Environmental Engineering Reference
In-Depth Information
compare too), most of the pairs at intermediate distances and very fewer at large
distances (because of the overall size of the study area). To mitigate these unequal
numbers of pair per distance lag, it is recommended to focus on lags distances equal
to the length of the first half (up to two thirds) of the smallest edge of the study area
as most spatial autocorrelation occurs at short distances and that the probability of
detecting it is also highest in the first spatial lag [
7
].
A plot of spatial autocorrelation coefficients against distance lags is called
a spatial correlogram. If the lags are based on distance only, the correlogram is
said to be an omnidirectional correlogram. When the data contain directionality, the
omnidirectional correlogram cannot reveal it and may, in fact, “mask” it. To detect
the presence of anisotropic spatial pattern (i.e., not having the same sill and range
according to direction), the samples need to be divided by distance class as well as
direction angle range (usually 0
,45
,90
,135
) to produce a set of directional
correlograms.
In areas where several processes influence ecological data, Moran's
I
that
assumes stationarity cannot be used. Instead, local indicator of spatial aggregation
statistics, LISA (e.g., local Moran, local Getis), can be used as they are computed at
each sampling location and allow the identification of subareas that have similar
high (“hot spots”) or low (“cold spots”) values [
7
].
Geostatistics
Spatial structure can be determined in terms of spatial autocorrelation as presented
above or as spatial variance according to distance as computed using variograms
which are part of the family of spatial statistical methods known as geostatistics [
7
,
58
]. Variograms represent a global method of scale-specific analysis that has been
used extensively in ecology to analyze spatial patterns [
58
]. Variograms model the
relationship between lag distance and semivariance and can be calculated using
continuous raster or point data. Semivariance is calculated as the sum of the squared
differences between pairs of locations separated by a given lag distance divided by
twice the number of pairs of locations at that particular lag distance [
7
]. From the
observed, or empirical, variogram, three parameters can be estimated to fit
a theoretical variogram: (1) range, or scale at which distance does not affect the
estimate of variance, (2) sill, or the variance of the data, and (3) nugget effect,
which represents the variability in the data that is not accounted for by spatial
structure [
58
]. Theoretical variogram models can identify whether there is
a directional trend in the data, that is, anisotropy as the spatial autocorrelation
values. In addition to describing the attributes of spatial structure, geostatistical
models can be used to “krige” (i.e., spatially interpolate) data [
58
] and to simulate
spatial patterns using a chosen variogram model, process model, and parameter
estimates [
74
].
