Title: Some Aspects on Mathematical Treatments of Uncertainty and Their Applications
1Some Aspects on Mathematical Treatments of
Uncertainty and Their Applications
- Luo Mao-Kang
- Institute of Mathematics
- Sichuan University
- Chengdu, 610064
- China
2- Outline
- Uncertainty
- Uncertainties in Research and Engineering
- Related Work
- Views and Ideas
3I. Uncertainty
- Uncertainties
- Impossible to be determinate by rules of the
objective world. - The Heisenberg Uncertainty Principle (1927)
- Position and momentum of a particle cannot be
accurately determined at the same time
4- The Rayleigh Criterion in Optics
- Resolution of an optical microscope
- in the best condition, .
5- The Time-Frequency Uncertainty Principle in
Communication - Signal
- Frequency spectrum of
- Frequency property of in a neighborhood of
time Observe and through
time-window and frequency window - Then the widths of these two windows
6- Uncertainty of age Time of birth cannot be
accurately defined, even time can be accurately
mensurated. - Unnecessary to be determinate Excessive
exactness causes disturbances of redundancy
information. - Concept Age Unnecessary to determine one's age
in seconds.
7- Concept Aged man Unnecessary to determine in
seconds whether a man has been aged or not, let
alone age cannot been accurately defined. - Concept Health Health consists of many
indexes, each of them is unnecessary to be very
accurate.
8- Two sorts of uncertainty often considered
- Randomness
- Causality ? Causal Law ? Formal Logic
- Randomness Uncertainties of causality,
Insufficient causality.
9- Fuzziness
- Age, Aged man, Health,
- Crispness Property stated by the Law of Excluded
Middle in formal logic. - Fuzziness Uncertainties of concepts,
Insufficient crispness.
10Luo Mao-Kang Connotation
- Crisp view
- Identify a concept with its extension
- (contrasted with its connotation)
- -- an ordinary set,
- then
11- Fuzzy view or is not clear or
crisp or trenchant, so a concept - is a mapping from to value range ,
-
- or to some kind of more general ordered structure
, - .
- That means
- Truth of possesses property may be a
degree different from both 0 and 1.
12II. Uncertainties in Research and Engineering
- Many problems of uncertainty have been considered
in classical mathematics, e.g., - Cybernetics ( -- Established in World War II,
uncertainties in harmonizing movements of
aircrafts and ground firepower to air, and wave
filtering in communication.) - Queueing Theory ( -- Established in the
beginning of 20th century, uncertainty of
communi-cation calls.)
13- Game Theory ( -- Uncertainty of behavior and
strategies of other antagonistic sides.) - Search Theory ( -- Established in World War II,
uncertainties of the positions of enemy
submarines when they were searched.)
14- More and more problems of uncertainty appear in
natural science, social science, technology - Information hiding,
- Weak signal detection,
- Low interception probability signal search,
- Information compression with high bit rate
- and low code rate,
- Gain and bandwidth of an amplifier,
- Early warning to enterprises under
- uncertain conditions,
15- Determination of time information and frequency
information, - Improvement of reliability and efficiency of
coding, - Natural language processing,
- Turbulent flow,
- Variation of sunspot,
- Atrial fibrillation,
- Rule of outbreak of contagious diseases,
- Pathogenesis of psychosis,
-
16- Both classical and non-classical mathe-matical
theories, methods and tools are possible to be
used into processing uncer-tainty. - Besides classical part, non-classical part
usually includes following branches - Fuzzy logic,
- Fuzzy control,
- Artificial nueral network,
- Genetic algorithm,
- Simulated annealing algorithm,
17- Tabu search algorithm,
- Rough set theory,
- Computing with words,
- Chaos theory,
- Fractal theory,
- Wavelet analysis,
- Data mining,
-
18III. Related Work
- On uncertainty, our previous work on (see
4,6,7,9,10,11) - Fuzzy set theory and topology,
- Fuzzy system and fuzzy control,
- Lattice theory,
- Locale theory (with dual objects of frames --
mathematical model of intuitional logic), - Domain theory (a branch of theoretical computer
science, model of denotational semantics in
formal semantics)
19- Including
- Multiple Choice Principle (Liu, 1977-1980),
- Stratified structure analysis (Liu, Luo,
1985-1998), - Dimension deduction (Liu, Li, 1991-1994 in this
aspect, to a class of associative functions by a
monotone 1-place function and addition, Ying-Ming
Liu and Zhong-Fu Li gave out a kind of
approximate representation in any requested
accuracy),
20- Self-adjusting of memberships and triangular
norms in fuzzy control (Li, Liu, 1999- based on
the results on dimension deduction mentioned
above), - Resolutions of problems of domain theory in Open
Problems in Topology (J. Van Mill and G.M. Reed,
North-Holland, 1990) (Liu, Liang, Kou, Luo,
1996-2003).
