# 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
ASReml User Guide.
`phi` affects the rate of decay of the correlation with increasing
distance. The parameter `nugt0` 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 `ν=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 MAT`k`
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 lt alpha lt pi/2`
and `delta gt 0`,
or by constraining `0 lt alpha < pi` and
either `0 lt delta lt 1`
or `delta gt 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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