pdNatural {nlme}

General Positive-Definite Matrix in Natural Parametrization

Description

This function is a constructor for the pdNatural class, representing a general positive-definite matrix, using a natural parametrization . If the matrix associated with object is of dimension n, it is represented by n(n+1)/2 parameters. Letting \sigma_{ij} denote the ij-th element of the underlying positive definite matrix and \rho_{ij}=\sigma_{i}/\sqrt{\sigma_{ii}\sigma_{jj}},\;i\neq j denote the associated "correlations", the "natural" parameters are given by \sqrt{\sigma_{ii}}, \;i=1,\ldots,n and \log((1+\rho_{ij})/(1-\rho_{ij})),\; i \neq j. Note that all natural parameters are individually unrestricted, but not jointly unrestricted (meaning that not all unrestricted vectors would give positive-definite matrices). Therefore, this parametrization should NOT be used for optimization. It is mostly used for deriving approximate confidence intervals on parameters following the optimization of an objective function. When value is numeric(0), an uninitialized pdMat object, a one-sided formula, or a vector of character strings, object is returned as an uninitialized pdSymm object (with just some of its attributes and its class defined) and needs to have its coefficients assigned later, generally using the coef or matrix replacement functions. If value is an initialized pdMat object, object will be constructed from as.matrix(value). Finally, if value is a numeric vector, it is assumed to represent the natural parameters of the underlying positive-definite matrix.

Usage

pdNatural(value, form, nam, data)

Arguments

value

an optional initialization value, which can be any of the following: a pdMat object, a positive-definite matrix, a one-sided linear formula (with variables separated by +), a vector of character strings, or a numeric vector. Defaults to numeric(0), corresponding to an uninitialized object.

form

an optional one-sided linear formula specifying the row/column names for the matrix represented by object. Because factors may be present in form, the formula needs to be evaluated on a data.frame to resolve the names it defines. This argument is ignored when value is a one-sided formula. Defaults to NULL.

nam

an optional vector of character strings specifying the row/column names for the matrix represented by object. It must have length equal to the dimension of the underlying positive-definite matrix and unreplicated elements. This argument is ignored when value is a vector of character strings. Defaults to NULL.

data

an optional data frame in which to evaluate the variables named in value and form. It is used to obtain the levels for factors, which affect the dimensions and the row/column names of the underlying matrix. If NULL, no attempt is made to obtain information on factors appearing in the formulas. Defaults to the parent frame from which the function was called.

Value

a pdNatural object representing a general positive-definite matrix in natural parametrization, also inheriting from class pdMat.

Author(s)

José Pinheiro and Douglas Bates [e-mail]

References

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer, esp. p. 162.

See Also

as.matrix.pdMat, coef.pdMat, pdClasses, matrix<-.pdMat

Examples

pdNatural(diag(1:3))