Computes the hypervolume contribution of each point of a set of points with
respect to a given reference point. Duplicated and dominated points have
zero contribution. By default, dominated points are ignored, that is, they
do not affect the contribution of other points. See the Notes below for
more details. For details about the hypervolume, see hypervolume().
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.- ignore_dominated
logical(1)
Whether dominated points are ignored when computing the contribution of nondominated points. The value of this parameter has an effect on the return values only if the input contains dominated points. Setting this toFALSEslows down the computation significantly. See the Notes below for a detailed explanation.
Value
numeric()
A numerical vector
Details
The hypervolume contribution of point \(\vec{p} \in X\) is defined as \(\text{hvc}(\vec{p}) = \text{hyp}(X) - \text{hyp}(X \setminus \{\vec{p}\})\). This definition implies that duplicated points have zero contribution even if not dominated, because removing one of the duplicates does not change the hypervolume of the remaining set. Moreover, dominated points also have zero contribution. However, a point that is dominated by a single (dominating) nondominated point reduces the contribution of the latter, because removing the dominating point makes the dominated one become nondominated.
Handling this special case is non-trivial and makes the computation more
expensive, thus the default (ignore_dominated=TRUE) ignores all dominated
points in the input, that is, their contribution is set to zero and their
presence does not affect the contribution of any other point. Setting
ignore_dominated=FALSE will consider dominated points according to the
mathematical definition given above, but the computation will be slower.
When the input only consists of mutually nondominated points, the value of
ignore_dominated does not change the result, but the default value is
significantly faster.
The current implementation uses a \(O(n\log n)\) dimension-sweep
algorithm for 2D. With ignore_dominated=TRUE, the 3D case uses the HVC3D
algorithm (Guerreiro and Fonseca 2018)
, which has \(O(n\log n)\) complexity.
Otherwise, the implementation uses the naive algorithm that requires
calculating the hypervolume \(|X|+1\) times.
References
Andreia P. Guerreiro, Carlos M. Fonseca (2018). “Computing and Updating Hypervolume Contributions in Up to Four Dimensions.” IEEE Transactions on Evolutionary Computation, 22(3), 449–463. doi:10.1109/tevc.2017.2729550 .
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))
#> [1] 0 3 3 0 0
# hvc[(5,1)] = 0 = duplicated
# hvc[(1,5)] = 3 = (4 - 1) * (6 - 5)
# hvc[(4,2)] = 3 = (5 - 4) * (5 - 2)
# hvc[(4,4)] = 0 = dominated
# hvc[(5,1)] = 0 = duplicated
hv_contributions(x, reference=c(6,6), ignore_dominated = FALSE)
#> [1] 0 3 2 0 0
# 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))
#> [1] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [8] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [15] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [22] 0.000 8.197 0.000 0.000 0.000 0.000 0.000
#> [29] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [36] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [43] 0.000 0.000 0.000 0.000 0.000 0.000 7959.940
#> [50] 1945.800 8147.132 0.000 0.000 0.000 0.000 0.000
#> [57] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [64] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [71] 26.255 0.000 0.000 0.000 0.000 0.000 0.000
#> [78] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [85] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [92] 0.000 3698.640 0.000 0.000 5.971 0.000 0.000
#> [99] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [106] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [113] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [120] 0.000 3069.000 779.240 0.000 0.000 0.000 0.000
#> [127] 0.000 0.000 0.000 0.000 0.000 41994.755 0.000
#> [134] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [141] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [148] 0.000 0.000 0.000 0.000 0.000 0.000 0.000
#> [155] 0.000 0.000 0.000 0.000 0.000 2294.064 0.000
#> [162] 0.000 0.000 0.000 0.000 0.000
# 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))
#> [1] 89283.920 255278.978 8.197 2242.660 7959.940 1945.800
#> [7] 8147.132 73.054 26.255 3698.640 5.971 193143.324
#> [13] 3069.000 779.240 41994.755 2294.064