General Positive-Definite Matrix in Natural Parameterization

DESCRIPTION:

This function is a constructor for the pdSymmNat class, representing a general positive-definite matrix, using a natural parameterization . 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 ij-th 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, if unconstrained = TRUE, the natural parameters are given by log(Sii), i=1,..,n and log((1+r(i,j))/(1-r(i,j))), i not equal to j. Note that, in this case, all natural parameters are individually unrestricted, but not jointly unrestricted (meaning that not all unrestricted vectors would give positive-definite matrices). If `unconstrained = FALSE', the S(i,j) are used to parameterize the returned object. This class does not define an unrestricted parameterization and, therefore, 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:

pdSymmNat(value, form, nam, data, unconstrained) 

OPTIONAL ARGUMENTS:

value
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.
form
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.
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 value is a vector of character strings. Defaults to NULL.
data
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.
unconstrained
an optional logical value indicating whether an unconstrained natural parameterization should be used. Defaults to TRUE.

VALUE:

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

SEE ALSO:

,

EXAMPLES:

pdSymmNat(diag(1:3))