Data Visualization in high-dimensional spaces is a great challenge and an important tool for decision-making. The method of Aggregation Trees was an important step in visualization in MaOPs, allowing a greater understanding of the problem. This method is based on the sequential aggregation of objectives, which is visually represented into a tree, based on a measure of conflict between pairs of (groups of) objectives. The method allows the visualization of a hierarchy for aggregation of the objectives, with possibility to create new constraints for the problem or reduce the number of objectives in a further analysis.

In the paper “A New Visualization Method in Many-Objective Optimization with Chord Diagram and Angular Mapping” (https://doi.org/10.1016/j.knosys.2017.09.035), the authors present a new method able to display high dimensional data. With the proposed method it is possible to represent more than one data set in a single two-dimensional graphic representation. The two methods presented (Chord Diagram and Angular Mapping), it is possible to observe the existence of conflict and harmony between objectives, as well as the formation of clusters, convergence and dispersion of points. It is also possible recognize the geometric format of each set of points.

In the Chord Diagram, the range of each coordinate is normalized and arranged in a circle and the coordinate values are connected by means of Bezier curves.

In the Angular Mapping, each point is seen as a vector and three features are collected of each vector: The Euclidean norm ρ, the smallest angle θ between the vector and the canonical basis and in which vector j of the canonical base this smallest angle occurs. With these three values, it is possible to observe the distribution of points in space, its angular proximity to each axis and its proximity to the origin of the Cartesian coordinate system.

In another paper called “Information to the Eye of the Beholder: Data Visualization for Many-Objective Optimization”, the authors combine the Chord Diagram, Angular Mapping and the Parallel Coordinates in a single two dimensional circular chart. An R script of this method is available in the download area of the MINDS website.