Cartogram visualization for nonlinear manifold learning models

Real-world applications of multivariate data analysis often stumble upon the barrier of interpretability. Simple data analysis methods are usually easy to interpret, but they risk providing poor data models. More involved methods may instead yield faithful data models, but limited interpretability. This is the case of linear and nonlinear methods for multivariate data visualization through dimensionality reduction. Even though the latter have provided some of the most exciting visualization developments, their practicality is hindered by the difficulty of explaining them in an intuitive manner. The interpretability, and therefore the practical applicability, of data visualization through nonlinear dimensionality reduction (NLDR) methods would improve if, first, we could accurately calculate the distortion introduced by these methods in the visual representation and, second, if we could faithfully reintroduce this distortion into such representation. In this paper, we describe a technique for the reintroduction of the distortion into the visualization space of NLDR models. It is based on the concept of density-equalizing maps, or cartograms, recently developed for the representation of geographic information. We illustrate it using Generative Topographic Mapping (GTM), a nonlinear manifold learning method that can provide both multivariate data visualization and a measure of the local distortion that the model generates. Although illustrated here with GTM, it could easily be extended to other NLDR visualization methods, provided a local distortion measure could be calculated. It could also serve as a guiding tool for interactive data visualization.