Alfred Inselberg

Visualization and data mining of high-dimensional data

Abstract Visualization provides insight through images and can be considered as a collection of application specific mappings: ProblemDomain→VisualRange. For the visualization of multivariate problems a multidimensional system of parallel coordinates (abbreviated as ∥-coords) is constructed which induces a one-to-one mapping between subsets of N -space and subsets of 2-space. The result is a rigorous methodology for doing and seeing N -dimensional geometry. Starting with an the overview of the mathematical foundations, it is seen that the display of high-dimensional datasets and search for multivariate relations among the variables is transformed into a 2-D pattern recognition problem. This is the basis for the application to Visual Data Mining which is illustrated with real dataset of Very Large Scale Integration (VLSI—“chip”) production. Then a recent geometric classifier is presented and applied to three real datasets. The results compared to those of 23 other classifiers have the least error. The algorithm has quadratic computational complexity in the size and number of parameters, provides comprehensible and explicit rules, does dimensionality selection—where the minimal set of original variables required to state the rule is found—and orders these variables so as to optimize the clarity of separation between the designated set and its complement. Finally, a simple visual economic model of a real country is constructed and analyzed in order to illustrate the special strength of ∥-coords in modeling multivariate relations by means of hypersurfaces.

Classification and visualization for high-dimensional 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(NjP j) in the number of variables N and dataset size jP j, ~ 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 of separation between the designated set and its complement.

Visualization and knowledge discovery for high dimensional data

The goal of the article is to present a multidimensional visualization methodology and its applications to visual and automatic knowledge discovery. Visualization provides insight through images and can be considered as a collection of application specific mappings: ProblemDomain/spl rarr/VisuaLRange. For the visualization of multivariate problems, a multidimensional system of parallel coordinates (/spl par/-coords) is constructed which induces a one-to-one mapping between subsets of N-space and subsets of 2-space. The result is a rigorous methodology for doing and seeing N-dimensional geometry. We start with an overview of the mathematical foundations where it is seen that from the display of high-dimensional datasets, the search for multivariate relations among the variables is transformed into a 2D pattern recognition problem. This is the basis for the application to visual knowledge discovery which is illustrated in the second part with a real dataset of VLSI production. Then a recent geometric classifier is presented and applied to 3 real datasets. The results compared to those of 23 other classifiers have the least error. The algorithm has quadratic computational complexity in the size and number of parameters, provides comprehensible and explicit rules, does dimensionality selection, and orders these variables so as to optimize the clarity of separation between the designated set and its complement. Finally a simple visual economic model of a real country is constructed and analyzed in order to illustrate the special strength of /spl par/-coords in modeling multivariate relations by means of hypersurfaces.

Don't panic ... just do it in parallel!

SummaryParallel coordinates is a methodology for visualizing N-dimensional geometry and multivariate problems. In this self-contained up-to-date overview the aim is to clarify salient points causing difficulties, and point out more sophisticated applications and uses in statistics which are marked by **. Starting from the definition of the parallel-axes multidimensional coordinate system, where a point in Euclidean N-space RN is represented by a polygonal line, it is found that a point↔line duality is induced in the Euclidean plane R2. This leads to the development in the projective, P2, rather than the Euclidean plane. Pointers on how to minimize the technical complications and avoid errors are provided. The representation (i.e. visualization) of 1-dimensional objects is obtained from the envelope of the polygonal lines representing the points on their points. On the plane R2 there is a inflection-point ↔ cusp, conics ↔ conics and other potentially useful dualities. A line ?⊂ RN is represented by N – 1 points with a pair of indices in [1, 2, ..., N]. This representation also enables the visualization and computation of proximity properties like the minimum distance between pairs of lines [18]. The representation of objects of dimension ≥ 2 is obtained recursively. Specifically, the representation of a p-flat, a plane of dimension 2 ≤ p ≤ N–1 in RN is obtained from the (p-1)-flats it contains, and which are obtained from the (p-2)-flats and so on all the way down from the points (0-dimensional); hence the recursion. A p-flat is represented by p-points each with (p+1) indices. This is the key message: **high-dimensional objects may be visualized recursively, in terms of their higher dimensional components, rather than directly from their points. Further, this process is robust so that “near” p-flats are also detected in the same way and very useful tight error bounds are available. The representation of a smooth hypersurface in RN is obtained as the envelope of the tangent hyperplanes. The set of points obtained in this way visually reveal properties like convexity, whether the surface is developable, or ruled. A simpler but ambiguous representation for hypersurfaces is also given together with modeling applications of an algorithm for computing and displaying interior, exterior or surface points.

