A Multi-Objective Approach to Visualize Proportions and Similarities Between Individuals by Rectangular Maps

In this paper we address the problem of visualizing the proportions and the similarities attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the proportions, and the closeness between portions represents the similarity between the individuals. By considering the most similar individuals as adjacent, we seek to represent adjacent individuals as adjacent portions in the rectangular map. Due to the impossibility of satisfying both area and adjacency requirements, this visualization problem is formulated as a three-objective Mixed Integer Nonlinear Problem. The first objective seeks to maximize the number of true adjacencies that the rectangular map is able to reproduce, the second one is to minimize the number of false adjacencies that the rectangular map adds, and the last one is to minimize the total deviation of the areas of the portions in the rectangular map from the given proportions. To guide the location of the rectangles, we have designed a tailored MultiDimensional Scaling for building rectangular maps. We study the tradeoff between the three objectives by solving the problem with their weighted summation. Our numerical results demonstrate that it is possible to provide a collection of rectangular maps with different tradeoffs between an accurate representation of the proportions by areas versus an accurate representation of the similarities by adjacencies.