships, we are unable to
employ the set of 12 structural measures originally proposed by Kruger (1979a) to evaluate
the degree of structural similarity between constellations. Instead, this is analysed using
SAMS/XRAG, which permits a range of simple descriptive statistics (e.g. the mean and
standard deviation (SD) of built-form unit area), measures of built-form density (e.g. the
number of built-form units per hectare) and spatial statistics (e.g. Moran’s I , a measure of
spatial autocorrelation) to be computed.
More specifically, we use a distance-weighted version of Moran’s I for point samples,
where the graph nodes (i.e., the geographical centroids of the corresponding land-cover
parcels) provide the point data. For the unique, pairwise combinations p of a set of n
observations on variable x, Moran’s I is given by:
I = n ∑ p w ij(x i − ¯x)(x j − ¯x)
( ∑
p w ) ∑ (3)
ij (x − ¯x)
2
where ¯x is the mean of the observations on x and w ij is a weighting factor applied to the
values of x for points i and j (Ebdon, 1985; Fotheringham et al., 2000). Here, w ij is most
commonly expressed as a reciprocal of the Euclidean distance (d) between points i and j
1
(e.g.
d ij
, 1 ,..., 1 ).
dij
2 dij
r
Moran’s I has bounds of −1.0 ≤ I ≤ 1.0. A value of I close to −1.0 indicates that
there is no spatial autocorrelation in terms of the values of variable x, while a value close
to 1.0 indicates that there is strong spatial autocorrelation. Values of I tending to 0.0 are
indicative of a random spatial distribution in the values of x. Assuming that values of x are
randomly distributed spatially, the expected value of I , E I , is given by:
E I = 1
(4)
n − 1
and its SD by:
√
n[(n
σ (E I ) =
2 + 3 − 3n)A + 3B 2 − nC] − k[(n 2 − n)A + 6B 2 − 2nC]
(5)
(n − 1)(n − 2)(n − 3)B 2
where A = ∑ p w2 ij , B = ∑ p w ij and C = ∑ i (∑ j w ij) 2 and k is the kurtosis of x (i.e.,
∑ (x− ¯x) 4
). Finally, the standard normal deviate (z I )ofI is given by:
nσ 4
z I = I − E i
(6)
σ I
If the calculated value of z I falls within the critical values of z I for a particular significance