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Fuzzy logic 7


Fuzzy logic Deeper into Fuzzy ... Rules Inference Defuzzification Fuzzy Operations Standard fuzzy operations are quite adequate in many practical applications of FIS, ... – PowerPoint PPT presentation

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Title: Fuzzy logic 7

Fuzzy logic
Deeper into Fuzzy Inference
Aleksandar Rakic rakic_at_etf.rs
  • Fuzzy Operations
  • Fuzzy Relations
  • Classical vs. Fuzzy Logic
  • Generalized Modus Ponens
  • Individual Rules Inference
  • Defuzzification

Fuzzy Operations
  • Standard fuzzy operations are quite adequate in
    many practical applications of FIS, but they do
    not utilize the real expressive power of fuzzy.
  • Alternatively, algebraic sum (1) and algebraic
    product (2) can be used for a definition of
    union and intersection of two fuzzy sets,

s t norms
  • General notation
  • Operator s is called an s-norm if it satisfies to
    the following axioms for any y, z, v and w ?
  • Operator t is called an t-norm (triangular norm)
    if it satisfies to the following axioms for any
    y, z, v and w ?0,1

Some s-norms for fuzzy union
  • For arbitrary fuzzy sets A and B, defined over
    the universe x ? X , whose membership values are
    denoted by y ?A(x) and z ?B(x)
    correspondingly, some s-norm definitions are
  • (S1) Drastic sum
  • (S2) Hamacher sum
  • (S3) Dubois-Prade class
  • (S4) Yager class

Some t-norms for intersection
  • For arbitrary fuzzy sets A and B, defined over
    the universe x ? X , whose membership values are
    denoted by y ?A(x) and z ?B(x)
    correspondingly, some t-norm definitions are
  • (T1) Drastic product
  • (T2) Hamacher product
  • (T3) Dubois-Prade class
  • (T4) Yager class

Some s t norm properties
  • s-norms are bounded below by max (standard fuzzy
    union) and bounded above by drastic sum (S1)
  • t-norms are bounded below by drastic product (T1)
    and bounded above by min (standard fuzzy
  • s-norms (a set of fuzzy disjunction operators)
    are often called triangular conorms or shortly
  • The alternative forms of operators AND and OR are
    called compensatory operators (they compensate
    the strictness of min and max operators proposed
    by L. A. Zadeh)

Fuzzy Complement
  • Operator c is called a fuzzy complement if it
    satisfies to the following axioms for any x and y
    ? 0,1
  • Some of the operators for the fuzzy complement
  • (C1) Sugenos complement
  • (C2) Yagers complement

Cartesian Product of Sets
  • For two arbitrary crisp (classical) sets X and
    Y,the Cartesian product of sets X and Y is a set
    of all ordered pairs (xi, yj ), xi?X, yj?Y that
  • Example X x1, x2, x3, x4 , Y y1, y2, y3
    the set X x Y consists of 12 ordered pairs
    (xi, yj ), i 1,2,3,4 j 1,2,3
  • Example Continuous sets X x1, x2, Y y1,

Crisp vs. Fuzzy Relations
  • A crisp binary relation ? on X1 and X2 if defined
    as a set of ordered pairs in X1 ? X2 that is
  • where P(x1,x2) is a property to which each pair
    (x1,x2)?? satisfies (or does not satisfy), or in
    means of membership
  • A fuzzy binary relation generalizes the concept
    of classical (crisp) relation introducing a
    degree of membership for each ordered couple
    (x1, x2) in universe of discourse X1 ? X2
  • means membership degree (or
    degree to which fuzzy relation holds true) of the
    ordered couple (x1, x2)in the fuzzy relation

Fuzzy Relations
  • Example Suppose that X1 and X2 are continuous
    segments 0,50 and 20,40, respectively. A
    fuzzy relation x1 is
    approximately equal to x2 may be defined by the
    membership function

Fuzzy relations enhance our capability to deal
with relational concepts expressed in a natural
language !
Fuzzy Relations
  • Example A fuzzy relation x1 is much larger than
    x2 may be defined by the membership function

Membership values on segment 0,1
Fuzzy relations are also fuzzy sets, and
fundamental properties of fuzzy sets hold for
fuzzy relations as well
Fuzzy Relations
  • Fuzzy set operations (complements, unions,
    intersections) are also applicable to fuzzy
  • Example Standard and Sugeno complement of the
    fuzzy relation x1 is much larger than x2 from
    the previous slide

Sugenos complement (parameter ? 2)
Standard complement
Composition ofFuzzy Relations
  • Composition of two crisp relations is relation
    TR1R2, which consists of those pairs (x, z),
    x?X, z?Z, of the Cartesian product X x Z that,
    via the given relations R1 and R2, share at least
    one element y?Y
  • Composition operation on fuzzy relations has two
    common forms
  • a) max-min composition
  • or in terms of OR(?)/AND(?) operations over
    membership functions
  • b) max-product (in general, max-star)
  • For example, max-product composition can have a

sign ? denotesany t-norm
Operations on Fuzzy Setsas Fuzzy Relations
  • Operations on fuzzy sets defined on different
    universal domains produce a fuzzy relation as a
    multidimensional fuzzy set
  • Example Operation A?B in universe of discourse X
    ? Y

Compound Prepositionsas Fuzzy Relations
  • Consider the compound prepositions in a fuzzy
    rules, e.g.
  • IF x1 is Ai1 AND x2 is Ai2 THEN
  • IF x1 is Ai1 OR x2 is Ai2 THEN
  • Compound fuzzy propositions are interpreted as
    fuzzy relations
  • (1) x1 is Ai1 AND x2 is Ai2

