rindicator: R1, R2 and R3-indicator
In jakobbossek/ecr3vis: Evolutionary Computation in R - Version 3

Description Usage Arguments Details Value References See Also

Functions r1, r2 and r3 calculate the R1, R2 and R3 indicator respectively for two sets of points x and y.

r1(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...)

r2(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...)

r3(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...)

`x`	[`matrix`] First numeric matrix of points (each colum contains one point).
`y`	[`matrix`] Second numeric matrix of points (each colum contains one point).
`ip`	[`numeric`] The utopia point of the true Pareto-front, i.e., each component of the point contains the best value if the other objectives are neglected. If `NULL` (default), the ideal point of the union of `x` and `y` is used.
`np`	[`numeric`] Nadir point of the true Pareto front or an approximation. If `NULL` (default), the nadir point of the union of `x` and `y` is used.
`lambda`	[`integer(1)`] Number of weight vectors to use for estimating the utility function.
`utility`	[`character(1)`] Name of the utility function to use. Must be one of “weighted-sum”, “tschebycheff” or “augmented-tschebycheff”. Default is “tschebycheff”.
`...`	[any] Not used.

Consider two Pareto-front (approximation) sets X = \{x_1, …, x_{|X|}\} and Y = \{y_1, …, y_{|Y|}\} for some optimization problem with m objectives. Furthermore, let U be a set of utility functions each of the form u : R^m \to R which maps each point in the objective space into a so-called measure of utility. Let p be a probability distribution over U. We denote by u^{*}(A) = \max_{x \in A} u(x) the maximal value reached by some utility function u on a set A.

The R1-indicator measures the probability that approximation set X is better than approximation set Y by integrating over the utility function u \in U. More precisely, it is defined as

R1(X, Y, U, p) = \int_{u \in U} C(X, Y, u)p(x)du

where C(X, Y, u) = 1 if u^{*}(X) > u^{*}(Y), C(X, Y, u) = 1/2 if u^{*}(X)=u^{*}(Y) and C(X, Y, u) = 0 otherwise is termed the outcome function.

Interpretation: if R1(X, Y, U, p) > 0.5, X is better than Y. Owing to the definition it holds that

R1(X, Y, U, p) = 1 - R1(X, Y, U, p)

and therefore X and Y cannot outperform each other simultaneously.

In practice this defintion is not of much use. Therefore, a finite discrete set of utility functions is used with the uniform probability distribution. Then, R1 can be written as

R1(X, Y) = \frac{1}{|U|}∑_{u \in U} C(X, Y, u).

The same approach is used for R2 and R3 which will be discussed in the following.

Classical utility functions involve weighted sum, Tschebycheff and augmented Tschbycheff. All three are availabe via the argument utility in the implementation. As an example, the set of (weighted) Tschebycheff utility functions is U_{∞} = (u_{λ}) where

u_{λ}(x) = -\max_{j=1, …, m} ≤ft(λ_j \cdot |x_j - r_j|\right)

where r \in R^m is the ideal point or an approximation thereof and λ \in R^m is a weight vector with λ_j ≥q 0 for j=1,…,m and ∑_{j=1}^{m} λ_j = 1. Further details are beyond the scope of this documentation (see [1] for in-depth information).

The R2-indicator instead considers the expected utility values. It is defined as

R2(X, Y, U, p) = E(u^{*}(X)) - E(u^{*}(Y)) = \int_{u \in U} (u^{*}(A) - u^{*}(B))p(u)du.

Interpretation: here, since the measure is defined as the difference of expectations approximation set X is considered better than set Y if R(X, Y) > 0.

Eventually, indicator R3 considers the ratios of best utility function values:

R3(X, Y, U, p) = E≤ft(\frac{u^{*}(Y) - u^{*}(X)}{u^{*}(Y)}\right) = \int_{u \in U} \frac{u^{*}(Y) - u^{*}(X)}{u^{*}(Y)}p(u)du.

For further details we refer the reader to Hansen and Jaszkiewicz [1].

Single numeric indicator value.

[1] M. P. Hansen and A. Jaszkiewicz. 1998. Evaluating the quality of approximations to the nondominated set. Imm-rep-1998-7. Institute of Mathematical Modeling, Technical University of Denmark.

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