hypervolume {moocore}

Hypervolume metric

Description

Compute the hypervolume metric with respect to a given reference point assuming minimization of all objectives. For 2D and 3D, the algorithm used (Fonseca et al. 2006; Beume et al. 2009) has O(n \log n) complexity. For 4D or higher, the algorithm (Fonseca et al. 2006) 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.

Author(s)

Manuel López-Ibáñez

References

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.

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.

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))