Environmental Engineering Reference
In-Depth Information
The most commonly used method of point pattern analysis is Ripley's
K
statistic [
72
]. New statistics have also been developed to compute local estimates
[
68
]. Ripley's
K
statistic, and derived statistics, can be applied in one-, two-, or
three-dimensional space to compare the degree of aggregation of points [
68
]. In
cases where two (or more) point processes are operating, it may be of interest to
assess whether one process influences the other and whether points of different
types tend to cluster together. Examples of ecologically relevant spatial point
processes include fire occurrence, plant occurrences [
67
], or the distribution of
animal nesting or denning sites [
72
]. The bivariate, or cross-
K
, Ripley's
K
test
assesses whether the co-occurrence of two types of points is clustered together
more or less than is expected by chance [
58
]. Using this technique, Lynch and
Moorcroft [
73
] examined co-occurrence of fire and insect outbreaks and found
that contrary to expectation, insect-caused forest mortality does not increase the
risk of forest fire.
Spatial Autocorrelation
Often, a researcher is interested in determining the scale and strength of spatial
autocorrelation of a variable as well as whether there is a directional trend in the
data (i.e., anisotropy). This can be achieved using spatial autocorrelation
coefficients such as Moran's
I
which computes the product of the deviations of
the values of the variable to its average according to various distance intervals (lags,
classes) standardized by the variance at that spatial lag [
7
]. Moran's
I
behaves like
a Pearson's correlation coefficient such that the null hypothesis is the absence of
spatial autocorrelation, positive autocorrelation (mostly a short distance) indicates
that values have comparable values, while negative values indicate that the values
are very dissimilar. Moran's
I
assumes that the underlying process is the same over
the entire study area (i.e., stationarity). Hence, the spatial autocorrelation
coefficients computed at various distance classes are average values. Spatial auto-
correlation in this sense can be referred to as a global spatial statistic that describes
an attribute of the data over the entire study area [
7
]. Significance of each coeffi-
cient can be computed based on an asymptotic
t
or randomization procedure. In
either case, stationarity is required. Moran's
I
is very sensitive to skewed data as the
mean will be biased and in consequence all the deviations values based on it will
also be biased. It is therefore recommended to check the distribution of the data
before computing spatial autocorrelation and if needed transform the data to obtain
a symmetric distribution.
Measures of spatial autocorrelation are also sensitive to sample size. When
autocorrelation is estimated using too few locations, (e.g.,
30 positions), spatial
patterns may not be detected, even though present. Similarly, depending on the
spacing among sampling locations, there will be different numbers of paired
comparisons at each lag distance. Typically, there are few pairs at short distances
due to the edge effects of the edge of the study area (no locations outside to
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