pdNatural {nlme} | R Documentation |
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 |
form |
an optional one-sided linear formula specifying the
row/column names for the matrix represented by |
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
|
data |
an optional data frame in which to evaluate the variables
named in |
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 bates@stat.wisc.edu
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))