gd: (Inverted) Generational Distance and Average Hausdorff...
In jakobbossek/ecr3vis: Evolutionary Computation in R - Version 3

Description Usage Arguments Details Value References See Also

These functions expect as mandatory arguments a point set X = \{x_1, …, x_{|X|}\} (parameter x) and and a set of reference points Y = \{y_1, …, y_{|Y|}\} (parameter y). For details on the optional argument p and modified see the section on details.

gd calculates the Generational Distance (GD).
igd calculates the Inverse Generational Distance (IGD).
gdp/igdp compute the (I)GD+ indicator [3].
ahd calculates the Average Hausdorff Distance.

gd(x, y, p = 2, modified = TRUE, ...)

igd(x, y, p = 2, modified = TRUE, ...)

gdp(x, y, p = 2, modified = TRUE, ...)

igdp(x, y, p = 2, modified = TRUE, ...)

ahd(x, y, p = 2, modified = TRUE, ...)

`x`	[`matrix`] Numeric matrix of points (each colum contains one point).
`y`	[`matrix`] The reference set as a numeric matrix of points (each colum contains one point).
`p`	[`numeric(1)`] Parameter p (see description).
`modified`	[`logical(1)`] Should the modified GD/IGD calculation by Schuetze et al. [2] be used? Default is `TRUE`.
`...`	[any] Not used.

The Generational Distance (GD) measures the distance of a point set X = \{r_1, …, r_{|X|}\}, e.g., a Pareto-front approximation, to a reference set R = \{r_1, …, r_{|R|}\}. Then GD is defined as

GD_p(A, R) = \frac{1}{|X|} ≤ft(∑_{i=1}^{|X|} d_i^p\right)^{1/p}

where d_i is the Euclidean distance of point x_i \in X to its nearest neigbor point in R. The Inverted Generational Distance works the other way around, i.e.,

IGD_p(A, R) = \frac{1}{|R|} ≤ft(∑_{i=1}^{|R|} \hat{d}_i^p\right)^{1/p}

where \hat{d}_i is the respective nearest neighbor distance of r_i to any point in X. Put differently, IGD_p(A, R) = GD_p(R, A). Functions gd and igd calcute these versions.

Schütze et al. [2] proposed a slight modification:

GD_p(A, R) = ≤ft(\frac{1}{|X|} ∑_{i=1}^{|X|} d_i^p\right)^{1/p}

where the average is taken before the power operation. IGD_p is apdated analogeously. This versions are calclated by gd and igd if the argument modified is set to TRUE.

Ishibushi et al. [3] proposed another modification which works on the formulation by Schütze et al. (see above). They modified the distance calculation:

GD_p^{+}(A, R) = ≤ft(\frac{1}{|X|} ∑_{i=1}^{|X|} d^{+^p}_i\right)^{1/p}

where d_i^{+} = \max\{x_i, z_i\}. This version can be calculated with the function gdp (the trailing p stands for “plus”).

Eventuelly, the function ahd calculates the Average Hausdorff Distance [2] which combines GD and IGD and is defined as

Δ_p(A, R) = \max\{GD_p(A, R), IGD_p(A, R)\}.

By default, ahd uses the modified versions of GD and IGD respectively (see argument modified).

IGDX [4] is a meaasure for decision space diversity. This is simply IGD; however, the input consists of the non-dominated solutions in decision space rather in objective space. Naturally, all implemented functions can be used as an “*X” version.

Single numeric indicator value.

[1] David A. Van Veldhuizen and David A. Van Veldhuizen. Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. Technical Report, Evolutionary Computation, 1999.

[2] Schütze, O., Esquivel, X.,Lara,A. ,Coello, C.A.C.: Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE Transactions on Evolutionary Computation 16, 504–522 (2012).

[3] Hisao Ishibuchi, Hiroyuki Masuda, Yuki Tanigaki, and Yusuke Nojima. Modified distance calculation in generational distance and inverted generational distance. In António Gaspar-Cunha, Carlos Henggeler Antunes, and Carlos Coello Coello, editors, Evolutionary Multi-Criterion Optimization, 110–125. Cham, 2015. Springer International Publishing.

[4] O. Schütze, M. Vasile, and C. A. C. Coello, Computing the Set of Epsilon-Efficient Solutions in Multiobjective Space Mission Design,

Other multi-objective performance indicators: cov(), df_get_indicators(), eps(), hv(), os(), r1(), rse()