Some Aspects on Mathematical Treatments of Uncertainty and Their Applications

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Some Aspects on Mathematical Treatments of
Uncertainty and Their Applications

• Luo Mao-Kang
• Institute of Mathematics
• Sichuan University
• Chengdu, 610064
• China

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• Outline
• Uncertainty
• Uncertainties in Research and Engineering
• Related Work
• Views and Ideas

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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

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• The Rayleigh Criterion in Optics
• Resolution of an optical microscope
• in the best condition, .

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• 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

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• 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.

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• 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.

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• Two sorts of uncertainty often considered
• Randomness
• Causality ? Causal Law ? Formal Logic
• Randomness Uncertainties of causality,
  Insufficient causality.

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• Fuzziness
• Age, Aged man, Health,
• Crispness Property stated by the Law of Excluded
  Middle in formal logic.
• Fuzziness Uncertainties of concepts,
  Insufficient crispness.

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Luo Mao-Kang Connotation

• Crisp view
• Identify a concept with its extension
• (contrasted with its connotation)
• -- an ordinary set,
• then

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• 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.

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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.)

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• 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.)

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• 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,

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• 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,

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• 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,

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• Tabu search algorithm,
• Rough set theory,
• Computing with words,
• Chaos theory,
• Fractal theory,
• Wavelet analysis,
• Data mining,

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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)

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• 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),

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• 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).

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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.

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• 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.

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• Based on fuzzy system and fuzzy control,
  associated with CMA and RLS, a fuzzy algorithm
  doubling the usual accuracy and convergence is
  presented.

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• 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.

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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.

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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)

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• 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

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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.

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• 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.

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• 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

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• 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.

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• 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.

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• 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,

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• 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?

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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.

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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.

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• 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.

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• 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.

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• 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.

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• 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.

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• 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.

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• 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.

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• Thank you!