pdNatural {nlme}  R Documentation 
This function is a constructor for the pdNatural
class,
representing a general positivedefinite 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 S(i,j) denote the ijth
element of the underlying positive definite matrix and
r(i,j) =
S(i,j)/sqrt(S(i,i)S(j,j)), i not equal to j denote the associated
"correlations", the "natural" parameters are given by
sqrt(Sii), i=1,..,n and
log((1+r(i,j))/(1r(i,j))), i not equal to j. Note that all
natural parameters are individually unrestricted, but not jointly
unrestricted (meaning that not all unrestricted vectors would give
positivedefinite 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 onesided
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 positivedefinite matrix.
pdNatural(value, form, nam, data)
value 
an optional initialization value, which can be any of the
following: a 
form 
an optional onesided 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 positivedefinite
matrix and unreplicated elements. This argument is ignored when

data 
an optional data frame in which to evaluate the variables
named in 
a pdNatural
object representing a general positivedefinite
matrix in natural parametrization, also inheriting from class
pdMat
.
JosÃ© Pinheiro and Douglas Bates bates@stat.wisc.edu
Pinheiro, J.C., and Bates, D.M. (2000) "MixedEffects Models in S and SPLUS", Springer, esp. p. 162.
as.matrix.pdMat
,
coef.pdMat
,
pdClasses
,
matrix<.pdMat
pdNatural(diag(1:3))