Parallel Coordinates: Visualization, Exploration and Classification of High-Dimensional Data

A dataset with M items has 2M subsets, any one of which may be the one we really want. With a good data display, our own fantastic pattern-recognition abilities can not only sort through this combinatorial explosion, but they can also extract insights fromthe visual patterns. These are the core reasons for data visualization. With parallel coordinates (abbrev. f-coords), the search for multivariate relations in highdimensional datasets is transformed into a 2-D pattern recognition problem. In this chapter, the guidelines and strategy for knowledge discovery using parallel coordinates are illustrated on various real datasets, one with 400 variables froma manufacturing process. A geometric classification algorithm based on f-coords is presented and applied to complex datasets. It has low computational complexity, providing the classification rule explicitly and visually.Theminimal set of variables required to state the rule are found and ordered by their predictive value. A visual economic model of a real country is constructed and analyzed to illustrate how multivariate relations can be modeled using hypersurfaces.The overview at the end provides a basic summary of f-coords and a prelude of what is on the way: the distillation of relational information into patterns that eliminate need for polygonal lines altogether.

Description of surfaces in parallel coordinates by linked planar regions

An overview of themethodology covers the representation (i.e. visualization) of multidimensional lines, planes, flats, hyperplanes, and curves. Starting with the visualization of hypercubes of arbitrary dimension the representation of smooth surfaces is developed in terms of linked planar regions. The representation of developable, ruled, non-orientable, convex and non-convex surfaces in R3 with generalizations to RN are presented enabling efficient visual detection of surface properties. The parallel coordinates methodology has been applied to collision avoidance algorithms for air traffic control (3 USA patents), computer vision (1 USA patent), data mining (1 USA patent), optimization and elsewhere.

♣ FT-1 The Plane with Parallel Coordinates

A point in the x 1 x 2 plane is represented in parallel coordinates (abbreviated ∥-coords) by a line in the xy-plane (Fig. 3.1). To find out what a line “looks like” in ∥-coords, a set of collinear points is selected (Fig. 3.2, top right), and the lines representing them intersect (top left) at a point!

Visualization and Data Mining for High Dimensional Datasets

Visualization provides insight through images and can be considered as a collection of application specific mappings: ProblemDomain → VisualRange. For the visualization of multivariate problems a multidimensional system of parallel coordinates (abbr. ∥-coords) is constructed which induces a one-to-one mapping between subsets of N-space and subsets of 2-space. The result is a rigorous methodology for doing and seeing N-dimensional geometry. Starting with an the overview of the mathematical foundations, it is seen that the display of high-dimensional datasets and search for multivariate relations among the variables is transformed into a 2-D pattern recognition problem. This is the basis for the application to Visual Data Mining which is illustrated with a real dataset of VLSI (Very Large Scale Integration — “chip”) production. Then a recent geometric classifier is presented and applied to 3 real datasets. The results compared to those of 23 other classifiers have the least error. The algorithm, has quadratic computational complexity in the size and number of parameters, provides comprehensible and explicit rules, does dimensionality selection — where the minimal set of original variables required to state the rule is found, and orders these variables so as to optimize the clarity of separation between the designated set and its complement. Finally a simple visual economic model of a real country is constructed and analyzed in order to illustrate the special strength of ∥-coords in modeling multivariate relations by means of hypersurfaces.

Visual analytics for high dimensional data

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 countrys 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.

Visual Analytics for High Dimensional Data: Very Late Added Paper

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 the combinatorial explosion by extracting insights from the visual patterns. This is the core reason for data visualization. With parallel coordinates, as illustrated here, the search for relations in multivariate data is transformed into a 2-D pattern recognition problem. 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 countrys economy reveals sensitivities, impact of constraints, trade-offs and economic sectors unknowingly competing for the same resources. A glimpse into this beautiful and powerful multidimensional geometry is shown at the end.