plot_eaf: Empirical attainment function (EAF)
In jakobbossek/ecr3vis: Evolutionary Computation in R - Version 3

Description Usage Arguments Value References See Also Examples

Plots the Empiriccal Attainment Function (EAF) given approximation sets of multi-objective stochastic algorithms.

The following description closely follows the excellent introduction into performance assessment of multi-objective optimizers by Knowles et al. [3]. (Evolutionary) multi-objective algorithms are stochastic. Hence, naturally, the result produced by such an algorithm can be described by a probability distribution. This distribution can be characterized by a random set

Z = \{z^j \in R^m \,|\, j = 1, …, |Z|\}

where the size |Z| is also random. The attainment function α_Z : R^m \to [0,1] is defined as

α_Z(z) = P(Z \preceq \{z\}) = P≤ft(z^1 \preceq z \lor … \lor z^{|Z|} \preceq z\right).

where z^j \preceq z denotes that z^j weakly dominates z. This can be interpreted as the probability to reach goal z in the sense that there is at least one objective vector in the solution set that weakly dominates, i.e., “attains”, z.

Given r independent runs of a stochastic multi-objective optimizer and denoting by X^i, 1 ≤q i ≤q r the corresponding Pareto-front approximation set, the empirical attainment function (EAF) is defined as

\hat{α}_r(z) = \frac{1}{r} ∑_{i=1}^{r} I(X^i \preceq \{z\})

where I(\cdot) is the indicator function evaluating to 1 if and only if the condition is true; 0 otherwise. This is a straight-forward estimation from empirical data and simply describes the frequency of attaining z within r runs.

The EAF can be used for visualization. Here, we plot the so-called k\%-attainment surface which splits “the goals that have been attained and the goals that have not been attained with a frequency of at least k percent” [3].

For further details we refer the reader to the technical report by Knowles, Thiele and Zitzler (reference [3]).

1	plot_eaf(df, obj.cols, percentiles = c(0, 50, 75, 100))

`df`	[`data.frame`] Data frame with columns at least those given via parameter `obj.cols`, “problem” and “algorithm”.
`obj.cols`	[`character(>= 2)`] Column names of the objective function values. Default is `c("y1", "y2")`.
`percentiles`	[`numeric`] Percentiles of the EAF that will be plotted as attainment surfaces.

A ggplot object.

[1] V. Grunert da Fonseca and C. M. Fonseca, The attainment-function approach to stochastic multiobjective optimizer assessment and comparison, in Experimental Methods for the Analysis of Optimization Algorithms (T. Bartz-Beielstein, M. Chiarandini, L. Paquete, and M. Preuss, eds.), ch. 5, pp. 103-130, Springer Berlin Heidelberg, 2010.

[2] Manuel López-Ibáñez, Luís Paquete, and Thomas Stützle. Exploratory Analysis of Stochastic Local Search Algorithms in Biobjective Optimization. In T. Bartz-Beielstein, M. Chiarandini, L. Paquete, and M. Preuss, editors, Experimental Methods for the Analysis of Optimization Algorithms, pages 209–222. Springer, Berlin, Germany, 2010. doi: 10.1007/978-3-642-02538-9_9.

[3] Knowles, J. D., Thiele, L. and Zitzler, E. A tutorial on the performance assessment of stochastive multiobjective optimizers. TIK-Report No. 214, Computer Engineering and Networks Laboratory, ETH Zurich, February 2006 (Revised version. First version, January 2005). doi: 10.3929/ethz-b-000023822.

Other multi-objective visualizations: plot_eaf_diff(), plot_heatmap(), plot_pcp(), plot_radar(), plot_scatter2d(), plot_scatter3d()

## Not run: 
data(emoas_on_zdt)
plot_eaf(emoas_on_zdt, obj.cols = c("y1", "y2"))
plot_eaf(emoas_on_zdt[emoas_on_zdt$algorithm == "nsga2", ], obj.cols = c("y1", "y2"))
plot_eaf(emoas_on_zdt[emoas_on_zdt$algorithm == "nsga2", ], obj.cols = c("y1", "y2"),
  percentiles = c(50, 100))

## End(Not run)