##' Second Main function of the CBM-RV package
##'
##' The CBM.RV.null function is the second main function of the CBM-RV
##' used package.
##'
##' @title CBM.RV.null main function
##' @param y matrix of K phenotypes data (one row-entry per individual), of
##' row dimension \eqn{n}, with \eqn{n} the number of subjects)
##' @param covariates matrix of covariates NOT-including intercept (dimension:
##' \eqn{n \times p}, with \eqn{p} the number of covariates)
##' @param kin represents the kinship matrix (dimension:
##' \eqn{n \times n})
##' @return A list containing varaince components estimates under the null model. The
##' output list contains the following arguments:
##' \itemize{
##' \item \code{inv.sqrt.Omega0}: the inverse of the phenotypes varainces-covariances matrix under the null model
##' \item \code{PolygenicVC}: the estimate of the polygenic varainces-covariances matrix
##' \item \code{Env.VC}: the estimate of the Environmental varainces-covariances matrix
##' \item \code{U}: the orthogonal matrix obtained form the spectral decomposition of the knship matrix: kinship = U S U^t
##' \item \code{S}: the diagonal matrix obtained form the spectral decomposition of the knship matrix: kinship = U S U^t
##' }
##' @author Karim Oualkacha
##' @export
CBM.RV.Null <- function(Y,
covariates = covariates,
kin = kin,
nb.running.guess = 3){
#---- need to estimate parameters/VCs under the null model----#
ss = eigen(kin)
U = ss$vectors
S = ss$values
vc.res = REML.VCs.under.null.BFGS(Y, covariates = covariates, U=U, S=S, nb.running.guess = 3)
inv.sqrt.Omega0<-eigen.Omega0(vc.res$Polygenic.VC, vc.res$Env.VC, kin)
res.null <- list(inv.sqrt.Omega0 = inv.sqrt.Omega0, PolygenicVC = vc.res$Polygenic.VC, Env.VC = vc.res$Env.VC, U=U, S=S)
return(res.null)
}
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