Preview

PDF, English Download (130MB) | Terms of use

Abstract

Multivariate data are data involve three or more variables. If the data are defined on a continuous domain, they can be considered as a mapping from Rn to Rm, where n is the dimensionality of the domain Ω ⊂ Rn, and m is the dimensionality of the codomain (or range) Ψ ⊂ Rm. Otherwise, if the data are not given on a continuous domain, the data are non-continuous, i.e., discrete.

This thesis provides novel visualization techniques that advance the analysis of both continuous multivariate data (with and without uncertainty) and non-continuous multivariate data in terms of hierarchies. The main content of this thesis is structured into two parts.

The first part aims at two cases: continuous multivariate data with m = n without uncertainty, and continuous multivariate data with n ≤ 3 with uncertainty. For m = n cases, a novel concept of equivalent regions is motivated by the equivalence property. A technique for obtaining these regions both in the domain and the codomain, including determination of their correspondence, is presented. This enables effective investigation of variation equivalence (similar variation of data values) within mappings, and between mappings in terms of comparative visualization. This approach is implemented for n = 2, and its utility is demonstrated using different examples. For uncertain continuous multivariate data, we derive respective mathematical models to extend scatterplots (representation using Cartesian coordinates for plotting values) and parallel coordinates (representation using parallel axes for plotting values) to uncertain continuous data. Two approaches to obtain uncertain continuous scatterplots are presented, a brute-force sampling-based approach and a convolution-based approach. At the same time, the sampling-based approach lends itself as well for introducing uncertainty into the definition of fibers (preimage of vector value) in bivariate data. The properties and the utility of this technique are demonstrated using specifically designed synthetic cases and simulated data.

The second part of this thesis focuses on a new area: hierarchical multivariate data, where a common hierarchy is shared by all attributes. In this part, a design study of hierarchical multivariate data visual analysis is conducted. The requirements of this task lead us to the extension of four hierarchical univariate concepts—the sunburst chart, the icicle plot, the circular treemap, and the bubble treemap—to the multivariate domain. Our study identifies several advantageous design variants, which are discussed with respect to previous approaches, and whose utility is evaluated with a user study and demonstrated for different analysis purposes and different types of data.