Parallel Coordinates: Visual Multidimensional Geometry and Its Applications

The concept of parallel coordinates was apparently first described by Maurice d’Ocagne in 1885. The idea was rediscovered almost seventy-five years later by the author of this book, Alfred Inselberg, who conceived of it primarily as a visualization tool. Since then, Inselberg has dedicated himself to developing the idea broadly and systematically and applying it in many contexts. This book is a kind of culmination of his work, and appears to be the first time all the ideas have appeared in book form.

Parallel coordinates provide a means of visualizing geometric objects in many dimensions. They are particularly useful for displaying multivariate data and supporting multivariate analysis. The name arises from the basic set-up: n parallel and equally spaced vertical axes are drawn to intersect a single standard horizontal axis. A point in n-dimensional space is represented by connected line segments joining vertices on the parallel vertical axes. The position of the vertex on the ith axis corresponds to the ith coordinate of the point. It is a simple idea with many interesting consequences.

This book is written as a textbook designed for use in a one semester course in applied mathematics or computer science. Although there is a considerable focus on multidimensional geometry per se, many readers are likely to concentrate on the parts that deal explicitly with visualization and exploratory data analysis. Getting insight into a multidimensional data set is a considerable challenge. I have used variations of techniques like projection pursuit and the grand tour to this end, and I am very curious how well Inselberg’s approach would have worked with my data.

There are several applications discussed in the book. These include an extended example in air traffic control and collision avoidance, (where Inselberg’s methods are used to display multiple flight trajectories), geometric modeling, computer vision, process control, statistics, automatic classification, and decision support.

Inselberg is a very enthusiastic advocate of his methods and he has written a big book, bursting with ideas. Sometimes, however, his direction and the connections between one section and another are not as clear as they could be. A little bit more “and here’s where we’re going in this chapter” would be most useful. Inselberg also relies a little too much on the passive voice — enough to be clearly noticeable and hence a bit distracting — but that’s a minor quibble.

The book comes with a special pathway marked through the book using the author’s “FastTrack” symbols to allow the reader to get to the essentials quickly. There are plenty of exercises as well as an accompanying CD with “interactive learning modules”. Prerequisites for the book include some familiarity with linear algebra, a bit of basic multivariable calculus, and willingness to work through extended geometrical arguments.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.