1 Some Aspects on Mathematical Treatments of Uncertainty and Their Applications Luo Mao-Kang...
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1 Some Aspects on Mathematical Treatments of Uncertainty and Their Applications Luo Mao-Kang Institute of Mathematics Sichuan University Chengdu, 610064
1 Some Aspects on Mathematical Treatments of Uncertainty and
Their Applications Luo Mao-Kang Institute of Mathematics Sichuan
University Chengdu, 610064 China
Slide 2
2 Outline: Uncertainty Uncertainties in Research and
Engineering Related Work Views and Ideas
Slide 3
3 I. Uncertainty Uncertainties: Impossible to be determinate
Impossible to be determinate: by rules of the objective world.
Heisenberg Uncertainty Principle The Heisenberg Uncertainty
Principle (1927): Position and momentum of a particle cannot be
accurately determined at the same time:
Slide 4
4 Rayleigh Criterion The Rayleigh Criterion in Optics:
Resolution of an optical microscope in the best condition,.
Slide 5
5 Time-Frequency Uncertainty Principle 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:
Slide 6
6 Uncertainty of age Uncertainty of age: Time of birth cannot
be accurately defined, even time can be accurately mensurated.
Unnecessary to be determinate Unnecessary to be determinate:
Excessive exactness causes disturbances of redundancy information.
Concept Age Concept Age: Unnecessary to determine one's age in
seconds.
Slide 7
7 Concept Aged man 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 Concept Health: Health
consists of many indexes, each of them is unnecessary to be very
accurate.
Slide 8
8 Two sorts of uncertainty often considered Two sorts of
uncertainty often considered: Randomness Randomness: Causality
Causal Law Formal Logic Randomness: Uncertainties of causality,
Insufficient causality.
Slide 9
9 Fuzziness Fuzziness: Age, Aged man, Health, Crispness:
Property stated by the Law of Excluded Middle in formal logic.
Fuzziness: Uncertainties of concepts, Insufficient crispness.
Slide 10
10 Crisp view: Identify a concept with its extension
(contrasted with its connotation) -- an ordinary set, then Luo
Mao-Kang: Connotation Luo Mao-Kang: Connotation
Slide 11
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.
Slide 12
12 II. 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. )
Slide 13
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. )
Slide 14
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,
Slide 15
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,
Slide 16
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,
Slide 17
17 Tabu search algorithm, Rough set theory, Computing with
words, Chaos theory, Fractal theory, Wavelet analysis, Data
mining,
Slide 18
18 III. 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 ) ;
Slide 19
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 ),
Slide 20
20 Self-adjusting of memberships and triangular norms in fuzzy
control ( Li, Liu, 1999-; based on the results on dimension
deduction mentioned above ), Open Problems in Topology 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 ).
Slide 21
21 Some 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.
Slide 22
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.
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.
Slide 25
25 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. Luo Mao-Kang: Population; Gene; Chromosome; Reproduction;
Crossover; mutation; fitness; offspring Luo Mao-Kang: Population;
Gene; Chromosome; Reproduction; Crossover; mutation; fitness;
offspring
Slide 26
26 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 4. Signal Detection for Frequency Hopping Based
on Power Ordered Sets (Certain Uncertain) Luo Mao-Kang: Usually,
polynomial, because the computation is executed on some kind of
indexes of parameters but not parameters themselves. Luo Mao-Kang:
Usually, polynomial, because the computation is executed on some
kind of indexes of parameters but not parameters themselves.
Slide 27
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
Slide 28
28 IV. 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.
Slide 29
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, adequate but
maybe not very accurate With these theories, methods and
algorithms, one can seek adequate but maybe not very accurate
resolutions for certain aims on uncertainty.
Slide 30
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:
Slide 31
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.
Slide 32
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.
Slide 33
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,
Slide 34
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?
Slide 35
35 Improvements of mathematical theories, methods and tools on
uncertainty in considering: manually interfered 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. Luo Mao-Kang: ( in international
congresses on cybernetics in recent years, more than a half of
papers involved fuzziness ) Luo Mao-Kang: ( in international
congresses on cybernetics in recent years, more than a half of
papers involved fuzziness )
Slide 36
36 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. Luo Mao-Kang: ( Some kinds of improveme nts of them have
considered relations among these objects, but still not enough )
Luo Mao-Kang: ( Some kinds of improveme nts of them have considered
relations among these objects, but still not enough )
Slide 37
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.
Slide 38
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 GainWinEquilibrium 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.
Slide 39
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.
Slide 40
40 one of 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.
soft computing can still do the job well under the framework of
game theory. In these cases, usually, soft computing can still do
the job well under the framework of game theory.
Slide 41
41 References 1.Theresa Beaubouef etc., Fuzzy Rough Set
Technique for uncertainty processing in Relational Database [J],
Intemational Journal of Intelligent System, 2000(5), 23-27. 2.L.
Davis, Genetic Algorighms and Simulated Annealing, Los Altos, CA:
Morgan Kaufmann Publishers, 1987. 3.T. Fogarty, Evolutionary
Computings, Berlin: Springer-Verlag, 1994. 4.He Wei and Liu
Yingming, Steenrod's theorem for locales, Math. Proc. Cambridge
Phil. Soc., 124(1998), Part 2, 305-307. 5.J. Van Mill and G. M.
Reed, Editors, Open Problems in Topology, North-Holland, Amsterdam,
1990.
Slide 42
42 7.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. 8.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. 9.L. Polkowski, Rough Sets -- Mathematical
Foundations, Physica-Verlag, 2002. 10.Bao-Ming Pu, Ying-Ming Liu,
Fuzzy topology I Neighborhood structure of a fuzzy point and Moore-
Smith convergence J.Math.Anal.Appl. 76(1980) 571-599 (with Pu,
Bao-Ming). 11.Bao-Ming Pu, Ying-Ming Liu, Fuzzy topology II Product
and quotient spaces (with B. Pu) J.Math.Anal.Appl. 77(1980) 20-39
(with Pu, Bao- Ming). 12.Ying-Ming Liu and Mao-Kang Luo, Fuzzy
Topology, World Sci. Publ., Singapore, 1998.
