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#' @title CovLP
#'
#' @export
#'
setClass("CovDepthWeighted", representation(depth = "character"), contains="CovRobust")
#'@title CovLp
#'
#'@description Weighted by \eqn{L^p} depth (outlyingness) multivariate location and scatter estimators.
#'
#' @param x The data as a matrix or data frame. If it is a matrix or data frame, then each row is viewed as one multivariate observation.
#' @param pdim The parameter of the weighted \eqn{L^pdim} depth
#' @param la parameter of a simple weight function w=a*x+b
#' @param lb parameter of a simple weight function w=a*x+b
#'
#'
#' @return loc: Robust Estimate of Location:
#' @return cov: Robust Estimate of Covariance:
#' @return Returns depth weighted covariance matrix.
#'
#'
#' @details
#'
#' Using depth function one can define a depth-weighted location and scatter estimators. In case of location estimator we have \deqn{ L(F)={\int{{x}{{w}_{1}}(D({x},F))dF({x})}}/{{{w}_{1}}(D({x},F))dF({x})}, } Subsequently, a depth-weighted scatter estimator is defined as \deqn{ S(F)=\frac{\int{({x}-L(F)){{({x}-L(F))}^{T}}{{w}_{2}}(D({x},F))dF({x})}}{\int{{{w}_{2}}(D({x},F))dF({x})}}, } where \eqn{ {{w}_{2}}(\cdot ) } is a suitable weight function that can be different from \eqn{ {{w}_{1}}(\cdot ) } .
#'
#' The \pkg{DepthProc} package offers these estimators for weighted \eqn{ {L}^{p} } depth. Note that \eqn{ L(\cdot ) } and \eqn{ S(\cdot ) } include multivariate versions of trimmed means and covariance matrices. Their sample counterparts take the form \deqn{ {{T}_{WD}}({{{X}}^{n}})={\sum\limits_{i=1}^{n}{{{d}_{i}}{{X}_{i}}}}/{\sum\limits_{i=1}^{n}{{{d}_{i}}}} , } \deqn{ DIS({{{X}}^{n}})=\frac{\sum\limits_{i=1}^{n}{{{d}_{i}}\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right){{\left( {{{X}}_{i}}-{{T}_{WD}}({{{X}}^{n}}) \right)}^{T}}}}{\sum\limits_{i=1}^{n}{{{d}_{i}}}}, } where \eqn{ {{d}_{i}} } are sample depth weights, \eqn{ {{w}_{1}}(x)={{w}_{2}}(x)=x } .
#'
#' @author Daniel Kosiorowski and Zygmunt Zawadzki from Cracow University of Economics.
#'
#' @export
#' @seealso \code{\link{depthContour}} and \code{\link{depthPersp}} for depth graphics.
#'
#' @examples
#' x = mvrnorm(n = 100, mu = c(0,0), Sigma = 3*diag(2))
#' cov_x = CovLP(x, 2, 1, 1)
#'
#' # EXAMPLE 2
#' data(under5.mort,inf.mort,maesles.imm)
#' data1990 = na.omit(cbind(under5.mort[,1],inf.mort[,1],maesles.imm[,1]))
#' CovLP(data1990)
#'
#'
#' @keywords
#' multivariate
#' nonparametric
#' robust
#' depth function
#'
CovLP = function(x, pdim=2, la=1, lb=1)
{
if(is.data.frame(x)) x = as.matrix(x)
cov = CovLPCPP(x, pdim, la, lb)
center = depthMedian(x, method = "LP", pdim = pdim, la = la, lb = lb)
method = "Depth Weighted Estimator"
new("CovDepthWeighted", cov = cov,
center = center,
det = det(cov),
n.obs = nrow(x),
X = x,
method = method,
call = match.call())
}
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