21Some work related to uncertainties in signal,
communication and control
- 1. Blind Equalization of Constant Modulus Signals
in Nonlinear Wireless Channels - Digital wireless communication systems
- Two major kinds of impairment to the channel
Noise and intersymbol interference (ISI). - ISI causes high bit error rate (BER).
- Equalization Filter designed for equalizing the
ISI.
22- In the case of multipoint mobile communi-cation,
multi-path and mobility cause the nonlinearity of
channels and the need to blind equalization. - Some knotty problems be often caused by using
usual equalizations in nonlinear channels.
23Luo Mao-Kang \beginpicture(500,250) \thicklines
\put(30,90)\smallr(k)\\ \put(75,95)\vector(1
,0)40\\ \put(52,63)\smallr(k)\\ \put(95,95)
\vector(0,-1)55\\ \put(116,85)\framebox(40,20
)c\footnotesizeM_1\\ \put(157,95)\vector(1
,0)35\\ \put(104,63)\smallr(k-1)\\ \put(172
,95)\vector(0,-1)55\\ \put(196,94)\ldots\\
\put(228,95)\vector(1,0)35\\ \put(263,85)\fr
amebox(40,20)c\footnotesize M_n-1\\ \put(
304,95)\line(1,0)35\\ \put(238,63)\smallr(k-
n1)\\ \put(339,95)\vector(0,-1)55\\ \put(60
,16)\framebox(310,24)bThe\ fuzzy\ blind\
equalizer\\ \put(140,-5)\smally(k-d)\\ \put
(207,12)\vector(0,-1)30\\ \endpicture
- Based on fuzzy system and fuzzy control,
associated with CMA and RLS, a fuzzy algorithm
doubling the usual accuracy and convergence is
presented.
24- 2. Real-time Quasi-Blind Adaptive Nonlinear
Equalization - Based on N-pseudo recursive fuzzy c-means
algorithm, a fuzzy controller is designed for a
nonlinear equalization, which is real-time,
quasi-blind and adap-tive, and it can neglect the
influence of nonlinear distortion.
25Luo Mao-Kang Population Gene Chromosome
Reproduction Crossover mutation fitness
offspring
- 3. Multi-user Detection Based on Genetic
Algorithm and Wavelet Analysis - In multi-user communication, multi-access
interference and far-near effect is its major
problems. The computational complexity of the
so-called optimum multi-user detector will
exponentially increase along with the increasing
of users.
26Luo Mao-Kang Usually, polynomial, because the
computation is executed on some kind of indexes
of parameters but not parameters themselves.
- Genetic algorithm, especially the one improved in
recent years, is a kind of heuristic algorithm
with lower compu-tational complexity, can
overcome the problem of exponential increase of
the search space. - (Usually, polynomial, because the computation is
executed on some kind of indexes of parameters
but not parameters themselves.) - 4. Signal Detection for Frequency Hopping Based
on Power Ordered Sets (Certain Uncertain)
27- 5. Speech Recognition Based on Genetic Algorithm
in Low Signal-to-Noise Ratio Communication - 6. Synchronization of Weak Signals Based on
Fractal Theory and Wavelet Analysis - 7. Early Warning to Enterprises under Uncertain
Conditions Based on Genetic Algorithm, Simulated
Annealing Algorithm and Neural Network
28IV. Views and Ideas
- Mathematical treatment of uncertainty
- Classical branches,
- Non-classical branches.
- There not exist a clear borderline of them in
research and applications on uncertainty. - Any parts of them can be combined even
syncretized together for a concrete aim on
uncertainty.
29- Soft computing A sort of widely used theories,
methods and algorithms on uncertainty - Usually considered to include
- Fuzzy logic, fuzzy control, artificial nueral
network, genetic algorithm, simu-lated annealing
algorithm, computing with words, - With these theories, methods and algorithms, one
can seek adequate but maybe not very accurate
resolutions for certain aims on uncertainty.
30- To use them,
- not necessary to have known too much details of a
certain concrete process, - but let these factors affect others under the
rules and limitations of the process itself, - and therefore obtain a last result.
- Because of these reasons, these theories, methods
and algorithms have some common characteristics
31- 1. Need not the continuities or convexities of
objective functions and constraints, even need
not analytic expressions. - 2. Possess characteristics of self-learning,
self-organizing, self-adaptive. - 3. Can be executed parallelly and
distri-butively. - 4. Usually be simpler, more universal and more
robust. - 5. Usually have lower costs on software, hardware
and time.
