Reliable Computing 10: 249–272, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Clouds, Fuzzy Sets, and Probability Intervals
ur Mathematik, Universit
at Wien, Strudlhofgasse 4, A-1090 Wien, Austria,
(Received: 14 October 2002; accepted: 22 February 2003)
Abstract. Clouds are a concept for uncertainty mediating between the concept of a fuzzy set and
that of a probability distribution. A cloud is to a random variable more or less what an interval is
to a number. We discuss the basic theoretical and numerical properties of clouds, and relate them to
histograms, cumulative distribution functions, and likelihood ratios.
We show how to compute nonlinear transformations of clouds, using global optimization and
constraint satisfaction techniques. We also show how to compute rigorous enclosures for the expecta-
tion of arbitrary functions of random variables, and for probabilities of arbitrary statements involving
random variables, even for problems involving more than a few variables.
Finally, we relate clouds to concepts from fuzzy set theory, in particular to the consistent possibility
and necessity measures of Jamison and Lodwick.
This paper proposes the concept of a cloud, combining aspects of fuzzy sets, interval
analysis, and probability theory in a way suitable for calculations.
The need for a conceptual basis for combining probabilistic uncertainty, due
to variability, and fuzzy uncertainty, due to missing information or missing preci-
sion in concepts, models, or measurements, has been known for a long time. The
deﬁciencies of Monte Carlo methods in the face of partial ignorance are well doc-
umented by Ferson et al. , . A number of alternatives have been explored.
The history until about 10 years ago, and the problems involved in the various
approaches known are excellently described from complementary point of views in
a thesis by Williamson  and a book by Walley . In the mean time, the basic
situation has not changed much, although computational advances make approach-
es using copulas (Springer ) or histograms (Moore ) more tractable; see,
e.g., , , , . However, their applicability is limited to problems in very
The scope of the present paper is to deﬁne and explain a new concept for cap-
turing a mix of probabilistic and fuzzy uncertainty that is able to handle also larger
uncertainty problems by being able to reduce calculations to global optimization
and constraint satisfaction problems, for which more and more efﬁcient algorithms
become available , , , .