Daniel Klötzl

Zusammenfassung

Seasonal-trend decomposition based on loess (STL) is a powerful tool to explore time series data visually. In this paper, we present an extension of STL to uncertain data, named uncertainty-aware STL (UASTL). Our method propagates multivariate Gaussian distributions mathematically exactly through the entire analysis and visualization pipeline. Thereby, stochastic quantities shared between the components of the decomposition are preserved. Moreover, we present application scenarios with uncertainty modeling based on Gaussian processes, e.g., data with uncertain areas or missing values. Besides these mathematical results and modeling aspects, we introduce visualization techniques that address the challenges of uncertainty visualization and the problem of visualizing highly correlated components of a decomposition. The global uncertainty propagation enables the time series visualization with STL-consistent samples, the exploration of correlation between and within decomposition's components, and the analysis of the impact of varying uncertainty. Finally, we show the usefulness of UASTL and the importance of uncertainty visualization with several examples. Thereby, a comparison with conventional STL is performed.

Zusammenfassung

Frequency-based decomposition of time series data is used in many visualization applications. Most of these decomposition methods (such as Fourier transform or singular spectrum analysis) only provide interaction via pre- and post-processing, but no means to influence the core algorithm. A method that also belongs to this class is Dynamic Mode Decomposition (DMD), a spectral decomposition method that extracts spatio-temporal patterns from data. In this paper, we incorporate frequency-based constraints into DMD for an adaptive decomposition that leads to user-controllable visualizations, allowing analysts to include their knowledge into the process. To accomplish this, we derive an equivalent reformulation of DMD that implicitly provides access to the eigenvalues (and therefore to the frequencies) identified by DMD. By utilizing a constrained minimization problem customized to DMD, we can guarantee the existence of desired frequencies by minimal changes to DMD. We complement this core approach by additional techniques for constrained DMD to facilitate explorative visualization and investigation of time series data. With several examples, we demonstrate the usefulness of constrained DMD and compare it to conventional frequency-based decomposition methods.

Zusammenfassung

We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.

Zusammenfassung

We present a new topological connection method for the local bilinear computation of Jacobi sets that improves the visual representation while preserving the topological structure and geometric configuration. To this end, the topological structure of the local bilinear method is utilized, which is given by the nerve complex of the traditional piecewise linear method. Since the nerve complex consists of higher-dimensional simplices, the local bilinear method (visually represented by the 1-skeleton of the nerve complex) leads to clutter via crossings of line segments. Therefore, we propose a homotopy-equivalent representation that uses different collapses and edge contractions to remove such artifacts. Our new connectivity method is easy to implement, comes with only little overhead, and results in a less cluttered representation.