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#' Compute Vorob'ev threshold, expectation and deviation. Also, displaying the
#' symmetric deviation function is possible. The symmetric deviation
#' function is the probability for a given target in the objective space to
#' belong to the symmetric difference between the Vorob'ev expectation and a
#' realization of the (random) attained set.
#'
#' @title Vorob'ev computations
#' @param x Either a matrix of data values, or a data frame, or a list of data
#' frames of exactly three columns. The third column gives the set (run,
#' sample, ...) identifier.
#' @template arg_refpoint
#' @return `vorobT` returns a list with elements `threshold`,
#' `VE`, and `avg_hyp` (average hypervolume)
#' @rdname Vorob
#' @author Mickael Binois
#' @examples
#' data(CPFs)
#' res <- vorobT(CPFs, reference = c(2, 200))
#' print(res$threshold)
#'
#' ## Display Vorob'ev expectation and attainment function
#' # First style
#' eafplot(CPFs[,1:2], sets = CPFs[,3], percentiles = c(0, 25, 50, 75, 100, res$threshold),
#' main = substitute(paste("Empirical attainment function, ",beta,"* = ", a, "%"),
#' list(a = formatC(res$threshold, digits = 2, format = "f"))))
#'
#' # Second style
#' eafplot(CPFs[,1:2], sets = CPFs[,3], percentiles = c(0, 20, 40, 60, 80, 100),
#' col = gray(seq(0.8, 0.1, length.out = 6)^0.5), type = "area",
#' legend.pos = "bottomleft", extra.points = res$VE, extra.col = "cyan",
#' extra.legend = "VE", extra.lty = "solid", extra.pch = NA, extra.lwd = 2,
#' main = substitute(paste("Empirical attainment function, ",beta,"* = ", a, "%"),
#' list(a = formatC(res$threshold, digits = 2, format = "f"))))
#' @concept eaf
#' @export
vorobT <- function(x, reference)
{
x <- check_eaf_data(x)
setcol <- ncol(x)
nobjs <- setcol - 1L
# First step: compute average hypervolume over conditional Pareto fronts
avg_hyp <- mean(sapply(split.data.frame(x[,1:nobjs], x[, setcol]),
hypervolume, reference = reference))
prev_hyp <- diff <- Inf # hypervolume of quantile at previous step
a <- 0
b <- 100
while (diff != 0) {
c <- (a + b) / 2
eaf_res <- eafs(x[,1:nobjs], x[,setcol], percentiles = c)[,1:nobjs]
tmp <- hypervolume(eaf_res, reference = reference)
if (tmp > avg_hyp) a <- c else b <- c
diff <- prev_hyp - tmp
prev_hyp <- tmp
}
return(list(threshold = c, VE = eaf_res, avg_hyp = avg_hyp))
}
#' @concept eaf
#' @rdname Vorob
#' @return `vorobDev` returns the Vorob'ev deviation.
#' @examples
#'
#' # Now print Vorob'ev deviation
#' VD <- vorobDev(CPFs, res$VE, reference = c(2, 200))
#' print(VD)
#' @export
vorobDev <- function(x, VE, reference)
{
if (is.data.frame(x)) x <- as.matrix(x)
if (is.null(VE)) VE <- vorobT(x, reference)$VE
setcol <- ncol(x)
nobjs <- setcol - 1L
# Hypervolume of the symmetric difference between A and B:
# 2 * H(AUB) - H(A) - H(B)
H2 <- hypervolume(VE, reference = reference)
x.split <- split.data.frame(x[,1:nobjs, drop=FALSE], x[,setcol])
H1 <- mean(sapply(x.split, hypervolume, reference = reference))
hv_union_VE <- function(y)
hypervolume(rbind(y[, 1:nobjs, drop=FALSE], VE), reference = reference)
VD <- 2 * sum(sapply(x.split, hv_union_VE))
nruns <- length(x.split)
return((VD / nruns) - H1 - H2)
}
#' @rdname Vorob
#' @references
#'
#' \insertRef{BinGinRou2015gaupar}{eaf}
#'
#' C. Chevalier (2013), Fast uncertainty reduction strategies relying on
#' Gaussian process models, University of Bern, PhD thesis.
#'
#' I. Molchanov (2005), Theory of random sets, Springer.
#'
#' @param VE,threshold Vorob'ev expectation and threshold, e.g., as returned
#' by [vorobT()].
#' @param nlevels number of levels in which is divided the range of the
#' symmetric deviation.
