Fuzzy Logic & Approximate Reasoning 1

2 Fuzzy Sets

Fuzzy Logic & Approximate Reasoning 3 References Journal: –IEEE Trans. on Fuzzy Systems. –Fuzzy Sets and Systems. –Journal of Intelligent & Fuzzy Systems – International 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, –S. N. Sivanandam, Introduction to Fuzzy Logic using MATLAB, Springer, –J.M. Mendel, Fuzzy Logic Systems for Engineering: A toturial, IEEE, –W. Siler, FUZZY EXPERT SYSTEMS AND FUZZY REASONING, John Wiley Sons, 2005 –G. Klir, Uncertainty and Informations, John Wiley Sons, –L.A. Zadeh, Fuzzy sets, Information and control, 8, , –L.A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I, II, III, Information Science 8, 1975 –L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems 90(1997), – – –IEEE Computational Intelligence Societỵ –International Fuzzy Systems Association –J.M. Mendel

Fuzzy Logic & Approximate Reasoning 4 Contents Fuzzy sets. Fuzzy Relations and Fuzzy reasoning Fuzzy Inference Systems Fuzzy Clustering Fuzzy Expert Systems Applications: Image Processing, Robotics, Control...

Fuzzy Logic & Approximate Reasoning 5 Fuzzy Sets: Outline Introduction: History, Current Level and Further Development of Fuzzy Logic Technologies in the U.S., Japan, and Europe Basic definitions and terminology Set-theoretic operations MF formulation and parameterization –MFs of one and two dimensions –Derivatives of parameterized MFs More on fuzzy union, intersection, and complement –Fuzzy complement –Fuzzy intersection and union –Parameterized T-norm and T-conorm Fuzzy Number Fuzzy Relations

Fuzzy Logic & Approximate Reasoning 6 History, State of the Art, and Future Development 1965Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory” 1970First Application of Fuzzy Logic in Control Engineering (Europe) 1975Introduction of Fuzzy Logic in Japan 1980Empirical Verification of Fuzzy Logic in Europe 1985Broad Application of Fuzzy Logic in Japan 1990Broad Application of Fuzzy Logic in Europe 1995Broad Application of Fuzzy Logic in the U.S. 1998Type-2 Fuzzy Systems 2000Fuzzy Logic Becomes a Standard Technology and Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance. 1965Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Foundation of the “Fuzzy Set Theory” 1970First Application of Fuzzy Logic in Control Engineering (Europe) 1975Introduction of Fuzzy Logic in Japan 1980Empirical Verification of Fuzzy Logic in Europe 1985Broad Application of Fuzzy Logic in Japan 1990Broad Application of Fuzzy Logic in Europe 1995Broad Application of Fuzzy Logic in the U.S. 1998Type-2 Fuzzy Systems 2000Fuzzy 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 7 Types of Uncertainty and the Modeling of Uncertainty Stochastic Uncertainty: XThe Probability of Hitting the Target Is 0.8 Lexical Uncertainty: X"Tall Men", "Hot Days", or "Stable Currencies" XWe Will Probably Have a Successful Business Year. XThe Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True. Stochastic Uncertainty: XThe Probability of Hitting the Target Is 0.8 Lexical Uncertainty: X"Tall Men", "Hot Days", or "Stable Currencies" XWe Will Probably Have a Successful Business Year. XThe 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 8 Possible Sources of Uncertainty and Imprecision There 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 9 Possible Sources of Uncertainty and Imprecision Other 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 10 Fuzzy Set Theory Conventional (Boolean) Set Theory: “Strong Fever” 40.1°C 42°C 41.4°C 39.3°C 38.7°C 37.2°C 38°C Fuzzy Set Theory: 40.1°C 42°C 41.4°C 39.3°C 38.7°C 37.2°C 38°C “More-or-Less” Rather Than “Either-Or” ! “Strong Fever”

Fuzzy Logic & Approximate Reasoning 11 Fuzzy Sets Sets with fuzzy boundaries A = Set of tall people

Fuzzy Logic & Approximate Reasoning 12 Membership Functions (MFs) Characteristics of MFs: –Subjective measures –Not probability functions

Fuzzy Logic & Approximate Reasoning 13 Fuzzy Sets A 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: Membership function (MF) Universe or universe of discourse Fuzzy set Universe or universe of discourse Universe or universe of discourse

Fuzzy Logic & Approximate Reasoning 14 Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and non-ordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {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 15 Fuzzy Sets with Cont. Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and non-ordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {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 16 Alternative Notation A fuzzy set A can be alternatively denoted as follows: X is discrete X is continuous Note that  and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

Fuzzy Logic & Approximate Reasoning 17 Fuzzy Partition Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

Fuzzy Logic & Approximate Reasoning 18 Linguistic Variables A 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 [ , T(  ), U], in which  : name of the variable, T(  ) : the term set of , universe of discourse U A 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 [ , T(  ), U], in which  : name of the variable, T(  ) : the term set of , universe of discourse U A Linguistic Variable Defines a Concept of Our Everyday Language!

