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.
pdNatural(value, form, nam, data)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.
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.
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.
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.
a pdNatural object representing a general positive-definite
matrix in natural parametrization, also inheriting from class
pdMat.
Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models in S and S-PLUS", Springer, esp. p. 162.
# NOT RUN {
pdNatural(diag(1:3))
# }
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