Visualization of Complex Dynamic Datasets by Means of Mathematical Optimization

In this paper we propose an optimization model and a solution approach to visualize datasets which are made up of individuals observed along different time periods. These individuals have attached a time-dependent magnitude and a dissimilarity measure, which may vary over time. Difference of convex optimization techniques, namely, the so-called Difference of Convex Algorithm, and nonconvex quadratic binary optimization techniques are used to heuristically solve the optimization model and develop this visualization framework. This way, the so-called Dynamic Visualization Map is obtained, in which the individuals are represented by geometric objects chosen from a catalogue. A Dynamic Visualization Map faithfully represents the dynamic magnitude by means of the areas of the objects, while it trades off three different goodness of fit criteria, namely the correct match of the dissimilarities between the individuals and the distances between the objects representing them, the spreading of such objects in the visual region, and the preservation of the mental map by ensuring smooth transitions along snapshots. Our procedure is successfully tested on dynamic geographic and linguistic datasets.