R/CovarianceFunctions.R

In DepthProc: Statistical Depth Functions for Multivariate Analysis

#' @title CovLP
#'
#' @description This class, derived from the virtual class "CovRobust" accomodates weighted by \eqn{ L ^ p } depth multivariate location and scatter estimator.
#'
#' @details 
#' 
#' See \code{\link{CovLP}} for the function used to calculate weighted by \eqn{ L ^ p } depth covariance matrix.
#' 
#' @export
#'
setClass("CovDepthWeighted", 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} ^ {p} dim } depth
#' @param la parameter of a simple weight function \eqn{ w = ax + b }
#' @param lb parameter of a simple weight function \eqn{ w = ax + 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
#' # EXAMPLE 1
#' 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)
  depth_params <- list(method = "LP", pdim = pdim, la = la, lb = lb)
  center <- depthMedian(x, depth_params)

  method <- "Depth Weighted Estimator"
  new("CovDepthWeighted", cov = cov, center = center, det = det(cov),
      n.obs = nrow(x), X = x, method = method, call = match.call())
}