# 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.