Banded Positive-Definite Matrix in Natural Parameterization
DESCRIPTION:
This function is a constructor for the
pdBandNat class,
representing a banded positive-definite matrix, using a natural
parameterization . If the matrix associated with
object is of
dimension n and order r, it is represented by
r*(2*n-r+1)/2 parameters. Letting
S(i,j) denote the ij-th element of the
underlying positive definite matrix (abs(i-j) < r)
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))), abs(i-j) < r. 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
pdBand 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.
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.
ord
an optional integer specifying the order of the banded
matrix represented by the returned
pdBandNat object, defined as
the number of nonzero diagonals in the upper-triangular part of the
matrix. Defaults to 2.
unconstrained
an optional logical value indicating whether an
unconstrained natural parameterization should be used. Defaults to
TRUE.
VALUE:
a
pdBandNat object representing a banded positive-definite
matrix in natural parameterization, also inheriting from class
pdMat
.