Transcript Slide 1

Neumaier Clouds
Yan Bulgak
[email protected]
October 30, MAR550, Challenger 165
Sets
• Defined by membership criterion:
If A a set, then for all
either
or
Fuzzy Sets
• Defined by a “degree of membership”
function:
(A, ) is a fuzzy set if A is a set and
• For each
membership
,
is the grade of
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
Clouds – Formal Definition
• A cloud over a set M is a mapping x that
associates with each
a (nonempty,
closed, bounded) interval
such that:
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
And Now For a Picture
Cloud x
M
x(
)
[0.333, 0.666]
x(
x(
)
)
[0.25, 0.75]
[0.0, 1.0]
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]
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
–
and
–
the level of in x;
are the upper and lower levels
and
are the upper and lower α-cuts
Continuous Cloud Example
A cloud over ℝ with α=0.6
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.
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
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
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
Expectation
• In general, computable only using linear
programming and global optimization
techniques.
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
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.
Confocal Cloud Example
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
Pretty Pictures
Normal
Distrib
3D
Cloud
Level Set
Sample Size Influence
Normal
Distrib
10 Sample Points
1000 Sample Points
Confidence Level Influence
Normal
Distrib
Confidence 99%
Confidence 99.9%
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
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
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
Neumaier, A. 2003. On the Structure of Clouds. Unpublished Manuscript.
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
Fin