epsilon {moocore} | R Documentation |
Epsilon metric
Description
Computes the epsilon metric, either additive or multiplicative.
Usage
epsilon_additive(x, reference, maximise = FALSE)
epsilon_mult(x, reference, maximise = FALSE)
Arguments
x |
|
reference |
|
maximise |
|
Details
The epsilon metric of a set A \subset \mathbb{R}^m
with respect to a
reference set R \subset \mathbb{R}^m
is defined as
epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 \leq i \leq m} epsilon(a_i, r_i)
where a
and b
are objective vectors of length m
.
In the case of minimization of objective i
, epsilon(a_i,b_i)
is
computed as a_i/b_i
for the multiplicative variant (respectively,
a_i - b_i
for the additive variant), whereas in the case of
maximization of objective i
, epsilon(a_i,b_i) = b_i/a_i
for the
multiplicative variant (respectively, b_i - a_i
for the additive
variant). This allows computing a single value for problems where some
objectives are to be maximized while others are to be minimized. Moreover, a
lower value corresponds to a better approximation set, independently of the
type of problem (minimization, maximization or mixed). However, the meaning
of the value is different for each objective type. For example, imagine that
objective 1 is to be minimized and objective 2 is to be maximized, and the
multiplicative epsilon computed here for epsilon(A,R) = 3
. This means
that A
needs to be multiplied by 1/3 for all a_1
values and by 3
for all a_2
values in order to weakly dominate R
.
The multiplicative variant can be computed as \exp(epsilon_{+}(\log(A),
\log(R)))
, which makes clear that the computation of the multiplicative
version for zero or negative values doesn't make sense. See the examples
below.
The current implementation uses the naive algorithm that requires
O(m \cdot |A| \cdot |R|)
, where m
is the number of objectives
(dimension of vectors).
Value
numeric(1)
A single numerical value.
Author(s)
Manuel López-Ibáñez
References
Eckart Zitzler, Lothar Thiele, Marco Laumanns, Carlos M. Fonseca, Viviane Grunert da Fonseca (2003). “Performance Assessment of Multiobjective Optimizers: an Analysis and Review.” IEEE Transactions on Evolutionary Computation, 7(2), 117–132. doi:10.1109/TEVC.2003.810758.
Examples
# Fig 6 from Zitzler et al. (2003).
A1 <- matrix(c(9,2,8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A2 <- matrix(c(8,4,7,5,5,6,4,7), ncol=2, byrow=TRUE)
A3 <- matrix(c(10,4,9,5,8,6,7,7,6,8), ncol=2, byrow=TRUE)
if (requireNamespace("graphics", quietly = TRUE)) {
plot(A1, xlab=expression(f[1]), ylab=expression(f[2]),
panel.first=grid(nx=NULL), pch=4, cex=1.5, xlim = c(0,10), ylim=c(0,8))
points(A2, pch=0, cex=1.5)
points(A3, pch=1, cex=1.5)
legend("bottomleft", legend=c("A1", "A2", "A3"), pch=c(4,0,1),
pt.bg="gray", bg="white", bty = "n", pt.cex=1.5, cex=1.2)
}
epsilon_mult(A1, A3) # A1 epsilon-dominates A3 => e = 9/10 < 1
epsilon_mult(A1, A2) # A1 weakly dominates A2 => e = 1
epsilon_mult(A2, A1) # A2 is epsilon-dominated by A1 => e = 2 > 1
# Equivalence between additive and multiplicative
exp(epsilon_additive(log(A2), log(A1)))
# A more realistic example
extdata_path <- system.file(package="moocore","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))
epsilon_additive(A1, ref)
epsilon_additive(A2, ref)
# Multiplicative version of epsilon metric
ref <- filter_dominated(rbind(A1, A2))
epsilon_mult(A1, ref)
epsilon_mult(A2, ref)