hv_contributions: Hypervolume contribution of a set of points

In moocore: Core Mathematical Functions for Multi-Objective Optimization

Hypervolume contribution of a set of points

Description

Computes the hypervolume contribution of each point of a set of points with respect to a given reference point. The hypervolume contribution of point \vec{p} \in X is \text{hvc}(\vec{p}) = \text{hyp}(X) - \text{hyp}(X \setminus \{\vec{p}\}). Dominated points have zero contribution but they may influence the contribution of other points. Duplicated points have zero contribution even if not dominated, because removing one of the duplicates does not change the hypervolume of the remaining set.

Usage

hv_contributions(x, reference, maximise = FALSE)

Arguments

x	matrix()|data.frame() Matrix or data frame of numerical values, where each row gives the coordinates of a point.
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.

Details

The current implementation uses the O(n\log n) dimension-sweep algorithm for 2D and the naive algorithm that requires calculating the hypervolume |X|+1 times for dimensions larger than 2.

For details about the hypervolume, see hypervolume().

Value

numeric()
A numerical vector

Author(s)

Manuel López-Ibáñez

References

\insertRef

FonPaqLop06:hypervolumemoocore

\insertRef

BeuFonLopPaqVah09:tecmoocore

See Also

hypervolume()

Examples

x <- matrix(c(5,1, 1,5, 4,2, 4,4, 5,1), ncol=2, byrow=TRUE)
hv_contributions(x, reference=c(6,6))
# hvc[(5,1)] = 0 = duplicated
# hvc[(1,5)] = 3 = (4 - 1) * (6 - 5)
# hvc[(4,2)] = 2 = (5 - 4) * (4 - 2)
# hvc[(4,4)] = 0 = dominated
# hvc[(5,1)] = 0 = duplicated

data(SPEA2minstoptimeRichmond)
# The second objective must be maximized
# We calculate the hypervolume contribution of each point of the union of all sets.
hv_contributions(SPEA2minstoptimeRichmond[, 1:2], reference = c(250, 0),
            maximise = c(FALSE, TRUE))

# Duplicated points show zero contribution above, even if not
# dominated. However, filter_dominated removes all duplicates except
# one. Hence, there are more points below with nonzero contribution.
hv_contributions(filter_dominated(SPEA2minstoptimeRichmond[, 1:2], maximise = c(FALSE, TRUE)),
                 reference = c(250, 0), maximise = c(FALSE, TRUE))

moocore documentation built on Aug. 8, 2025, 6:12 p.m.