Myriam Elizabeth Saavedra López - Repositorio Digital USFQ ...

for r ≥ 0 and in the general d-dimensional case, g (r) = K′ (r)
dπdrd−1 for r ≥ 0.
When r is large, the lim g (r) = 1, means there is independent between the two compared points
r→∞
for large r. To be more precise, for this equation, the point process must have some distributional
properties (called mixing properties).
If there is a finite distance rcorr with g (r) = 1 for r ≥ rcorr then rcorr is called range of correlation.
This means that there are no correlations between point positions at larger distances.
It is clear that in the Poisson process or CSR case g (r) = 1 for r ≥ 0, i.e. which means that the
location of any point is entirely independent of the locations of the other points. Again, in typical
non-Poisson cases a characteristic behaviour of g (r) may be found in the cluster process if g (r) > 1
and regular process if g (r) < 1 , in particular for small radii in both cases.
The Second Moment Characteristics are related to statistical concepts such as spatial autocorrelation
and semi-variogram because all these functions study the distances (space) between variables (spatial
data) and what is it happening between these variables when these distances are short or long (in-
teraction and/or dependence). Therefore all these functions support the first law of geography that
“everything is related to everything else, but near things are more related than distant things” (Waldo
Tobler). See the following descriptions:
1. Spatial Autocorrelation, according to Zuur, states that pairs of subjects (points) that are close
to each other are more likely to have values that are more similar, and pair of subjects far apart
forms each other are more likely to have values that are less similar. It is important to mention
that the Function Ripley’s K is one of the indices of spatial autocorrelation.
2. According to Cressei (1993, 58), semi-variogram has been called the variogram divided by 2,
by Matheron (1962), therefore they are commonly referenced. The semi-variogram is a plot of
semivariance as a function of distance. The semi-variance measures the dissimilarity of sub-
jects within a single variable, compared to covariance which measures the similarity of one or
more variables. It is not normalized and values are not as constrained as are most correlation
coefficients (Zuur).
3.6 Corrected Estimates
According to Illian et al (2008, 180-183), the spatial statistics analysis with stationary point processes
faces a difficult problem at its window edges: data are given for a bounded observation window W
only, but the pattern is (implicitly) assumed to be infinite and the summary characteristic to be
estimated is defined independently of W and should not show any traces of W . However, natural
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