Abstract Uncertainty in experimental measurements makes it challenging to determine which features are intrinsic to the phenomenon and which are most stable and reliable. Granular materials, such as sand, form a complex system in which the forces between individual particles influence the material's macroscopic behavior. However, these forces are also subject to uncertainty, as repeated measurements can yield different results. In this paper, we investigate how to model and visually analyze the uncertain structure of forces in granular materials. We adopt different perspectives on uncertainty by considering it as variance, probability, or additional variable. For a nuanced analysis of granular material data, we propose combining visualizations that represent these perspectives into a single visual analytics approach. We integrate uncertainty‐aware spatial visualizations that convey the probability of features, visualizations of derived measures and their variance over changes in the packing fraction, and overviews of varying probability thresholds. Finally, we evaluate our proposed approach in a case study conducted together with geotechnical engineers for the example of a 2D ensemble of photoelastic disks.

Abstract Varying the input parameters of simulations or experiments often leads to different classes of results. Parameter sensitivity analysis in this context includes estimating the sensitivity to the individual parameters, that is, to understand which parameters contribute most to changes in output classifications and for which parameter ranges these occur. We propose a novel visual parameter sensitivity analysis approach based on Voronoi cell interfaces between the sample points in the parameter space to tackle the problem. The Voronoi diagram of the sample points in the parameter space is first calculated. We then extract Voronoi cell interfaces which we use to quantify the sensitivity to parameters, considering the class label information of each sample's corresponding output. Multiple visual encodings are then utilized to represent the cell interface transitions and class label distribution, including stacked graphs for local parameter sensitivity. We evaluate the approach's expressiveness and usefulness with case studies for synthetic and real‐world datasets.

Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models, where each segment of the partitioning represents a region of similar model behavior. Having computed a partitioning, one is commonly interested in understanding how large the segments are and which segments lie next to each other. While visual representations of 2D domain partitionings that reveal sizes and neighborhoods are straightforward, this is no longer the case when considering multi-dimensional domains of three or more dimensions. We propose an algorithm for computing 2D embeddings of multi-dimensional partitionings. The embedding shall have the following properties: It shall maintain the topology of the partitioning and optimize the area sizes and joint boundary lengths of the embedded segments to match the respective sizes and lengths in the multi-dimensional domain. We demonstrate the effectiveness of our approach by applying it to different use cases, including the visual exploration of 3D spatial domain segmentations and multi-dimensional parameter space partitionings of simulation ensembles. We numerically evaluate our algorithm with respect to how well sizes and lengths are preserved depending on the dimensionality of the domain and the number of segments.

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 $n$n simulation parameters, we obtain multi-field data consisting of $n$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.

One common task in time series analysis is the visual investigation of spectra such as Fourier spectra or wavelet spectra to identify dominating frequencies. In this article, we present the propagation of data uncertainty to the spectra and its visualization. We consider the Fourier and continuous wavelet transformations, which are two common spectral analysis methods. Deriving the propagation for time series that can be modeled as a Gaussian process leads to a combination of weighted non-central chi-squared distributions in the spectrum. Percentile-based visualizations explicitly encode the non-normal uncertainty in the 1D Fourier and 2D wavelet spectrum. We enrich the visualization by including correlations, sensitivity, and signal-to-noise analysis. For visual exploration, we combine the different visualizations into an interactive approach that allows for investigating the uncertain time series in the temporal and spectral domains. Finally, we show the usefulness of our approach by applying it to several real-world data sets and by a qualitative interview study with visualization experts.

Abstract Given a time‐varying scalar field, the analysis of correlations between different spatial regions, i.e., the linear dependence of time series within these regions, provides insights into the structural properties of the data. In this context, regions are connected components of the spatial domain with high time series correlations. The detection and analysis of such regions is often performed globally, which requires pairwise correlation computations that are quadratic in the number of spatial data samples. Thus, operations based on all pairwise correlations are computationally demanding, especially when dealing with ensembles that model the uncertainty in the spatio‐temporal phenomena using multiple simulation runs. We propose a two‐step procedure: In a first step, we map the spatial samples to a 3D embedding based on a pairwise correlation matrix computed from the ensemble of time series. The 3D embedding allows for a one‐to‐one mapping to a 3D color space such that the outcome can be visually investigated by rendering the colors for all samples in the spatial domain. In a second step, we generate a hierarchical image segmentation based on the color images. From then on, we can visually analyze correlations of regions at all levels in the hierarchy within an interactive setting, which includes the uncertainty‐aware analysis of the region's time series correlation and respective time lags.