Overview and State-of-the-Art of Uncertainty Visualization - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 10

became known as Dempster-Shafer theory (DST) or evidence theory. DST is capable
of dealing with interval-based probabilities, such that belief or probability measures
are equal to the ranges of admissible probabilities. As it turns out, belief measures are
equivalent to Choquet capacities of order inf and plausibility measures are equivalent
to alternating capacities of order inf .
The comparison of membership functions of fuzzy sets and probabilities was
investigated in 1978 by Michio Sugeno and found to be not directly possible. This
led to the generalization of additive measures analogous to generalization such that
crisp sets generalize to fuzzy sets, and additive measures generalize to (non-additive)
fuzzymeasures or monotonemeasures. The Sugeno integral was then introducedwith
respect to a monotone measure. That same year, Lofti Zadeh defined a possibility
function associated with each fuzzy set that is numerically a membership function,
and a possibility measure that is a supremum of the possibility function in each set
of concern, for both crisp and fuzzy sets. This is one of several interpretations of the
“theory of graded possibilities”. Its connection to DST is that constant plausibility
measures are equivalent to possibility measures and constant belief measures are
necessity measures.
In summary, the three most utilized uncertainty theories are the Classical
Probability Theory, the Dempster-Shafer Theory, and Possibility Theory and can
be divided into two classes. The first class uses additive measures in which the addi-
tion equal to the union expresses no interaction between events and can be thought
of as classical probability combined with measure theory. The second class uses
non-additive measures, in which addition greater than the union expresses positive
interaction between events, such as synergy, cooperation, coalition, enhancement
or amplification, while addition less than the union expresses negative interaction
between events such as incompatibility, rivalry, inhibition, downgrading, or con-
densation. This class combines one of many uncertainty theories with generalized
measure theory.

1.4 Evaluation

Visualization research is too often neglected by industry and other potential expert
users. One of the reasons is the lack of a proper evaluation of the results. This
lack of evaluation was especially obvious in historical visualization fields such as
volume rendering or fluid flowvisualization. In themore recent domain of uncertainty
visualization, researchers have made a significant effort into the assessment of the
proposed techniques. The types of evaluation may be classified into three groups:

• Theoretical evaluation: the method is analyzed to see if it follows established

graphical design principles,

• Low-level visual evaluation: a psychometric visual user study is performed to

evaluate low-level visual effects of the method,