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Visually-Motivated Characterizations of Point Sets Embedded in High-Dimensional Geometric Spaces
The proposed research exploits an idea of John Tukey that was never published. Called scagnostics (a Tukey neologism for "scatterplot diagnostics"), the original idea leads to a more general characterization of high-dimensional point sets using visually-based geometric and graph-theoretic measures. These measures comprise a canonical set of 9 features of pointwise data typically observed by experienced statisticians. Computing these measures on all possible 2D axis-parallel orthogonal projections in a p-dimensional space results in a p(p- 1)/2 × 9 matrix of measures. The objective of the proposed research is to generalize scagnostics to a new approach called Visual-Model-Based Transformations (VMBT). Visually-based transforms, together with multivariate analyses, can reveal visual patterns that are of interest to analysts. When interesting patterns are discovered in transform-space, one can invert the map and infer patterns in the raw data space.
Scagnostics exploits an important aspect of visualizations. A visualization can be thought of as a visual representation of an underlying mathematical model. Even simple charts of raw data rest on a model that helps (one hopes) to reveal some interesting aspect of the data. We often take these models for granted when we view familiar graphs. However, understanding mathematical models underlying visualizations can help us to devise more effective models for revealing structure in more complex datasets. Visual-Model-Based Transformations are a class of models that may prove especially effective for this purpose. Such models are motivated by visual structures perceived and processed by analysts. Given this visual motivation behind their design, visual models are likely to reveal features of data that are quite different from those appearing in common statistical and scientific graphics.