  • ,
  • where t 0,1?0,1 ? 0,1 is a t-norm
  • (2) x1 is Ai1 OR x2 is Ai2

  • ,
  • where s 0,1?0,1 ? 0,1 is a s-norm

Fuzzy Implicationsas Fuzzy Relations
  • Fuzzy rule implication is fuzzy relation,which
    can be expressed as a conjunction
  • IF x is Ai THEN y is Bi ? Ai ? Bi ? Ai ? Bi
    t µAi , µBi
  • (fuzzy relation between input and output on a
    base of linguistic terms is expressed as the
    conjunction i.e. Ai AND Bi, i 1,,n)
  • (A) Mamdani implication (correlation minimum,
  • (B) Larsen implication (correlation product,
  • As an implication ("entailment of Bi by Ai" or
    "Ai entails Bi"),fuzzy rules represents human
    abilities of imprecise reasoning.
  • The foundation of a fuzzy implication rule is the
    narrow sense of fuzzy logic (generalization of a
    classical binary logic to the multivalue logic)

Some Fuzzy Implications
  • Selected fuzzy implication (FI) operators
  • (FI1) Zadehs classical maximum FI
  • (FI1) Dienes-Rescher implication
  • (FI2) Lukasiewicz implication
  • (FI3) Godelian FI
  • (FI4) Standard sequence implication

equivalent to (FI1) when µBi (y) ? µAi(x)
Classical Logic
  • In the propositional logic the inference can be
    depicted as follows
  • If the resulting premises are both true, then the
    conclusion is also true (valid deduction the
    truth conclusion is inferred or deduced from
    truth premises).
  • Among basic inference rules (the premises
    logically imply rules conclusions) we can
    mention the following
  • (a) Modus Ponens (b)
    Modus Tollens
  • (Lat. method of affirming) (Lat.
    method of denying)
  • sign ? represents the relation if-then
  • P and Q are propositions (variables)
  • symbol (therefore) is placed before

Fuzzy Logic
  • Fuzzy Logic generalizes the notion of truth
    values in classical logic, and provides a
    background for reasoning (inferencing) when
    corresponding conditions are only partially
  • Approximate reasoning (fuzzy rule IF x is A
    THEN y is B)
  • 1. Possibility distribution of the variable x
  • 2. Implication possibility from x to y
  • Possibility distribution of y
  • Fuzzy implication use a compositional rule of
    inference for calculation of output results.

Generalized Modus Ponens
  • Consider a given rule IF x is A THEN y is B,
    where A and B arefuzzy sets defined on universes
    X and Y, correspondingly,
  • and a fact x is A (A and A are not
    necessarily identical)
  • Generalized Modus Ponens (GMP)
  • Rule (premise 1) x is A ? y is B
  • Fact (premise 2) x is A
  • Infer (result of fuzzy inference engine) y is
  • B A ? R is composition of premise A and
    fuzzy relation R, which represents an implication
    (x is A) ? (y is B)
  • In approximate reasoning, GMP is an inference
    mechanism that allows to obtain imprecise
    conclusion from imprecise (vague) fact.

GMP Compositional Operators
  • The inference process can be implemented
    differently usingdifferent compositional
    operators (most commonly used)
  • a) max-min compositional operator
  • b) max-product compositional operator
  • In general, GMP states that for the premise x is
    A and the fuzzy relation R(x, y) ? A ? B (the
    rule IF x is A THEN y is B), the inferred set B
    (conclusion y is B ) is calculated as follows
  • where X is a domain of the variable x (antecedent
    domain), and the letter "t" denotes a t-norm.
  • As a result, µB (y) represents a possibility
    distribution of output variable over its domain
    (universe of discourse).

Individual Rules Inference
  • Consider a fuzzy rule base (totality of n rules)
    IF x is Ai THEN y is Bi, where Ai ?X and Bi ?Y
    are fuzzy sets x and y are input and output
    variables, respectively (can be expanded to
    multiple input/output case)
  • Individual Rules Inference (IRI)
  • a) calculation of Ri (x, y) ? (Ai ? Bi ) for
    each i-th rule
  • b) calculation of Bi?Y (fired THEN-part set)
    as a GMP composition of input A and
    implication Ai ? Bi
  • c) generation of a fuzzy output B?Y as a
    aggregation (combination) of separate Bi, i.e.
  • c1) Mamdani aggregation , or
  • c2) Gödel aggregation

Individual Rules Inference
  • One of the commonly used inference engines is a
    Mamdani Engine (Minimum Engine), which is based
  • 1) OR/AND operators Mamdani s-norm/t-norm
  • 2) Implication operator Mamdani t-norm
  • 3) IRI with aggregation operator Mamdani s-norm
  • There are other inference engines, i.e. Zadehs
    Engine and Lukasiewiczs Engine both use standard
    minimum as t-norm, but IRI is used with Gödel
    aggregation, and each use its own implication
    Zadehs and Lukasiewiczs, respectively.

  • Suppose the inferred fuzzy output y is B is
    having the aggregated membership function µB
  • Defuzzification is conversion of fuzzy output
    (possibility distribution of the output µB (y))
    to precise (crisp) value y
  • How to defuzzify?
  • One of criteria is plausibility y should lie
    approximately in the middle of the support
    region of B(y) and have a high degree of
    membership in B(y)

  • Among the major defuzzification techniques we can
  • a) Mean of Maximum (Middle of Maxima) method
  • b) Center of Gravity (Centroid, Center of Area)
    method (CoG)
  • MoM calculates the average of all variable values
    having maximum membership degree.
  • CoG defuzzified output is calculated as weighted
    average over the whole universe Y of output
  • where µB (y) is aggregated membership function
    of the inferred fuzzy output.
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