Vorob'ev computations

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.

Usage

vorobT(x, reference)

vorobDev(x, VE, reference)

symDifPlot(
  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)
)

Arguments

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.

reference

(numeric())
Reference point as a vector of numerical values.

VE, threshold

Vorob'ev expectation and threshold, e.g., as returned by vorobT().

nlevels

number of levels in which is divided the range of the symmetric deviation.

ve.col

plotting parameters for the Vorob'ev expectation.

xlim, ylim, main

Graphical parameters, see plot.default().

legend.pos

the position of the legend, see legend(). A value of "none" hides the legend.

col.fun

function that creates a vector of n colors, see heat.colors().

Value

vorobT returns a list with elements threshold, VE, and avg_hyp (average hypervolume)

vorobDev returns the Vorob'ev deviation.

References

M Binois, D Ginsbourger, O Roustant (2015). “Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations.” European Journal of Operational Research, 243(2), 386–394. doi:10.1016/j.ejor.2014.07.032 .

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.

Author

Mickael Binois

Examples

data(CPFs)
res <- vorobT(CPFs, reference = c(2, 200))
print(res$threshold)
#> [1] 44.14062

## 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"))))


# Now print Vorob'ev deviation
VD <- vorobDev(CPFs, res$VE, reference = c(2, 200))
print(VD)
#> [1] 3017.13
# Now display the symmetric deviation function.
symDifPlot(CPFs, res$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)