1. Neumaier Clouds Yan Bulgak yan@ramas.com October 30, MAR550, Challenger 165

2. Sets Defined by membership criterion: If A a set, then for all either or

3. Fuzzy Sets Defined by a “degree of membership” function: (A, ) is a fuzzy set if A is a set and For each, is the grade of membership

4. α-Cut For fuzzy set (A, ), and define the α-cut of A to be If α represents the degree of confidence, then the α-cut is that subset of A of which we are α certain

5. Clouds – Formal Definition A cloud over a set M is a mapping x that associates with each a (nonempty, closed, bounded) interval such that:

6. Clouds – Informal Definition “Clouds allow the representation of incomplete stochastic information in a clearly understandable and computationally attractive way” “A cloud is a new, easily visualized concept for uncertainty with well-defined semantics, mediating between the concept of a fuzzy set and a probability distribution” Translation: It’s great, it’ll change the world, I need more research money

7. Cloud x And Now For a Picture M [0.333, 0.666] [0.25, 0.75] [0.0, 1.0] x( ) Note: [0.333, 0.666 ]∪ [0.25, 0.75]∪ [0.0, 1.0] = [0.0, 1.0] and ]0,1[ ⊆ [0.0, 1.0] ⊆ [0,1]

8. Some More Terms To examine the structure of a cloud we can consider the following concepts: Let x be the cloud over M with mapping – the level of in x; and are the upper and lower levels – and are the upper and lower α-cuts

9. Continuous Cloud Example A cloud over ℝ with α=0.6

10. Discrete Random Variables and Clouds A cloud x is discrete if it has finitely many levels. There exists a 1-1 correspondence between discrete clouds and histograms (proven by Neumaier) Since random variables are well understood in the context of histograms, we can interpret an r.v. as a cloud.

11. Continuous R.V. and Thin Clouds A cloud x is thin if = for all If x is a r.v. with a continuous CDF then defines a thin cloud x with the property that a random variable belongs to x iff it has the same distribution as

12. Potential Clouds Let be r.v. with values in M, and let be bounded below. Then defines a cloud x with whose α-cuts are level sets of V Note: a level set of function V for some constant c is V is called the potential function and, are potential level maps

13. Functions of a Cloudy R.V. Let be a r.v. with values in M. Let z be a r.v. defined by with If x, z are clouds that satisfy, then

14. Expectation In general, computable only using linear programming and global optimization techniques.

15. Open Problems Computer implementation of theorems and techniques discussed above (partially solved) Combining clouds x, y to form a new cloud z with precise control of dependence. Requires the use of copulas Optimal expectation computation for joint clouds (2, then any n), given dependence information Find closed form solutions to special cases

16. Practical Use Used in a proposal to the ESA (European Space Agency) for robust system design. This is a recent development (2007) The clouds used in this study were confocal clouds, defined by ellipsoidal potential functions.

17. Confocal Cloud Example

18. Examples Preface The examples on the next slides were generated from a normal distribution with a fixed mean on a [-5,5] 2 mea The functions and refer to and respectively The ellipsoidal potential map is given by

19. Pretty Pictures Normal Distrib 3D Cloud Level Set

20. Sample Size Influence Normal Distrib 10 Sample Points1000 Sample Points

21. Confidence Level Influence Normal Distrib Confidence 99%Confidence 99.9%

22. Bottom Line So what’s this all about, really, once you get right down to it? We create a collection of intervals in such a way as to reflect our understanding of the confidence levels and stochastic implications of the input data The potential function approach distances us from the problems of adding many intervals

23. Issues In the defining paper, Neumaier lists among the advantages of clouds the ease of constructing them from data; in the ESA report, he describes the problem as very difficult in general and presents workarounds. Which observation is right? If x, y are clouds, what is x + y? Dependence issues

24. References Kreinovich V., Berleant D., Ferson S., and Lodwick A. 2005. Combining Interval, Probabilistic, and Fuzzy Uncertainty: Foundations, Algorithms, Challenges: An Overview. Pennsylvania. http://www.cs.utep.edu/vladik/2005/tr05-09.pdf Neumaier, A. 2004. Clouds, Fuzzy Sets and Probability Intervals. Reliable Computing 10, 249-272 http://www.mat.univie.ac.at/~neum/ms/cloud.pdf http://www.mat.univie.ac.at/~neum/ms/cloud.pdf Neumaier, A. 2003. On the Structure of Clouds. Unpublished Manuscript. http://www.mat.univie.ac.at/~neum/ms/struc.pdf http://www.mat.univie.ac.at/~neum/ms/struc.pdf Neumaier A., M. Fuchs, E. Dolejsi, T. Csendes, J. Dombi, B. Bánhelyi, Z. Gera. 2007. Application of clouds for modeling uncertainties in robust space system design, Final Report, ARIADNA Study 05/5201, European Space Agency (ESA). http://www.mat.univie.ac.at/~neum/ms/ESAclouds.pdf http://www.mat.univie.ac.at/~neum/ms/ESAclouds.pdf

25. Fin