Geography Reference
In-Depth Information
h e discussion on i rst- and second-order ef ects in Section 7.1 is relevant in this con-
text as it deals with the distinction between variation in the mean and spatial depen-
dence. h is section also has links with the introduction to TPS, whereby the TPS
function (Equation 9.3) is shown to comprise a trend component and the component
(here termed the 'random' part) modelled by the basis function
R
(
x
-
x
i
) (in the present
case, the variogram is used instead to model this component, as described below).
In practical terms, as we estimate parameters, namely the mean and variance, of a
distribution, we estimate parameters of the RF model using the data. h ese parame-
ters, like the mean and variance, summarize the variable. h e mean and variance of a
distribution are useful only if the distribution is approximately normal and, similarly,
the parameters of the RF model are only meaningful in certain conditions. Where the
properties of the variable of interest are the same, or at least similar in some sense,
across the region of interest we can employ a stationary model. In other words, we can
use the same model parameters at all locations. If the properties of the variable are
clearly spatially variable then a standard RF model may not be appropriate. h ere are
dif erent degrees of stationarity, but for present purposes we will only consider one,
intrinsic stationarity. h ere are two requirements of intrinsic stationarity. Firstly, the
mean is constant across the region of interest. In other words, the expected value of the
variable does not depend on the location,
x
:
(9.7)
EZ
{
(
x
)}
=
m
(
x
) for all
x
h e mean is therefore assumed to be the same for all locations. Secondly, the expected
squared dif erence between paired RFs (i.e. the observations) (summarized by the
variogram, g(
h
)) should depend only on the separation distance and direction (the lag
h
)
between the RFs and not on the location of the RFs:
1
(9.8)
2
g
(
h
)
=
E[{
ZZ
(
x
)
-
(
x
+
h
)} ] for all
h
2
where
x
+
h
indicates a distance (and direction)
h
from location
x
.
In terms of the data, the expected semivariance should be the same for all observa-
tions separated by a particular lag, irrespective of where the paired observations are
located. In practical terms, the geostatistical approach can be applied irrespective of
these conditions, but the results will clearly be suboptimal if the data depart markedly
from the conditions. In some cases the mean is allowed to vary from place to place, but
stay constant within a moving window. h is is known as quasi stationarity (Webster
and Oliver, 2007).
9.7.1
Variogram
Analysis of the degree to which values dif er according to how far apart they are can be
conducted by computing the variogram (or semivariogram). With reference to the
variogram, the term 'lag' is used to describe the distance and direction by which
Search WWH ::
Custom Search