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

vorob_t(x, sets, reference, maximise = FALSE)

vorob_dev(x, sets, reference, ve = NULL, maximise = FALSE)

Arguments

x

matrix()|data.frame()
Matrix or data frame of numerical values, where each row gives the coordinates of a point. If sets is missing, the last column of x gives the sets.

sets

integer()
A vector that indicates the set of each point in x. If missing, the last column of x is used instead.

reference

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

maximise

logical()
Whether the objectives must be maximised instead of minimised. Either a single logical value that applies to all objectives or a vector of logical values, with one value per objective.

ve

matrix()
Vorob'ev expectation, e.g., as returned by vorob_t().

Value

vorob_t returns a list with elements threshold, ve, and avg_hyp (average hypervolume)

vorob_dev returns the Vorob'ev deviation.

References

Mickaël Binois, David Ginsbourger, Olivier 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.

Ilya Molchanov (2005). Theory of Random Sets. Springer.

Author

Mickael Binois

Examples

data(CPFs)
res <- vorob_t(CPFs, reference = c(2, 200))
res$threshold
#> [1] 44.14062
res$avg_hyp
#> [1] 8943.333
# Now print Vorob'ev deviation
vd <- vorob_dev(CPFs, ve = res$ve, reference = c(2, 200))
vd
#> [1] 3017.13