Uncertainty-Aware Seasonal-Trend Decomposition Based on Loess.

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.

Uncertainty-Aware Multidimensional Scaling.

We present an extension of multidimensional scaling (MDS) to uncertain data, facilitating uncertainty visualization of multidimensional data. Our approach uses local projection operators that map high-dimensional random vectors to low-dimensional space to formulate a generalized stress. In this way, our generic model supports arbitrary distributions and various stress types. We use our uncertainty-aware multidimensional scaling (UAMDS) concept to derive a formulation for the case of normally distributed random vectors and a squared stress. The resulting minimization problem is numerically solved via gradient descent. We complement UAMDS by additional visualization techniques that address the sensitivity and trustworthiness of dimensionality reduction under uncertainty. With several examples, we demonstrate the usefulness of our approach and the importance of uncertainty-aware techniques.

Constrained Dynamic Mode Decomposition.

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.

Relaxed Dot Plots: Faithful Visualization of Samples and Their Distribution.

We introduce relaxed dot plots as an improvement of nonlinear dot plots for unit visualization. Our plots produce more faithful data representations and reduce moiré effects. Their contour is based on a customized kernel frequency estimation to match the shape of the distribution of underlying data values. Previous nonlinear layouts introduce column-centric nonlinear scaling of dot diameters for visualization of high-dynamic-range data with high peaks. We provide a mathematical approach to convert that column-centric scaling to our smooth envelope shape. This formalism allows us to use linear, root, and logarithmic scaling to find ideal dot sizes. Our method iteratively relaxes the dot layout for more correct and aesthetically pleasing results. To achieve this, we modified Lloyd's algorithm with additional constraints and heuristics. We evaluate the layouts of relaxed dot plots against a previously existing nonlinear variant and show that our algorithm produces less error regarding the underlying data while establishing the blue noise property that works against moiré effects. Further, we analyze the readability of our relaxed plots in three crowd-sourced experiments. The results indicate that our proposed technique surpasses traditional dot plots.

Unsupervised and Generic Short-Term Anticipation of Human Body Motions.

Various neural network based methods are capable of anticipating human body motions from data for a short period of time. What these methods lack are the interpretability and explainability of the network and its results. We propose to use Dynamic Mode Decomposition with delays to represent and anticipate human body motions. Exploring the influence of the number of delays on the reconstruction and prediction of various motion classes, we show that the anticipation errors in our results are comparable to or even better for very short anticipation times ( < 0 . 4 sec) than a recurrent neural network based method. We perceive our method as a first step towards the interpretability of the results by representing human body motions as linear combinations of previous states and delays. In addition, compared to the neural network based methods large training times are not needed. Actually, our methods do not even regress to any other motions than the one to be anticipated and hence it is of a generic nature.