eps: Binary \varepsilon-indicator
In jakobbossek/ecr3vis: Evolutionary Computation in R - Version 3

Description Usage Arguments Details Value Note References See Also

For two point sets x and y the function returns the value of the binary \varepsilon-indicator [1].

1	eps(x, y, ...)

`x`	[`matrix`] First numeric matrix of points (each colum contains one point).
`y`	[`matrix`] Second numeric matrix of points (each colum contains one point).
`...`	[any] Not used.

The \varepsilon-indicator, often denoted as I_{\varepsilon}, requires the concept of \varepsilon-dominance. A vector x \in R^m, for some \varepsilon > 0, \varepsilon-dominates another vector y \in R^m, x \preceq_{\varepsilon} y, if and only if

\Leftrightarrow x_i ≤q \varepsilon y_i \quad \forall i = 1,…,m.

Equipped with this, Zitzler et al. [1] define the (multiplicative) binary \varepsilon-indicator for two point sets X = \{x_1, …, x_{|X|}\} and Y = \{y_1, …, y_{|Y|}\} as follows:

I_{\varepsilon}(X, Y) = \inf_{\varepsilon > 0}\{y \in Y \mid \exists x \in X: x \preceq_{\varepsilon} y\}.

It means that I_{\varepsilon}(X, Y) is the smallest \varepsilon such that there exists a point in X that dominates a point y \in Y in the \varepsilon-dominance sense. It can be calculated the following way

I_{\varepsilon}(X, Y) = \max_{y \in Y} \min_{x \in X} \max_{1 ≤ i ≤ m} \frac{x_i}{y_i}.

Given a reference set R, e.g., the known true Pareto-front or a good approximation of it, the unary version is simply

I_{\varepsilon}(X) := I_{\varepsilon}(R, X).

Function eps implements the binary \varepsilon-indicator. It should be obvious how to calculate the unary indicator.

Single numeric indicator value.

Keep in mind that this function assumes all objectives to be minimized. In case at least one objective is to be maximized, the data needs to be transformed accordingly in advance.

[1] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, V. G. Da Fonseca, Performance assessment of multiobjective optimizers: An analysis and review, IEEE Transactions on evolutionary computation 7 (2) (2003) 117–132.

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