Mathematical structures of data and their implications for visualization

Abstract

We have briefly investigated how mathematical structures on data can be used to define conditions on the visualization mapping from data to displays. The first three conditions that we discussed are that D: U → V map:

• The algebraic structure of U to the algebraic structure of V (i.e., D is linear).

• The metric structure of U to the metric structure of V (i.e., D is isometric).

• The lattice structure of U to the lattice structure of V (i.e., D is a lattice isomorphism).

Thus for each of three kinds of mathematical structures on the data and display models U and V, the conditions state that D should define a correspondence between the structure of U and the structure of V. This is an interesting similarity of form between these three conditions. The fourth condition is a bit different, relating the structure of the repertoire E to the structure of a symmetry group G on the display model V.

While these ideas are certainly not fully developed, they define an interesting approach to visualization. They suggest the possibility of expressing properties of human perception and visualization requirements in terms of mathematical structures, and deriving visualization mappings by mathematical analysis. We note that the problems of symbolic integration and theorem proving were once solved heuristically (i.e., solved by applying expert “rules of thumb”), and are now solved systematically. It may be possible to solve the problem of designing visualizations in a similarly systematic way.