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Theoretical and Computational Details of the Semivariogram
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Let
be a spatial random field (SRF) with
measured values
at respective locations
,
. You use the VARIOGRAM procedure because you want to gain insight into the spatial continuity and structure of
. A good measure of the spatial continuity of
is defined by means of the variance of the difference
, where
and
are locations in
. Specifically, if you consider
and
to be spatial increments such that
, then the variance function based on the increments
is independent of the actual locations
,
. Most commonly, the continuity measure used in practice is one half of this variance, better known as the semivariance function:
or, equivalently,
The plot of semivariance as a function of
is the semivariogram. In extension to its meaning, you might commonly see the term semivariogram used instead of the term semivariance, as well.
Assume that the SRF
is free of nonrandom (or systematic) surface trends. Then, the expected value
of
will be a constant for all
, and the semivariance expression is simplified to the following:
Given the preceding assumption, you can compute an estimate
of the semivariance
from a finite set of points in a practical way by using the formula
where the sets
contain all the neighboring pairs at distance
:
and
is the number of such pairs
.
The expression for
is called the empirical semivariance (Matheron; 1963). This is the quantity that PROC VARIOGRAM computes, and its corresponding plot is the empirical semivariogram. According to Cressie (1993, p. 96), the estimate
has approximate variance
The empirical semivariance
is also referred to as classical. This name is used so that it can be distinguished from the robust semivariance estimate
and the corresponding robust semivariogram. The robust semivariance was introduced by Cressie and Hawkins (1980) and is described by Cressie (1993, p. 75) as:
In the preceding expression the parameter
is defined as:
Note that if your data include a surface trend, then the empirical semivariance
is not an estimate of the theoretical semivariance function
. Instead, rather than the spatial increments variance, it represents a different quantity known as pseudo-semivariance, and its corresponding plot is a pseudo-semivariogram. In principle, pseudo-semivariograms do not provide measures of the spatial continuity. They can thus lead to misinterpretations of the
spatial structure, and are consequently unsuitable for the purpose of spatial prediction. For further information, see the detailed discussion in the section Empirical Semivariograms and Surface Trends. Under certain conditions you might be able to gain some insight about the spatial continuity with a pseudo-semivariogram. This case is presented in Analysis without Surface Trend Removal.