Parallel coordinates are better than they look!

With parallel coordinates (abbr. ||-coords) the perceptual barrier imposed by our 3-dimensional habitation is breached enabling the visualization of multidimensional problems. The representation of N-dimensional points by polygonal lines is deceptively simple and additional ideas are needed to represent multivariate relations. In this talk, a panorama of highlights from the foundations to the most recent results, and interlaced with applications, are intuitively developed. This is also an opportunity to demystify some subtleties. By learning to untangle patterns from ||-coords displays (Fig. 1, 2) a powerful knowledge discovery process has evolved. It is illustrated on a real dataset together with guidelines for exploration and good query design. Realizing that this approach is intrinsically limited (see Fig. 3 -- left) leads to a deeper geometrical insight, the recognition of M-dimensional objects recursively from their (M-- 1)-dimensional subsets (Fig. 3 -- right). Behind this striking cognitive success lies a special family of planes unique to ||-coords, the superplanes, whose points are represented by straight (rather than polygonal) lines. It emerges that any linear N-dimensionsal relation is represented by (N-- 1) indexed points. Points representing lines have two indices, points representing planes 3 indices and so on. In turn, powerful geometrical algorithms (e.g. for intersections, containment, proximities) and applications including classification Fig. 4 emerge. The classifier's power is demonstrated by obtaining a rule for the recognition of hostile vehicles from afar by their noise signature. A smooth surface in 3-D is the envelope of its tangent planes each of which is represented by 2 points Fig. 6. As a result, a surface in 3-D is represented by two planar regions and in N-dimensions by (N-- 1) regions. This is equivalent to representing a surface by its normal vectors, rather than projections as in standard surface descriptions. Developable surfaces are represented by curves Fig. 7 revealing the surfaces' characteristics. Convex surfaces in any dimension are recognized by the hyperbola-like (i.e. having two assymptotes) regions from just one orientation Fig. 5 -- right, Fig. 8, Fig. 10 -- right. Non-orientable surfaces (i.e. like the Möbius strip) yield stunning patterns Fig. 9 unlocking new geometrical insights. Non-convexities like folds, bumps, coiling, dimples and more are no longer hidden Fig. 10 -- left and are detected from just one orientation. Evidently this representation is preferable for some applications even in 3-D. By the way, many of these results were first discovered visually and then proved mathematically; in the true spirit of Geometry. These state of the art examples show what has been achieved on the representation of complex relations and how they generalize to N-dimensions. The patterns persist in the presence of errors deforming in ways revealing the type and magnitude of the errors and that's good news for the applications. New vistas for multidimensional visualization are emerging. The processing is performed directly on the data and is not display bound opening the way for the exploration of massive datasets. Only the results are displayed in patterns where information is immensely concentrated (see again Fig. 3 - just the point is needed) and without any display clutter. These are the "graphs" of multidimensional relations within the data. The challenge is to speed up the recursive algorithm, employing among others, intelligent agents to rapidly identify relational properties. For some applications it will be worthwhile to display the patterns of partial results, during the processing, to enable computational steering. We stand on the threshold of cracking the gridlock of multidimensional visualization.