Visualization of Uncertainty

The output of a simulation is best comprehended through the agency and methods of visualization, but a vital component of good science is knowledge of uncertainty. While great strides have been made in the quantification of uncertainty, especially in simulation, there is still a notable gap: there is no widely accepted means of simultaneously viewing the data and the associated uncertainty in one pane. Visualization saturates the screen, using the full range of color, shadow, opacity and tricks of perspective to display even a single variable. There is no room in the visualization expert's repertoire left for uncertainty. We present a method of visualizing uncertainty without sacrificing the clarity and power of the underlying visualization that works as well in 3-D and time-varying visualizations as it does in 2-D. At its heart, it relies on a principal tenet of continuum mechanics, replacing the notion of value at a point with a more diffuse notion of density as a measure of content in a region. First, the uncertainties calculated or tabulated at each point are transformed into a piecewise continuous field of uncertainty density . We next compute a weighted Voronoi tessellation of a user specified N convex polygonal/polyhedral cells such that each cell contains the same amount of uncertainty as defined by . The problem thus devolves into minimizing . Computation of such a spatial decomposition is O(N*N ), and can be computed iteratively making it possible to update easily over time as well as faster. The polygonal mesh does not interfere with the visualization of the data and can be easily toggled on or off. In this representation, a small cell implies a great concentration of uncertainty, and conversely. The content weighted polygons are identical to the cartogram familiar to the information visualization community in the depiction of things voting results per stat. Furthermore, one can dispense with the mesh or edges entirely to be replaced by symbols or glyphs at the generating points (effectively the center of the polygon). This methodology readily admits to rigorous statistical analysis using standard components found in R and thus entirely compatible with the visualization package we use (Visit and/or ParaView), the language we use (Python) and the UVCDAT environment that provides the programmer and analyst workbench. We will demonstrate the power and effectiveness of this methodology in climate studies. We will further argue that our method of defining (or predicting) values in a region has many advantages over the traditional visualization notion of value at a point.