Description Usage Arguments Details Value Author(s) References Examples

Functions to compute the inverted generational distance (IGD and IGD+) and the averaged Hausdorff distance between nondominated sets of points.

1 2 3 4 5 |

`data` |
( |

`reference` |
( |

`maximise` |
( |

`p` |
( |

The generational distance (GD) of a set *A* is defined as the distance
between each point *a \in A* and the closest point *r* in a
reference set *R*, averaged over the size of *A*. Formally,

*GD(A,R) = (1/|A|) * ( sum_{a in A} min_{r in R} d(a,r)^p )^(1/p)*

where the distance in our implementation is the Euclidean distance:

*d(a,r) = sqrt( sum_{k=1}^M (a_k - r_k)^2)*

The inverted generational distance (IGD) is calculated as *IGD_p(A,R) = GD_p(R,A)*.

The modified inverted generational distanced (IGD+) was proposed by
\citetIshMasTanNoj2015igd to ensure that IGD+ is weakly Pareto compliant,
similarly to `epsilon_additive()`

or `epsilon_mult()`

. It modifies the
distance measure as:

*d^+(r,a) = sqrt(sum_{k=1}^M (max {r_k - a_k, 0 })^2)*

The average Hausdorff distance (*Δ_p*) was proposed by
\citetSchEsqLarCoe2012tec and it is calculated as:

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

IGDX \citepZhoZhaJin2009igdx is the application of IGD to decision vectors
instead of objective vectors to measure closeness and diversity in decision
space. One can use the functions `igd()`

or `igd_plus()`

(recommended)
directly, just passing the decision vectors as `data`

.

There are different formulations of the GD and IGD metrics in the literature
that differ on the value of *p*, on the distance metric used and on
whether the term *|A|^{-1}* is inside (as above) or outside the exponent
*1/p*. GD was first proposed by \citetVelLam1998gp with *p=2* and
the term *|A|^{-1}* outside the exponent. IGD seems to have been
mentioned first by \citetCoeSie2004igd, however, some people also used the
name D-metric for the same concept with *p=1* and later papers have
often used IGD/GD with *p=1*. \citetSchEsqLarCoe2012tec proposed to
place the term *|A|^{-1}* inside the exponent, as in the formulation
shown above. This has a significant effect for GD and less so for IGD given
a constant reference set. IGD+ also follows this formulation. We refer to
\citetIshMasTanNoj2015igd and \citetBezLopStu2017emo for a more detailed
historical perspective and a comparison of the various variants.

Following \citetIshMasTanNoj2015igd, we always use *p=1* in our
implementation of IGD and IGD+ because (1) it is the setting most used in
recent works; (2) it makes irrelevant whether the term *|A|^{-1}* is
inside or outside the exponent *1/p*; and (3) the meaning of IGD becomes
the average Euclidean distance from each reference point to its nearest
objective vector). It is also slightly faster to compute.

GD should never be used directly to compare the quality of approximations to
a Pareto front, as it often contradicts Pareto optimality. We
recommend IGD+ instead of IGD, since the latter contradicts Pareto
optimality in some cases (see examples below), but we implement IGD here
because it is still popular due to historical reasons. We are not aware of
any proof of whether *Δ_p(A,R)* contradicts or not Pareto
optimality, thus it must be used with care.

(`numeric(1)`

) A single numerical value.

Manuel López-Ibáñez

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | ```
# Example 4 from Ishibuchi et al. (2015)
ref <- matrix(c(10,0,6,1,2,2,1,6,0,10), ncol=2, byrow=TRUE)
A <- matrix(c(4,2,3,3,2,4), ncol=2, byrow=TRUE)
B <- matrix(c(8,2,4,4,2,8), ncol=2, byrow=TRUE)
plot(ref, xlab=expression(f[1]), ylab=expression(f[2]),
panel.first=grid(nx=NULL), pch=23, bg="gray", cex=1.5)
points(A, pch=1, cex=1.5)
points(B, pch=19, cex=1.5)
legend("topright", legend=c("Reference", "A", "B"), pch=c(23,1,19),
pt.bg="gray", bg="white", bty = "n", pt.cex=1.5, cex=1.2)
cat("A is better than B in terms of Pareto optimality,\n however, IGD(A)=",
igd(A, ref), "> IGD(B)=", igd(B, ref),
", which contradicts it.\nBy contrast, IGD+(A)=",
igd_plus(A, ref), "< IGD+(B)=", igd_plus(B, ref), ", which is correct.\n")
# A less trivial example.
extdata_path <- system.file(package="eaf","extdata")
path.A1 <- file.path(extdata_path, "ALG_1_dat.xz")
path.A2 <- file.path(extdata_path, "ALG_2_dat.xz")
A1 <- read_datasets(path.A1)[,1:2]
A2 <- read_datasets(path.A2)[,1:2]
ref <- filter_dominated(rbind(A1, A2))
igd(A1, ref)
igd(A2, ref)
# IGD+ (Pareto compliant)
igd_plus(A1, ref)
igd_plus(A2, ref)
# Average Haussdorff distance
avg_hausdorff_dist(A1, ref)
avg_hausdorff_dist(A2, ref)
``` |

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