A dataset with M items has 2M subsets anyone of which may be the one satisfying our objective. With a good data display and interactivity our fantastic pattern-recognition defeats this combinatorial explosion by extracting insights from the visual patterns. This is the core reason for data visualization. With parallel coordinates the search for relations in multivariate data is transformed into a 2-D pattern recognition problem. Together with criteria for good query design, we illustrate this on several real datasets (financial, process control, credit-score, one with hundreds of variables) with stunning results. A geometric classification algorithm yields the classification rule explicitly and visually. The minimal set of variables, features, are found and ordered by their predictive value. A model of a country's economy reveals sensitivities, impact of constraints, trade-offs and economic sectors unknowingly competing for the same resources. An overview of the methodology provides foundational understanding; learning the patterns corresponding to various multivariate relations. These patterns are robust in the presence of errors and that is good news for the applications. A topology of proximity emerges opening the way for visualization in Big Data.

A geometrically motivated classi er is presented and applied, with both training and testing stages, to 3 real datasets.Our results compared to those from 23 other classi ers have the least error.The algorithm is based on parallel coordinates and : ~has worst-case computational complexity O(N 2 jPj 2 ) in the number of variables N and dataset size jPj, ~provides comprehensible and explicit rules, ~does dimensionality selection { where the minimal set of original variables (not transformed new variables as in Principal Component Analysis) required to state the rule is found, and ~orders these variables so as to optimize the clarity o f separation between the designated set and its complement.

The paper provides a review of the field of multidimensional data visualisation and discusses some promising methodologies. It considers some crucial problems and directions. The emphasis is more on concepts and foundations rather than ad hoc methods. Visualization is considered as a collection of transformations from problem domains to a perceptual domain, usually visual. The paper discusses the extension of visualisation from the pixel to icons.>

A methodology for visualizing analytic and synthetic geometry in R/sup N/ is presented. It is based on a system of parallel coordinates which induces a nonprojective mapping between N-dimensional and two-dimensional sets. Hypersurfaces are represented by their planar images which have some geometrical properties analogous to the properties of the hypersurface that they represent. A point from to line duality when N=2 generalizes to lines and hyperplanes enabling the representation of polyhedra in R/sup N/. The representation of a class of convex and non-convex hypersurfaces is discussed, together with an algorithm for constructing and displaying any interior point. The display shows some local properties of the hypersurface and provides information on the point's proximity to the boundary. Applications are discussed.>

By means of a system of parallel coordinates, a nonprojective mapping from R/sup N/ to R/sup 2/ is obtained for any positive integer N. In this way multivariate data and relations can be represented in the Euclidean plane (embedded in the projective plane). Basically, R/sup 2/ with Cartesian coordinates is augmented by N parallel axes, one for each variable. The N joint variables of a robotic device can be represented graphically by using parallel coordinates. Two key reasons motivate this kind of approach: first, some properties of the relation are better perceived visually from the parallel coordinate representation; and secondly, new algorithms and data structures can be obtained from this representation. The main features of parallel coordinates are described, and an example is presented of their use for configuration space representation of a mechanical are. The purpose of the example is to show the potential of the representation for the cases where Cartesian coordinates cannot be used.>

With a system of parallel coordinates , objects in R N can be represented with planar “ graphs ” (i.e., planar diagrams) for arbitrary N [21]. In R 2 , embedded in the projective plane, parallel coordinates induce a point ← → line duality. This yields a new duality between bounded and unbounded convex sets and hstars ( a generalization of hyperbolas ), as well as a duality between convex union (convex merge) and intersection. From these results, algorithms for constructing the intersection and convex merge of convex polygons in O ( n ) time and the convex hull on the plane in O (log n ) for real-time and O ( n log n ) worst-case construction, where n is the total number of points, are derived. By virtue of the duality, these algorithms also apply to polygons whose edges are a certain class of convex curves. These planar constructions are studied prior to exploring generalizations to N -dimensions. The needed results on parallel coordinates are given first.