Hypervolume metric

Computes the hypervolume metric with respect to a given reference point assuming minimization of all objectives.

Usage

hypervolume(data, reference, maximise = FALSE)

Arguments

data

(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() | logical(1))
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

A single numerical value.

Details

The algorithm 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.

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. IEEE Press, Piscataway, NJ. 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 .

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