Fuzzy Logic & Approximate Reasoning 1. 2 Fuzzy Sets

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  • Fuzzy Logic & Approximate Reasoning*Fuzzy Logic & Approximate Reasoning

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Sets

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*ReferencesJournal:IEEE Trans. on Fuzzy Systems.Fuzzy Sets and Systems.Journal of Intelligent & Fuzzy SystemsInternational Journal of Uncertainty, Fuzziness and Knowledge-Based Systems ...Conferences:IEEE Conference on Fuzzy Systems.IFSA World Congress....Books and Papers:Z.Chi et al, Fuzzy Algorithms with applications to Image Processing and Pattern Recognition, World Scientific, 1996.S. N. Sivanandam, Introduction to Fuzzy Logic using MATLAB, Springer, 2007.J.M. Mendel, Fuzzy Logic Systems for Engineering: A toturial, IEEE, 1995.W. Siler, FUZZY EXPERT SYSTEMS AND FUZZY REASONING, John Wiley Sons, 2005G. Klir, Uncertainty and Informations, John Wiley Sons, 2006.L.A. Zadeh, Fuzzy sets, Information and control, 8, 338-365, 1965.L.A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I, II, III, Information Science 8, 1975L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems 90(1997), 111-127.......http://www.type2fuzzylogic.org/IEEE Computational Intelligence Society http://ieee-cis.org/International Fuzzy Systems Association http://www.isc.meiji.ac.jp/~ifsatkym/J.M. Mendel http://sipi.usc.edu/~mendel

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*ContentsFuzzy sets.Fuzzy Relations and Fuzzy reasoningFuzzy Inference SystemsFuzzy ClusteringFuzzy Expert SystemsApplications: Image Processing, Robotics, Control...

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Sets: OutlineIntroduction: History, Current Level and Further Development of Fuzzy Logic Technologies in the U.S., Japan, and Europe Basic definitions and terminologySet-theoretic operationsMF formulation and parameterizationMFs of one and two dimensionsDerivatives of parameterized MFsMore on fuzzy union, intersection, and complementFuzzy complementFuzzy intersection and unionParameterized T-norm and T-conormFuzzy NumberFuzzy Relations

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*History, State of the Art, and Future Development1965Seminal Paper Fuzzy Logic by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the Fuzzy Set Theory1970First Application of Fuzzy Logic in Control Engineering (Europe)1975Introduction of Fuzzy Logic in Japan 1980Empirical Verification of Fuzzy Logic in Europe1985Broad Application of Fuzzy Logic in Japan1990Broad Application of Fuzzy Logic in Europe1995Broad Application of Fuzzy Logic in the U.S.1998Type-2 Fuzzy SystemsFuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Types of Uncertainty and the Modeling of UncertaintyStochastic Uncertainty:The Probability of Hitting the Target Is 0.8

    Lexical Uncertainty:"Tall Men", "Hot Days", or "Stable Currencies"We Will Probably Have a Successful Business Year.The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True.Most Words and Evaluations We Use in Our Daily Reasoning Are Not Clearly Defined in a Mathematical Manner. This Allows Humans to Reason on an Abstract Level!

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Possible Sources of Uncertaintyand ImprecisionThere are many sources of uncertainty facing any control system in dynamic real world unstructured environments and real world applications; some sources of these uncertainties are as follows: Uncertainties in the inputs of the system due to:The sensors measurements being affected by high noise levels from various sources such a electromagnetic and radio frequency interference, vibration, etc.The input sensors being affected by the conditions of observation (i.e. their characteristics can be changed by the environmental conditions such as wind, sunshine, humidity, rain, etc.).

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Possible Sources of Uncertaintyand ImprecisionOther sources of Uncertainties include: Uncertainties in control outputs which can result from the change of the actuators characteristics due to wear and tear or due to environmental changes. Linguistic uncertainties as words mean different things to different people. Uncertainties associated with the change in the operation conditions due to varying load and environment conditions.

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Set TheoryConventional (Boolean) Set Theory:

    40.1C42C41.4C39.3C38.7C37.2C38CFuzzy Set Theory:

    40.1C42C41.4C39.3C38.7C37.2C38CMore-or-Less Rather Than Either-Or !Strong Fever

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy SetsSets with fuzzy boundariesA = Set of tall people

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Membership Functions (MFs)Characteristics of MFs:Subjective measuresNot probability functions

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy SetsA fuzzy set A is characterized by a member set function (MF), A, mapping the elements of A to the unit interval [0, 1].Formal definition:A fuzzy set A in X is expressed as a set of ordered pairs:

    Membershipfunction(MF)

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Sets with Discrete UniversesFuzzy set C = desirable city to live inX = {SF, Boston, LA} (discrete and non-ordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}Fuzzy set A = sensible number of childrenX = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2),(6, .1)}

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Sets with Cont. UniversesFuzzy set C = desirable city to live inX = {SF, Boston, LA} (discrete and non-ordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}Fuzzy set A = sensible number of childrenX = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2),(6, .1)}

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Alternative NotationA fuzzy set A can be alternatively denoted as follows:X is discreteX is continuousNote that S and integral signs stand for the union of membership grades; / stands for a marker and does not imply division.

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy PartitionFuzzy partitions formed by the linguistic values young, middle aged, and old:

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Linguistic VariablesA linguistic variable is a variable whose values are not numbers but words or sentences in a natural or artificial language (Zadeh, 1975a, p. 201) Linguistic variable is characterized by [c , T(c), U], in which c : name of the variable, T(c) : the term set of c , universe of discourse UA Linguistic Variable Defines a Concept of Our Everyday Language!

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy HedgesSuppose you had already defined a fuzzy set to describe a hot temperature. Fuzzy set should be modified to represent the hedges "Very" and "Fairly: very hot or fairly hot.

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*MF Terminology

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Convexity of Fuzzy SetsAlternatively, A is convex if all its a-cuts are convex.A fuzzy set A is convex if for any l in [0, 1],

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Set-Theoretic OperationsSubset:

    Complement:

    Union:

    Intersection:

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Set-Theoretic Operations

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*MF FormulationTriangular MF:Trapezoidal MF:Generalized bell MF:Gaussian MF:

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*MF Formulation

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*MF FormulationSigmoidal MF:Extensions:Abs. differenceof two sig. MFProductof two sig. MF

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy ComplementGeneral requirements:Boundary: N(0)=1 and N(1) = 0Monotonicity: N(a) > N(b) if a < bInvolution: N(N(a) = aTwo types of fuzzy complements:Sugenos complement:

    Yagers complement:

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy ComplementSugenos complement:Yagers complement:

    Fuzzy Logic & Approximate Reasoning

  • Fuzzy Logic & Approximate Reasoning*Fuzzy Intersection: T-normBasic requirements:Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = aMonotonicity: T(a, b) < T(c, d) if a < c and b < dCommutativity: T(a, b) = T(b, a)Associativity: T(a, T(