R/fts.freqdom.R

In kidzik/freqdom.fda: Functional Time Series: Dynamic Functional Principal Components

#' Creates an object of class  \code{fts.freqdom}. 
#'
#' This class is used to describe a frequency domain operator (for example a spectral
#' density operator) on selected frequencies. Formally we consider an object of class
#' \code{\link{freqdom}} and add some basis functions. Depending on the context, we
#' have different interpretations for the new object.
#' 
#' (I) In order to define an operator which maps between two functions spaces, the we
#' interpret \code{F$operators} as coefficients in the basis function expansion of
#' the kernel of some finite rank operators \deqn{\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}).} The kernels are \eqn{f_k(u,v)=\boldsymbol{b}_1^\prime(u)\, F_k\, \boldsymbol{b}_2(v)}, where \eqn{\boldsymbol{b_1}(u)=(b_{11}(u),\ldots, b_{1d_1}(u))^\prime} and \eqn{\boldsymbol{b_2}(u)=(b_{21}(u),\ldots, b_{2d_1}(u))^\prime} are the basis functions provided by the arguments  basisX and basisY, respectively. Moreover, we consider frequencies \eqn{\{\omega_1,\ldots, \omega_K\}\subset[-\pi,\pi]}. The object this function creates corresponds to the mapping \eqn{\omega_k \mapsto f_k(u,v)}.
#' 
#' (II) We may ignore basisX, and represent the linear mapping
#' \deqn{\mathcal{F}_k:\mathrm{span}(\code{basisY})+\mathrm{i}\, \mathrm{span}(\code{basisY})\to C^{d_1},}
#' by considering \eqn{f_k(v):=F_k\,\boldsymbol{b}_2(v)} and \eqn{\mathcal{F}_k(x)=\int f_k(v)x(v)dv}.
#' 
#' (III) We may ignore basisY, and represent the linear mapping
#' \deqn{\mathcal{F}_k: C^{d_1}\to\mathrm{span}(\code{basisX})+\mathrm{i}\, \mathrm{span}(\code{basisX}),}
#' by considering \eqn{f_k(u):=\boldsymbol{b}_1^\prime(u)F_k} and \eqn{\mathcal{F}_k(y)=f_k(u)y}.
#' 
#' @title Creates an object of class  \code{fts.freqdom}. 
#' 
#' @param F an object of class freqdom.
#' @param basisX an object of class \code{basis.fd} (see \code{\link[fda]{create.basis}})
#' @param basisY an object of class \code{basis.fd} (see \code{\link[fda]{create.basis}})
#' @return Returns an object of class  \code{fts.freqdom}. An object of class
#' \code{fts.freqdom} is a list containing the following components:
#' * \code{operators} \eqn{\quad} returns the array \code{F$operators}.
#' * \code{basisX} \eqn{\quad} returns \code{basisX} as given in the argument.
#' * \code{basisY} \eqn{\quad} returns \code{basisY} as given in the argument.
#' * \code{freq} \eqn{\quad} returns the vector \code{F$freq}.
#' @seealso The multivariate equivalent in the \code{freqdom} package: \code{\link[freqdom]{freqdom}}
#' @export
#' @import fda
#' @import freqdom
#' @keywords classes
fts.freqdom = function (F, basisX, basisY=basisX)
{
	
	if(!is.freqdom(F)) 
	stop("F must be an object of class freqdom")  

	if(!is.basis(basisX))
	stop("basisX must be a functional basis")  

	if(!is.basis(basisY))
	stop("basisY must be a functional basis")  

	res = F
	res$basisX=basisX
	res$basisY=basisY
	class(res) = c("fts.freqdom","freqdom")
	res
}