#' @param ve.col plotting parameters for the Vorob'ev expectation.
#' @param xlim,ylim,main Graphical parameters, see
#' [`plot.default()`][graphics::plot.default()].
#' @param legend.pos the position of the legend, see
#' [`legend()`][graphics::legend()]. A value of `"none"` hides the legend.
#' @param col.fun function that creates a vector of `n` colors, see
#' [`heat.colors()`][grDevices::heat.colors()].
#' @examples
#' # Now display the symmetric deviation function.
#' symDifPlot(CPFs, res$VE, res$threshold, nlevels = 11)
#' # Levels are adjusted automatically if too large.
#' symDifPlot(CPFs, res$VE, res$threshold, nlevels = 200, legend.pos = "none")
#'
#' # Use a different palette.
#' symDifPlot(CPFs, res$VE, res$threshold, nlevels = 11, col.fun = heat.colors)
#' @concept eaf
#' @export
# FIXME: Implement "add=TRUE" option that just plots the lines,points or
# surfaces and does not create the plot nor the legend (but returns the info
# needed to create a legend), so that one can use the function to add stuff to
# another plot.
symDifPlot <- function(x, VE, threshold, nlevels = 11,
ve.col = "blue", xlim = NULL, ylim = NULL,
legend.pos = "topright", main = "Symmetric deviation function",
col.fun = function(n) gray(seq(0, 0.9, length.out = n)^2))
{
# FIXME: These maybe should be parameters of the function in the future.
maximise <- c(FALSE, FALSE)
xaxis_side <- "below"
yaxis_side <- "left"
log <- ""
xlab <- colnames(x)[1]
ylab <- colnames(x)[2]
las <- par("las")
sci.notation <- FALSE
nlevels <- min(length(unique.default(x[, 3])) - 1, nlevels)
threshold <- round(threshold, 4)
seq.levs <- round(seq(0, 100, length.out = nlevels), 4)
levs <- sort.int(unique.default(c(threshold, seq.levs)))
attsurfs <- compute_eaf_as_list(x, percentiles = levs)
# Denote p_n the attainment probability, the value of the symmetric
# difference function is p_n if p_n < alpha (Vorob'ev threshold) and 1 - p_n
# otherwise. Therefore, there is a sharp transition at alpha. For example,
# for threshold = 44.5 and 5 levels, we color the following intervals:
#
# [0, 25) [25, 44.9) [44.9, 50) [50, 75) [75, 100]
#
# with the following colors:
#
# [0, 25) [25, 50) [50, 75) [25, 50) [0, 25)
max.interval <- max(which(seq.levs < max(100 - threshold, threshold)))
colscale <- seq.levs[1:max.interval]
# Reversed so that darker colors are associated to higher values
names(colscale) <- rev(col.fun(max.interval))
cols <- c(names(colscale[colscale < threshold]),
rev(names(colscale[1:max(which(colscale < 100 - threshold))])),
"#FFFFFF") # To have white after worst case
names(levs) <- cols
# FIXME: We should take the range from the attsurfs to not make x mandatory.
xlim <- get_xylim(xlim, maximise[1], data = x[,1])
ylim <- get_xylim(ylim, maximise[2], data = x[,2])
extreme <- get_extremes(xlim, ylim, maximise, log = log)
plot(xlim, ylim, type = "n", xlab = "", ylab = "",
xlim = xlim, ylim = ylim, log = log, axes = FALSE, las = las,
main = main,
panel.first = {
plot_eaf_full_area(attsurfs, extreme = extreme, maximise = maximise, col = cols)
# We place the axis after so that we get grid lines.
plot_eaf_axis (xaxis_side, xlab, las = las, sci.notation = sci.notation)
plot_eaf_axis (yaxis_side, ylab, las = las, sci.notation = sci.notation,
line = 2.2)
plot_eaf_full_lines(list(VE), extreme, maximise,
col = ve.col, lty = 1, lwd = 2)
})
# Use first.open to print "(0,X)", because the color for 0 is white.
intervals <- seq_intervals_labels(seq.levs, first.open = TRUE)
intervals <- intervals[1:max.interval]
names(intervals) <- names(colscale)
#names(intervals) <- names(colscale[1:max.interval])
if (is.na(pmatch(legend.pos, "none")))
legend(legend.pos, legend = c("VE", intervals), fill = c(ve.col, names(intervals)),
bg="white", bty="n", xjust=0, yjust=0, cex=0.9)
box()
}
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