hv_contributions {moocore} | R Documentation |
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 |
|
reference |
|
maximise |
|
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
Carlos M. Fonseca, Luís Paquete, Manuel López-Ibáñez (2006). “An improved dimension-sweep algorithm for the hypervolume indicator.” In Proceedings of the 2006 Congress on Evolutionary Computation (CEC 2006), 1157–1163. doi:10.1109/CEC.2006.1688440.
Nicola Beume, Carlos M. Fonseca, Manuel López-Ibáñez, Luís Paquete, Jan Vahrenhold (2009). “On the complexity of computing the hypervolume indicator.” IEEE Transactions on Evolutionary Computation, 13(5), 1075–1082. doi:10.1109/TEVC.2009.2015575.
See Also
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))