Exploring Visualization Methods for Complex Variables

Applications of complex variables and related manifolds appear throughout mathematics and science. Here we review a family of basic methods for applying visualization concepts to the study of complex variables and the properties of specific complex manifolds. We begin with an outline of the methods we can employ to directly visualize poles and branch cuts as complex functions of one complex variable. $CP^2$ polynomial methods and their higher analogs can then be exploited to produce visualizations of Calabi-Yau spaces such as those modeling the hypothesized hidden dimensions of string theory. Finally, we show how the study of N-boson scattering in dual model/string theory leads to novel cross-ratio-space methods for the treatment of analysis in two or more complex variables

Exploring Uncertainty in Geodemographics with Interactive Graphics

Geodemographic classifiers characterise populations by categorising geographical areas according to the demographic and lifestyle characteristics of those who live within them. The dimension-reducing quality of such classifiers provides a simple and effective means of characterising population through a manageable set of categories, but inevitably hides heterogeneity, which varies within and between the demographic categories and geographical areas, sometimes systematically. This may have implications for their use, which is widespread in government and commerce for planning, marketing and related activities. We use novel interactive graphics to delve into OAC – a free and open geodemographic classifier that classifies the UK population in over 200,000 small geographical areas into 7 super-groups, 21 groups and 52 sub-groups. Our graphics provide access to the original 41 demographic variables used in the classification and the uncertainty associated with the classification of each geographical area on-demand. It also supports comparison geographically and by category. This serves the dual purpose of helping understand the classifier itself leading to its more informed use and providing a more comprehensive view of population in a comprehensible manner. We assess the impact of these interactive graphics on experienced OAC users who explored the details of the classification, its uncertainty and the nature of between – and within – class variation and then reflect on their experiences. Visualization of the complexities and subtleties of the classification proved to be a thought-provoking exercise both confirming and challenging users’ understanding of population, the OAC classifier and the way it is used in their organisations. Users identified three contexts for which the techniques were deemed useful in the context of local government, confirming the validity of the proposed methods

On the Optimization of Visualizations of Complex Phenomena

The problem of perceptually optimizing complex visualizations is a difficult one, involving perceptual as well as aesthetic issues. In our experience, controlled experiments are quite limited in their ability to uncover interrelationships among visualization parameters, and thus may not be the most useful way to develop rules-of-thumb or theory to guide the production of high-quality visualizations. In this paper, we propose a new experimental approach to optimizing visualization quality that integrates some of the strong points of controlled experiments with methods more suited to investigating complex highly-coupled phenomena. We use human-in-the-loop experiments to search through visualization parameter space, generating large databases of rated visualization solutions. This is followed by data mining to extract results such as exemplar visualizations, guidelines for producing visualizations, and hypotheses about strategies leading to strong visualizations. The approach can easily address both perceptual and aesthetic concerns, and can handle complex parameter interactions. We suggest a genetic algorithm as a valuable way of guiding the human-in-the-loop search through visualization parameter space. We describe our methods for using clustering, histogramming, principal component analysis, and neural networks for data mining. The experimental approach is illustrated with a study of the problem of optimal texturing for viewing layered surfaces so that both surfaces are maximally observable

Visual and interactive exploration of point data

Point data, such as Unit Postcodes (UPC), can provide very detailed information at fine scales of resolution. For instance, socio-economic attributes are commonly assigned to UPC. Hence, they can be represented as points and observable at the postcode level. Using UPC as a common field allows the concatenation of variables from disparate data sources that can potentially support sophisticated spatial analysis. However, visualising UPC in urban areas has at least three limitations. First, at small scales UPC occurrences can be very dense making their visualisation as points difficult. On the other hand, patterns in the associated attribute values are often hardly recognisable at large scales. Secondly, UPC can be used as a common field to allow the concatenation of highly multivariate data sets with an associated postcode. Finally, socio-economic variables assigned to UPC (such as the ones used here) can be non-Normal in their distributions as a result of a large presence of zero values and high variances which constrain their analysis using traditional statistics. This paper discusses a Point Visualisation Tool (PVT), a proof-of-concept system developed to visually explore point data. Various well-known visualisation techniques were implemented to enable their interactive and dynamic interrogation. PVT provides multiple representations of point data to facilitate the understanding of the relations between attributes or variables as well as their spatial characteristics. Brushing between alternative views is used to link several representations of a single attribute, as well as to simultaneously explore more than one variable. PVT’s functionality shows how the use of visual techniques embedded in an interactive environment enable the exploration of large amounts of multivariate point data

Exploring Causal Influences

Recent data mining techniques exploit patterns of statistical independence in multivariate data to make conjectures about cause/effect relationships. These relationships can be used to construct causal graphs, which are sometimes represented by weighted node-link diagrams, with nodes representing variables and combinations of weighted links and/or nodes showing the strength of causal relationships. We present an interactive visualization for causal graphs (ICGs), inspired in part by the Influence Explorer. The key principles of this visualization are as follows: Variables are represented with vertical bars attached to nodes in a graph. Direct manipulation of variables is achieved by sliding a variable value up and down, which reveals causality by producing instantaneous change in causally and/or probabilistically linked variables. This direct manipulation technique gives users the impression they are causally influencing the variables linked to the one they are manipulating. In this context, we demonstrate the subtle distinction between seeing and setting of variable values, and in an extended example, show how this visualization can help a user understand the relationships in a large variable set, and with some intuitions about the domain and a few basic concepts, quickly detect bugs in causal models constructed from these data mining techniques

Using treemaps for variable selection in spatio-temporal visualisation

We demonstrate and reflect upon the use of enhanced treemaps that incorporate spatial and temporal ordering for exploring a large multivariate spatio-temporal data set. The resulting data-dense views summarise and simultaneously present hundreds of space-, time-, and variable-constrained subsets of a large multivariate data set in a structure that facilitates their meaningful comparison and supports visual analysis. Interactive techniques allow localised patterns to be explored and subsets of interest selected and compared with the spatial aggregate. Spatial variation is considered through interactive raster maps and high-resolution local road maps. The techniques are developed in the context of 42.2 million records of vehicular activity in a 98 km(2) area of central London and informally evaluated through a design used in the exploratory visualisation of this data set. The main advantages of our technique are the means to simultaneously display hundreds of summaries of the data and to interactively browse hundreds of variable combinations with ordering and symbolism that are consistent and appropriate for space- and time- based variables. These capabilities are difficult to achieve in the case of spatio-temporal data with categorical attributes using existing geovisualisation methods. We acknowledge limitations in the treemap representation but enhance the cognitive plausibility of this popular layout through our two-dimensional ordering algorithm and interactions. Patterns that are expected (e.g. more traffic in central London), interesting (e.g. the spatial and temporal distribution of particular vehicle types) and anomalous (e.g. low speeds on particular road sections) are detected at various scales and locations using the approach. In many cases, anomalies identify biases that may have implications for future use of the data set for analyses and applications. Ordered treemaps appear to have potential as interactive interfaces for variable selection in spatio-temporal visualisation. Information Visualization (2008) 7, 210-224. doi: 10.1057/palgrave.ivs.950018