rse: Riesz s-energy
In jakobbossek/ecr3vis: Evolutionary Computation in R - Version 3

Description Usage Arguments Details Value References See Also

Calculate the Riesz s-energy measure for a set of points.

1	rse(x, s, ...)

`x`	[`matrix`] Numeric matrix of points (each colum contains one point).
`s`	[`numeric`] Degree of uniformity s ≥q 0 (see description).
`...`	[any] Not used.

The Riesz s-energy is designed as a measure for the evenness of a set of points X = \{x_1, …, x_{|X|}\}. It is formally defined as

R_s(X) = ∑_{x \in X} ∑_{y \in X, y \neq x} k_s(x, y)

where function

k_s(x, y) = d(x, y)^{-s}, s > 0

and

k_s(x, y) = -\log(d(x, y)), s = 0

is the so-called Riesz s-kernel and d(x,y) is the Euclidean distance between x and y. The parameter s ≥q 0 steers the desired degree of uniformity of the distribution with increasing emphasis on uniformity for s \to ∞. See [1] for an application of the Riesz s-energy in multi-objective evolutionary optimization and [2, 3] for a mathematically rigorous introduction into the general idea.

Single numeric indicator value.

[1] J. G. Falcón-Cardona, H. Ishibuchi and C. A. C. Coello, Riesz s-energy-based Reference Sets for Multi-Objective optimization," 2020 IEEE Congress on Evolutionary Computation (CEC), 2020, pp. 1-8, doi: 10.1109/CEC48606.2020.9185833.

[2] D.P. Hardinand and E.B. Saff, Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds, Advances in Mathematics, vol. 193, no. 1, pp. 174–204, 2005.

[3] D. P. Hardin and E. B. Saff, Discretizing Manifolds via Minimum Energy Points, Notices of the AMS, vol. 51, no. 10, pp. 1186–1194, 2004.

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