Fuzzy Logic & Approximate Reasoning 19 Fuzzy Hedges Suppose 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 20 MF Terminology

Fuzzy Logic & Approximate Reasoning 21 Convexity of Fuzzy Sets Alternatively, A is convex if all its  -cuts are convex. A fuzzy set A is convex if for any l in [0, 1],

Fuzzy Logic & Approximate Reasoning 22 Set-Theoretic Operations Subset: Complement: Union: Intersection:

Fuzzy Logic & Approximate Reasoning 23 Set-Theoretic Operations

Fuzzy Logic & Approximate Reasoning 24 MF Formulation Triangular MF: T rapezoidal MF: Generalized bell MF: Gaussian MF:

Fuzzy Logic & Approximate Reasoning 25 MF Formulation

Fuzzy Logic & Approximate Reasoning 26 MF Formulation Sigmoidal MF: Extensions: Abs. difference of two sig. MF Product of two sig. MF

Fuzzy Logic & Approximate Reasoning 27 Fuzzy Complement General requirements: –Boundary: N(0)=1 and N(1) = 0 –Monotonicity: N(a) > N(b) if a < b –Involution: N(N(a) = a Two types of fuzzy complements: –Sugeno’s complement: –Yager’s complement:

Fuzzy Logic & Approximate Reasoning 28 Fuzzy Complement Sugeno’s complement: Yager’s complement:

Fuzzy Logic & Approximate Reasoning 29 Fuzzy Intersection: T-norm Basic requirements: –Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a –Monotonicity: T(a, b) < T(c, d) if a < c and b < d –Commutativity: T(a, b) = T(b, a) –Associativity: T(a, T(b, c)) = T(T(a, b), c) Four examples: –Minimum: Tm(a, b) = min{a, b}. –Algebraic product: Ta(a, b) = a.b –Bounded product: Tb(a, b) = max{0, a+b-1} –Drastic product: Td(a, b)

Fuzzy Logic & Approximate Reasoning 30 T-norm Operator Minimum: T m (a, b) Algebraic product: T a (a, b) Bounded product: T b (a, b) Drastic product: T d (a, b)

Fuzzy Logic & Approximate Reasoning 31 Fuzzy Union: T-conorm or S-norm Basic requirements: –Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a –Monotonicity: S(a, b) < S(c, d) if a < c and b < d –Commutativity: S(a, b) = S(b, a) –Associativity: S(a, S(b, c)) = S(S(a, b), c) Four examples: –Maximum: Sm(a, b) = max{a, b} –Algebraic sum: Sa(a, b) = a+b-a.b –Bounded sum: Sb(a, b) = min{a+b, 1}. –Drastic sum: Sd(a, b) =

Fuzzy Logic & Approximate Reasoning 32 T-conorm or S-norm Maximum: S m (a, b) Algebraic sum: S a (a, b) Bounded sum: S b (a, b) Drastic sum: S d (a, b)

Fuzzy Logic & Approximate Reasoning 33 Fuzzy Number A fuzzy number A must possess the following three properties: 1. A must must be a normal fuzzy set, 2. The alpha levels must be closed for every, 3. The support of A,, must be bounded.

Fuzzy Logic & Approximate Reasoning 34 Fuzzy Number 1 Membership function is the suport of z 1 is the modal value  is an  -level of,  (0,1]  ’’

Fuzzy Logic & Approximate Reasoning 35 Fuzzy Relations A fuzzy relation  is a 2 D MF:  :{ ((x, y),   (x, y)) | (x, y)  X  Y} Examples:  x is close to y (x & y are real numbers)  x depends on y (x & y are events)  x and y look alike (x & y are persons or objects)  Let X = Y = IR + and R(x,y) = “y is much greater than x” The MF of this fuzzy relation can be subjectively defined as: if X = {3,4,5} & Y = {3,4,5,6,7}

Fuzzy Logic & Approximate Reasoning 36 Extension Principle The image of a fuzzy set A on X under f(.) is a fuzzy set B: where y i = f(x i ), i = 1 to n If f(.) is a many-to-one mapping, then

Fuzzy Logic & Approximate Reasoning 37 Example –Application of the extension principle to fuzzy sets with discrete universes Let A = 0.1 / / / / / 2 and f(x) = x 2 – 3 Applying the extension principle, we obtain: B = 0.1 / / / / /1 = 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / 1 = 0.8 / / / 1 where “V” represents the “max” operator, Same reasoning for continuous universes

Fuzzy Logic & Approximate Reasoning 38 Transition From Type-1 to Type-2 Fuzzy Sets Blur the boundaries of a T1 FS Possibility assigned—could be non-uniform Clean things up Choose uniform possibilities—interval type-2 FS

Fuzzy Logic & Approximate Reasoning 39 Interval Type-2 Fuzzy Sets: Terminology-1

Fuzzy Logic & Approximate Reasoning 40

Fuzzy Logic & Approximate Reasoning 41 Questions