R/deepreg2d.R

In zzawadz/DepthProc: Statistical Depth Functions for Multivariate Analysis

#' @title Simple deepest regression method.
#' @description This function calculates deepest regression estimator for simple regression.
#'
#' @param x Independent variable.
#' @param y Dependent variable.
#'
#' @details
#' Function originates from an original algorithm proposed by Rousseeuw and Hubert. Let \eqn{ {Z} ^ {n} = {(x_1, y_1), ..., (x_n, y_n)} \subset {{R} ^ {d}} } denotes a sample considered from a following semiparametric model: \eqn{ {{y}_{l}} = {{a}_{0}} + {{a}_{1}}{{x}_{1l}} + ... + {{a}_{(d - 1)l}}{{x}_{(d - 1)l}} + {{\varepsilon }_{l}}, {l = 1, ..., n} }, we calculate a depth of a fit \eqn{ \alpha = (a_{0}, ..., a_{d - 1}) } as \eqn{ RD(\alpha, {{Z} ^ {n}}) = {u \ne 0}\min\sharp {l: \frac{ {{r}_{l}}(\alpha) }{ {{u} ^ {T}}{{x}_{l}} } < 0, l = 1, ..., n} }, where \eqn{ r(\cdot) } denotes the regression residual, \eqn{ \alpha = (a_{0}, ..., a_{d - 1}) }, \eqn{ {u} ^ {T}{x}_{l} \ne 0 }. \code{The deepest regression estimator} \eqn{ DR(\alpha, {{Z} ^ {n}}) } is defined as \eqn{ DR(\alpha, {{Z} ^ {n}}) = {\alpha \ne 0}\,\arg\max\,RD(\alpha, {{Z} ^ {n}}) }
#'
#' @references
#' Rousseeuw J.P., Hubert M. (1998), Regression Depth, \emph{Journal of The American Statistical Association}, vol.94.
#'
#' @author Daniel Kosiorowski, Mateusz Bocian, Anna Wegrzynkiewicz and Zygmunt Zawadzki from Cracow University of Economics.
#'
#' @examples
#'
#' # EXAMPLE 1
#' data(pension)
#' plot(pension)
#' abline(
#'   lm(Reserves ~ Income, data = pension),
#'   lty = 3,
#'   lwd = 2) # lm
#' abline(
#'   deepReg2d(pension[, 1], pension[, 2]),
#'   lwd = 2) # deepreg2d
#'
#' # EXAMPLE 2
#' data(under5.mort)
#' data(inf.mort)
#' data(maesles.imm)
#' data2011 <- na.omit(
#'     cbind(under5.mort[, 22], inf.mort[, 22],
#'     maesles.imm[, 22]))
#'
#' x <- data2011[, 3]
#' y <- data2011[, 2]
#' plot(
#'   x, y,
#'   cex = 1.2,
#'   ylab = "infant mortality rate per 1000 live birth",
#'   xlab = "against masles immunized percentage",
#'   main = "Projection Depth Trimmed vs. LS regressions"
#' )
#' abline(lm(x ~ y), lwd = 2, col = "black") # lm
#' abline(
#'   deepReg2d (x, y),
#'   lwd = 2, col = "red"
#' ) # trimmed reg
#'
#' legend(
#'   "bottomleft",
#'   c("LS", "DeepReg"),
#'   fill = c("black", "red"),
#'   cex = 1.4,
#'   bty = "n"
#' )
#' @export
#'
deepReg2d <- function(x, y) {
  y <- y[order(x)]
  x <- sort(x)
  tmp <- depth2dcpp(x, y)
  new("DeepReg2d", coef = tmp[2:1], depth = tmp[3])
}