This set of functions implements an nlme library pdMat class to allow
tensor product smooths to be estimated by lme as called by gamm. Tensor product smooths
have a penalty matrix made up of a weighted sum of penalty matrices, where the weights are the smoothing
parameters. In the mixed model formulation the penalty matrix is the inverse of the covariance matrix for
the random effects of a term, and the smoothing parameters (times a half) are variance parameters to be estimated.
It's not possible to transform the problem to make the required random effects covariance matrix look like one of the standard
pdMat classes: hence the need for the pdTens class. A notLog2 parameterization ensures that
the parameters are positive.
These functions (pdTens, pdConstruct.pdTens,
pdFactor.pdTens, pdMatrix.pdTens, coef.pdTens and summary.pdTens)
would not normally be called directly.
pdTens(value = numeric(0), form = NULL,
nam = NULL, data = sys.frame(sys.parent()))Initialization values for parameters. Not normally used.
A one sided formula specifying the random effects structure. The formula should have
an attribute S which is a list of the penalty matrices the weighted sum of which gives the inverse of the
covariance matrix for these random effects.
a names argument, not normally used with this class.
data frame in which to evaluate formula.
A class pdTens object, or its coefficients or the matrix it
represents or the factor of
that matrix. pdFactor returns the factor as a vector (packed
column-wise) (pdMatrix always returns a matrix).
If using this class directly note that it is worthwhile scaling the
S matrices to be of `moderate size', for example by dividing each
matrix by its largest singular value: this avoids problems with lme
defaults (smooth.construct.tensor.smooth.spec does this automatically).
This appears to be the minimum set of functions required to implement a new pdMat class.
Note that while the pdFactor and pdMatrix functions return the inverse of the scaled random
effect covariance matrix or its factor, the pdConstruct function is
sometimes initialised with estimates of the scaled covariance matrix, and
sometimes intialized with its inverse.
Pinheiro J.C. and Bates, D.M. (2000) Mixed effects Models in S and S-PLUS. Springer
The nlme source code.