Variance Models

Converting Correlation matrices to variance matrices

The base correlation identifiers are converted to the corresponding homogeneous and heterogeneous variance models by appending V and H to the base identifiers respectively. This convention holds for most models. However, no V or H should be appended to the base identifiers for the other variance models (from DIAG on).

In summary, to specify
  • a correlation model, use the base identifier, for example
    EXP .1 is an exponential correlation model,
  • an homogeneous variance model, append a V to the base identifier and provide an additional initial value for the variance, for example,
    EXPV .1 .3 is an exponential variance model,
  • a heterogeneous variance model, append an H to the base identifier and provide additional initial values for the diagonal variances, for example, 3 0 CORUH .1 .3 .4 .2 is a 3 x 3 matrix with uniform correlations of 0.1 and heterogeneous variances 0.3, 0.4 and 0.2.

    There are important considerations when combining variance matrices to define a variance structure.

    The algebraic forms of the homogeneous and heterogeneous variance models are determined as follows. Let C denote the correlation matrix for a particular correlation model. Then S=vC is the corresponding homogeneous variance matrix where v is the variance parameter. For example, the homogeneous variance model corresponding to the ID correlation model has variance matrix S=vI specified IDV in the ASReml command file) and one parameter.

    The initial values for the variance parameters are listed after the initial values for the correlation parameters. For example, in AR1V 0.3 0.5, 0.3 is the initial spatial correlation parameter and 0.5 is the initial variance parameter value. p Similarly, the heterogeneous variance matrix corresponding to C$ is S=DCD where D is a diagonal matrix whose values are the square roots of the variance parameters. For example, the heterogeneous variance model corresponding to ID is IDH which is the same as the DIAG structure and has variance matrix S=DID. Similarly, the CORGH model is equivalent to a US model although parameterized differently,

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