32- But however, they have still some problems
- 1. They are not mature, still being improved
continually. - 2. Their interior action mechanisms and theoretic
bases are still in studying. - 3. They cannot ensure their reso-lution being
optimum.
33- Many research results on the interior action
mechanisms and the theoretic bases of soft
computing, e.g., study on - search mechanism,
- convergency,
- conver-gent speed,
- complexity,
- effectiveness,
- solvability,
-
34- Consideration
- Relations are often more important than other
factors in the executions of soft computing
algorithms. - Considering their limitations, can we
- Introduce theories and methods of
- Ordered structure, algebra even topology
- Combining with that of probability theory and
stochastic process into the study on soft
computing?
35Luo Mao-Kang (in international congresses on
cybernetics in recent years, more than a half of
papers involved fuzziness)
- Improvements of mathematical theories, methods
and tools on uncertainty in considering - 1. Fuzzy controlor can be used as a universal
approximator for most of control process,
especially effectual in manually interfered
processes. - Determinations of membership functions and
triangular norms often consume much workload in a
design of fuzzy controlor. - Use nueral network, genetic algorithm to adjust
and optimize membership function and triangular
norms in fuzzy control, will decrease this
workload and optimize the result.
36Luo Mao-Kang (Some kinds of improvements of
them have considered relations among these
objects, but still not enough)
- 2. In genetic algorithm and simulated annealing
algorithm, crossover, mutation and perturbation
are often impartially executed for all chosen
objects with same randomicity. This kind of
operation push the result close to global
optimum, but - (1) Decrease the convergent speed,
- (2) Maybe waste some useful infor-mation about
the differences among these objects especially
their relations.
37- Some kind of mathematical structure, such as
various partially ordered sets, lattices and so
on, can be introduced to describe these relations
and differences, and then design different
crossover, mutation and perturbation to increase
the convergent speed and the probability of
closing to the global optimum.
38- 3. All the methods of programming,
queueing theory, game theory, decision theory and
so on, are established for some kinds of
optimizations, therefore, getting to - Gain or Win or Equilibrium
- via competition are often their major aims.
- Many branches of soft computing are just
designed for competition and/or equili-brium. - Introduce soft computing into these
braches of operational research, combine them
into various mixtures for different conditions
and aims.
39- For example, electronic warfare is a kind of very
complex confrontation. - Besides manual operations, automatic operation
occupies more important position, and hence the
most of confrontation strategies are
self-adaptive.
40- Game theory is one of the most often used
branches in the mathematical aspect of this
problem. - But in game theory, existence of a solution of a
game needs some strict conditions, and they often
cannot be completely satisfied in real
confrontations. - In these cases, usually, soft computing can still
do the job well under the framework of game
theory.
41- References
- Theresa Beaubouef etc., Fuzzy Rough Set Technique
for uncertainty processing in Relational Database
J, Intemational Journal of Intelligent System,
2000(5) , 23-27. - L. Davis, Genetic Algorighms and Simulated
Annealing, Los Altos, CA Morgan Kaufmann
Publishers, 1987. - T. Fogarty, Evolutionary Computings, Berlin
Springer-Verlag, 1994. - He Wei and Liu Yingming, Steenrod's theorem for
locales, Math. Proc. Cambridge Phil. Soc.,
124(1998), Part 2, 305-307. - J. Van Mill and G. M. Reed, Editors, Open
Problems in Topology, North-Holland, Amsterdam,
1990.
42- Zhong-Fu Li, Ying-Ming Liu, An approach to the
management of uncertainty in expert systems,
Analysis and Management of Uncertainty Theory
and Applications, Eds. B.M.Ayyub, 1991, Elsevier,
Amsterdam, 133-140. - Zhong-Fu Li, Ying-Ming Liu, Approximate
represen-tation of a class of associative
functions by a monotone 1-place function and
addition, Science in China, Ser. A, 37(1994),
No.7, 769-779. - L. Polkowski, Rough Sets -- Mathematical
Foundations, Physica-Verlag, 2002. - Bao-Ming Pu, Ying-Ming Liu, Fuzzy topology
INeighborhood structure of a fuzzy point and
Moore-Smith convergence,J.Math.Anal.Appl.,76(1980)
,571-599 (with Pu, Bao-Ming). - Bao-Ming Pu, Ying-Ming Liu, Fuzzy topology
IIProduct and quotient spaces (with B.
Pu),J.Math.Anal.Appl.,77(1980),20-39 (with Pu,
Bao-Ming). - Ying-Ming Liu and Mao-Kang Luo, Fuzzy Topology,
World Sci. Publ., Singapore, 1998.
43