An overview of quantitative representations of uncertainty and formal methods for uncertainty quantification will be provided. First, the main categories of uncertainty, aleatory variability and epistemic uncertainty, will be explained. Then the major representations of uncertainty and formal methods for uncertainty quantification will be briefly discussed. In addition to the traditional probabilistic representation, alternative methods will be shortly explained, such as set-based representations (interval analysis), fuzzy sets (possibility theory), disjunctive random sets (theory of belief functions) and probability intervals (theory of imprecise probabilities). The various representations will be motivated and illustrated using illustrative examples, and guidance for further reading will be provided.

In the second part modeling of uncertainties in spatial and spatio-temporal fields using the mathematical concept of discrete random fields will be presented. It will be shown how the propagation of uncertainties from the raw data, i.e. given scalar, vector and tensor fields, to uncertainties of derived quantities and local features can be statistically modeled and numerically computed.

• Major categories of uncertainty: aleatory variability and epistemic uncertainty
• Quantitative representations of uncertainty (motivation, examples, further reading)
• statistical representations (variance of PDFs, random fields)
• set-based representations (interval analysis)
• fuzzy set representations (possibility theory)
• representations by disjunctive random sets (theory of belief functions)>
• representations by probability intervals (theory of imprecise probabilities)
• Modelling uncertainties in spatial or spatio-temporal fields by discrete random fields
• general idea
• local probabilities for presence of features
• consideration/neglection of spatial correlations
• examples: iso-surfaces, critical points, vortex cores

The aim of this session is to provide access to the complex field of quantitative representation and treatment of uncertainties. The attendees shall become aware that various approaches are used today, in addition to traditional statistical methods. This knowledge shall enable them to better understand uncertainty representations and quantifications used in application domains.

Hints to selected references will enable the attendees to deepen this knowledge.

• Didier Dubois, "Uncertainty Theories: A Unified View," CNRS - 2011 Spring School “Belief Functions Theory and Applications”, http://www.gipsa-lab.grenoble-inp.fr/summerschool/bfta/includes/Uncertainty-Dubois.pptx.pdf
• Kai Pöthkow and Hans-Christian Hege, "Positional uncertainty of isocontours: condition analysis and probabilistic measures," IEEE Trans. Vis. Comput. Graph., 17:10, pp. 1393–1406, October 2011.
• Kai Pöthkow, Britta Weber, and Hans-Christian Hege, "Probabilistic marching cubes," Comput. Graph. Forum 30:3, pp. 931-940, June 2011
• Christoph Petz, Kai Pöthkow, and Hans-Christian Hege, "Probabilistic local features in uncertain vector fields with spatial correlation," Comput. Graph. Forum 31:3, pp. 1325-1334, June 2012.

Zuse Institute Berlin, Germany

Hans-Christian Hege is director of the Visualization and Data Analysis department at Zuse Institute Berlin. He studied physics and mathematics, and performed research in computational physics and quantum field theory at Freie Universität Berlin. In 1989 he joined the High-Performance Computing division at ZIB and in 1991 he started the Scientific Visualization department in the Numerical Mathematics division. His department performs research in visual data analysis and develops software such as Amira.

He co-founded three companies in the field of computer graphics and visualization. He taught at Free University Berlin, Universitat Pompeu Fabra, Barcelona, and German Film School. His research interests include visual computing and applications in life and natural sciences. He co-authored about 250 publications and acted as a co-chair, IPC member, and reviewer for various conferences in the field.