epsilon {eaf}

Epsilon metric

Description

Computes the epsilon metric, either additive or multiplicative.

Usage

epsilon_additive(data, reference, maximise = FALSE)

epsilon_mult(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

(matrix | data.frame)
Reference set as a matrix or data.frame 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.

Details

The epsilon metric of a set A with respect to a reference set R is defined as

epsilon(A,R) = \max_{r \in R} \min_{a \in A} \max_{1 \leq i \leq n} epsilon(a_i, r_i)

where a and b are objective vectors and, 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 computation of the multiplicative version for negative values doesn't make sense.

Computation of the epsilon indicator requires O(n \cdot |A| \cdot |R|), where n is the number of objectives (dimension of vectors).

Value

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.

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)

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

# A more realistic 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))
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)