Interactive Visual Analysis of Spatial Sensitivities

TL;DR

The paper addresses spatially resolved sensitivity analysis for 3D simulation ensembles by computing per-voxel sensitivity volumes for each input parameter using Sobol, delta, or DGSA measures. It introduces an interactive visual analytics pipeline that projects multi-field sensitivity data onto a data-driven space-filling curve and combines Horizon Graphs with line charts, complemented by a parallel coordinates plot, a 3D surface view, and a parameter-dependency heatmap to link parameter effects with spatial regions. Key contributions include adapting data-driven SFCs to preserve locality across multiple sensitivity fields, evaluating sensitivity measures and SFC options, and validating the approach on synthetic and real medical datasets with domain expert feedback. The framework enables efficient identification of influential parameters, spatial regions of high sensitivity, and qualitative dependencies, supporting targeted simulations and informed decision-making in spatially complex applications.

Abstract

Sensitivity analyses of simulation ensembles determine how simulation parameters influence the simulation's outcome. Commonly, one global numerical sensitivity value is computed per simulation parameter. However, when considering 3D spatial simulations, the analysis of localized sensitivities in different spatial regions is of importance in many applications. For analyzing the spatial variation of parameter sensitivity, one needs to compute a spatial sensitivity scalar field per simulation parameter. Given $n$ simulation parameters, we obtain multi-field data consisting of $n$ scalar fields when considering all simulation parameters. We propose an interactive visual analytics solution to analyze the multi-field sensitivity data. It supports the investigation of how strongly and in what way individual parameters influence the simulation outcome, in which spatial regions this is happening, and what the interplay of the simulation parameters is. Its central component is an overview visualization of all sensitivity fields that avoids 3D occlusions by linearizing the data using an adapted scheme of data-driven space-filling curves. The spatial sensitivity values are visualized in a combination of a Horizon Graph and a line chart. We validate our approach by applying it to synthetic and real-world ensemble data.

Figures (15)

• Figure 1: Workflow of our approach: In a preprocessing step, the sensitivity volumes and the space-filling curve are calculated from the simulation ensemble. These data are used as input to the interactive analysis. The parallel coordinates plot (PCP) and spatial sensitivity visualization are linked and visualize the sensitivity volumes. Selections in those visualizations can be shown in more detail in the surface rendering and the parameter dependency visualization that shows the ensemble's simulation output.
• Figure 2: Interactive visual analysis of spatial sensitivities. Parallel coordinates (a) provide an overview of the sensitivities of input parameters and allow for brushing: A sensitivity visualization (b) shows the different sensitivity volumes over a 1D mapping using a data-driven space-filling curve. Both visualizations are linked to a surface rendering (d) that provides 3D spatial context. A parameter dependency visualization (c) supports an in-detail analysis of the parameters' influence on the simulation outcome.
• Figure 3: Different visual encodings for the multi-field sensitivity data over a space-filing curve: While a line plot might cause overplotting (a), Horizon Graphs (b) become small with limited vertical space. Combining both visualizations (c) reduces these issues. Brushing (gray area) further links the plots vertically as well as to the other visualizations.
• Figure 4: Creation of Horizon Graphs for sensitivity visualization. After plotting an area chart (a), the plot is horizontally divided into bands, where each band is color-coded and has a height of $1$ (b) in case of DGSA and a height of $0.2$ for normalized sensitivity measures. Then, each band is moved to the baseline, leading to a constant height, independent of the occurring data values (c).
• Figure 5: Parameter dependency visualization of synthetic dataset. Simulation output is shown as colors in a 2D heatmap spanned by the parameter values (horizontal axis) of parameter $P_1$ (a, b) and parameter $P_2$ (c) and the space-filling curve (vertical axis). a Missing values can be caused by spatial selections and are indicated by gaps, which complicate the interpretation of patterns. b Filling the gaps provides an undisturbed view on occurring patterns like the increase of the values with increasing parameter values. c Showing a different spatial selection over parameter $P_2$ reveals that the spatial regions of high intensity vary with changes in $P_2$.