Neumaier Clouds

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Neumaier Clouds

• Yan Bulgak
• yan_at_ramas.com

October 30, MAR550, Challenger 165
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Sets

• Defined by membership criterionIf A a set, then
  for all either or

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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

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a-Cut

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

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Clouds Formal Definition

• A cloud over a set M is a mapping x that
  associates with each a (nonempty,
  closed, bounded) interval such that

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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 Its great, itll change the world,
I need more research money
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And Now For a Picture
M
0.333, 0.666
0.25, 0.75
0.0, 1.0
Note 0.333, 0.666 ? 0.25, 0.75? 0.0, 1.0
0.0, 1.0and 0,1 ? 0.0, 1.0 ? 0,1
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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
  a-cuts

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Continuous Cloud Example

• A cloud over R with a0.6

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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.

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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

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Potential Clouds

• Let be r.v. with values in M, and let be
  bounded below. Thendefines a cloud x with
  whose a-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

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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

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Expectation

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

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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

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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.

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Confocal Cloud Example
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Examples Preface

• The examples on the next slides were generated
  from a normal distribution with a fixed mean on a
  -5,52 mea
• The functions and refer to
  and respectively
• The ellipsoidal potential map is given by

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Pretty Pictures
Normal Distrib
3D Cloud
Level Set
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Sample Size Influence
Normal Distrib 10 Sample Points 1000 Sample Points


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Confidence Level Influence
Normal Distrib Confidence 99 Confidence 99.9


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Bottom Line

• So whats 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

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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

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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,
  249-272http//www.mat.univie.ac.at/neum/ms/cloud
  .pdf
• Neumaier, A. 2003. On the Structure of Clouds.
  Unpublished Manuscript.http//www.mat.univie.ac.a
  t/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
  

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Fin