Hypervolume metric

Compute the hypervolume metric with respect to a given reference point assuming minimization of all objectives. For 2D and 3D, the algorithm used FonPaqLop06:hypervolume,BeuFonLopPaqVah09:tec has \(O(n \log n)\) complexity. For 4D or higher, the algorithm FonPaqLop06:hypervolume has \(O(n^{d-2} \log n)\) time and linear space complexity in the worst-case, but experimental results show that the pruning techniques used may reduce the time complexity even further.

Usage

hypervolume(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(1)
A single numerical value.

References

Author

Manuel López-Ibáñez

Examples

data(SPEA2minstoptimeRichmond)
# The second objective must be maximized
# We calculate the hypervolume of the union of all sets.
hypervolume(SPEA2minstoptimeRichmond[, 1:2], reference = c(250, 0),
            maximise = c(FALSE, TRUE))
#> [1] 7911376