Variance Models
Matern class
ASReml uses an extended
Matern class which accomodates geometric anisotropy and a choice
of metrics for
random fields observed in two dimensions.
This extension, described in detail in Haskard (2006), is parameterised
with five parameters:
phi (range) and nu (smoothness) control the shape of the correlation function
delta (distortion scaling ratio) and alpha (angle of rotation) governs geometric
anisotropy, and
lambda (1=cityblock, 2=euclidean) specifies the choice of metric.
The algebraic details are available in the
User Guide
phi affects the rate of decay of the correlation with increasing
distance. The parameter nu>0 controls the analytic smoothness of the
underlying process.
Larger nu correspond to smoother
processes. ASReml uses numerical derivatives for nu when its current value is outside the interval [0.2,5].
When nu=m+0.5
with m a non-negative integer,
the correlation function is the product of an exponential model and a
polynomial of degree m.
Thus nu=0.5 yields
the exponential correlation function,
and \nu=1 yields Whittle's
elementary correlation function,
(Webster and Oliver, 2001).
nu = 1.5
generates the correlation function of a random field which is
continuous and once differentiable.
As nu approaches infinity, the correlation function
tends to the gaussian correlation function.
The metric parameter lambda is not estimated by ASReml; it is usually set
to 2 for Euclidean distance. Setting lambda=1 provides the
cityblock metric, which together with nu=0.5 models a separable
AR1.AR1 process. Cityblock metric may be appropriate when the dominant
spatial processes are aligned with rows/columns as occurs in field experiments.
Geometric
anisotropy is discussed in most geostatistical books (Webster and Oliver, 2001,
Diggle et al, 2003)
but rarely are the anisotropy angle or ratio estimated
from the data. Similarly the smoothness parameter nu is often set
a-priori (Kammann and Wand, 2003, Diggle etal, 2003).
However Stein (1999)
and Haskard (2006)
demonstrate that nu can be reliably estimated even
for modest sized data-sets, subject to caveats regarding the
sampling design.
Syntax
The syntax for the Matern class in ASReml is given by MATk
where k is the number of parameters to be
specified; the remaining parameters take their default values.
Use the !G qualifier to control whether a specified parameter
is estimated or fixed.
The order of the parameters in ASReml, with their defaults, is
(phi, nu=0.5, delta=1, alpha=0, lambda=2). For example, if we
wish to fit a Matern model with only phi estimated and the
other parameters set at their defaults then we use MAT1.
MAT2
allows nu to be estimated or fixed at some other value
(for example MAT2 .2 1 !GPF).
The parameters phi
and nu
are highly correlated so it may be better to manually cover a grid of nu values.
We note that there is non-uniqueness in the anisotropy parameters
of this metric
since inverting delta and adding
pi/2
to alpha
gives the same distance. This non-uniqueness can be removed by
constraining 0 < alpha < pi/2
and delta > 0,
or by constraining 0 < alpha < pi and
either 0 < delta < 1
or delta > 1.
With lambda=2, isotropy occurs when
delta=1, and then the rotation angle
alpha is irrelevant:
correlation contours are circles, compared with ellipses in general.
With lambda=1, correlation contours are diamonds.
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