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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.

Value

numeric()
A numerical vector

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().

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

Author

Manuel López-Ibáñez

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 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     4.380     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  6397.052
#>  [50]  1945.800  3386.197     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    15.840     0.000     0.000     0.066     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 12428.431     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