This paper provides a brief overview of uncertainty visualization along with some fundamental considerations on uncertainty propagation and modeling. Starting from the visualization pipeline, we discuss how the different stages along this pipeline can be affected by uncertainty and how they can deal with this and propagate uncertainty information to subsequent processing steps. We illustrate recent advances in the field with a number of examples from a wide range of applications: uncertainty visualization of hierarchical data, multivariate time series, stochastic partial differential equations, and data from linguistic annotation.

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Nonlinear programming is a complex methodology where a problem is mathematically expressed in terms of optimality while imposing constraints on feasibility. Such problems are formulated by humans and solved by optimization algorithms. We support domain experts in their challenging tasks of understanding and troubleshooting optimization runs of intricate and high-dimensional nonlinear programs through a visual analytics system. The system was designed for our collaborators’ robot motion planning problems, but is domain agnostic in most parts of the visualizations. It allows for an exploration of the iterative solving process of a nonlinear program through several linked views of the computational process. We give insights into this design study, demonstrate our system for selected real-world cases, and discuss the extension of visualization and visual analytics methods for nonlinear programming. Graphic abstract

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.

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