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