Topological and Statistical Methods for Complex Data by Janine Bennett Fabien Vivodtzev & Valerio Pascucci

Author:Janine Bennett, Fabien Vivodtzev & Valerio Pascucci
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

The second data set consists of three artificial functions obtained from sums of Gaussians. Each component function has three minima and maxima. The location of the minima and maxima are only slightly perturbed among the three function, cf. Fig. 3a for an illustration of one of the functions. The Jacobi sets are calculated and colored as in the previous figure, however since Jacobi sets are only defined for k ≤ d, we calculate them separately for each two-pairings of variables, shown in the three Fig. 3b–d. Pareto sets based on all three functions are also shown in these figures with the same coloring as in Fig. 2, but with additional yellow for Pareto optima.

Note that the Pareto sets for each pairing of two functions can be calculated through the intersection of the corresponding Jacobi sets and the Pareto set based on all three functions. Hence, it was sufficient to only calculate the Pareto set once. Also note how the Jacobi sets from all three figures, if combined, form the border of the components in the Pareto set. Our assumption and additional direction of research is that this introduces, together with our main proof, a new approach to calculate Pareto sets for a large number of functions through the combined Jacobi sets for small number of functions.

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