Some Aspects on Mathematical Treatments of Uncertainty and Their Applications

1. Some 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

3. I. 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.

10. Luo 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.

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.)

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, ……

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);

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).

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.

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.

23. Luo Mao-Kang: \begin{picture}(500,250) \thicklines \put(30,90){\small$r(k)$}\\ \put(75,95){\vector(1,0){40}}\\ \put(52,63){\small$r(k)$}\\ \put(95,95){\vector(0,-1){55}}\\ \put(116,85){\framebox(40,20)[c]{\footnotesize$M_1$}}\\ \put(157,95){\vector(1,0){35}}\\ \put(104,63){\small$r(k-1)$}\\ \put(172,95){\vector(0,-1){55}}\\ \put(196,94){$\ldots$}\\ \put(228,95){\vector(1,0){35}}\\ \put(263,85){\framebox(40,20)[c]{\footnotesize $M_{n-1}$}}\\ \put(304,95){\line(1,0){35}}\\ \put(238,63){\small$r(k-n+1)$}\\ \put(339,95){\vector(0,-1){55}}\\ \put(60,16){\framebox(310,24)[b]{$The\ fuzzy\ blind\ equalizer$}}\\ \put(140,-5){\small$y(k-d)$}\\ \put(207,12){\vector(0,-1){30}}\\ \end{picture} 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.

25. Luo 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.

26. Luo 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

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.

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?

35. Luo 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.

36. Luo 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 I：Neighborhood 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 II：Product 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. Thank you!