nlme (version 3.1-86)

pdNatural: 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

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

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

Run this code
pdNatural(diag(1:3))

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