Complex Fuzzy Sets. Verkfræðideild A brief extraction from the article: Complex Fuzzy Logic by Daniel Ramot et. al,

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1 Complex Fuzzy Sets Sveinn Ríkarður Jóelsson Háskóli Íslands Verkfræðideild A brief extraction from the article: Complex Fuzzy Logic by Daniel Ramot et. al, IEEE Transactions on Fuzzy Systems, vol. 11, no. 4, Augus 2003 p. 1

2 Why? Why use complex number representation where a simple 2 element vector variable will work just as well? Ease of representation Ease of calculations Physical accuracy of the complex representations(e.g. one of the parameters represent a phase) Utility of complex algebra in the specific application p. 2

3 Definition of a Complex Fuzzy Set A complex fuzzy set S is defined: on a universe of discourse U is characterized by a membership function µ S (x) µ S (x) = r S (x)e jω S(x) where j = 1, r S (x) and ω S (x) are both real-valued, and r S (x) [0, 1]. The complex fuzzy set S may be represented as the set of ordered pairs: S = {(x, µ S (x)) x U} p. 3

4 Interpretation Considering the form µ S (x) it has two terms: Amplitude r S (x) and phase ω S (x). Just like any other real-valued membership function µ T (x) [0, 1] can be regarded having: Amplitude µ T (x) and phase ω T (x) = 0 So a complex fuzzy variable has a membership grade r S (x) and a membership phase ω S (x). p. 4

5 Membership phase The meaning of membership phase is application specific but a a few properties are required for the basic fuzzy set operators, we will consider the two complex fuzzy sets A and B: p. 5

6 Membership phase The meaning of membership phase is application specific but a a few properties are required for the basic fuzzy set operators, we will consider the two complex fuzzy sets A and B: Union A B : Intersection A B: µ A S B (x) = [r A (x) r B (x)] e jω A S B(x) µ A T B (x) = [r A (x) r B (x)] e jω A T B(x) Where the and are some t-conorm and t-norm operators respectively. p. 5

7 Membership Phase It remains to define ω A T B and ω A S B. There are a few axioms to follow when deciding this and for that we need the following definitions: Let A and B be two complex fuzzy sets in U, with the complex fuzzy union/intersection of A and B denoted A B and A B, specified by a functions u and i as follows: u : {a a C, a 1} {b b C, b 1} {d d C, d 1} i : {a a C, a 1} {b b C, b 1} {d d C, d 1} p. 6

8 Phase Union u must at least uphold the following axioms: Axiom 1 (boundary conditions): u(a, 0) = a. Axiom 2 (monotonicity): b d u(a, b) u(a, d). Axiom 3 (commutativity): u(a, b) = u(b, a). Axiom 4 (associativity): u(a, u(a, d)) = u(u(a, b), d). p. 7

9 Phase Union u must at least uphold the following axioms: Axiom 1 (boundary conditions): u(a, 0) = a. Axiom 2 (monotonicity): b d u(a, b) u(a, d). Axiom 3 (commutativity): u(a, b) = u(b, a). Axiom 4 (associativity): u(a, u(a, d)) = u(u(a, b), d). In some cases it may be desireable that u also satisfy the following reuirements. Axiom 5 (continuity): u is a continuious function. Axiom 6 (superidempotency): u(a, a) > a. Axiom 7 (strict monotinicity): a c and b d u(a, b) u(c, d). p. 7

10 Phase Intersection i must at least uphold the follwing axioms: Axiom 1 (bondary conditions): if b = 1, i(a, b) = a Axiom 2 (monotonicity): b d i(a, b) i(a, d) Axiom 3 (commutativity): i(a, b) = i(b, a). Axiom 4 (associativity): i(a, i(b, d)) = i(i(a, b), d). p. 8

11 Phase Intersection i must at least uphold the follwing axioms: Axiom 1 (bondary conditions): if b = 1, i(a, b) = a Axiom 2 (monotonicity): b d i(a, b) i(a, d) Axiom 3 (commutativity): i(a, b) = i(b, a). Axiom 4 (associativity): i(a, i(b, d)) = i(i(a, b), d). In some cases, it may be desirable that i aslo satisfy the follwoing requirements. Axiom 5 (continutiy): i is a continous function. Axiom 6 (subidempotency): i(a, a) < a. Axiom 7 (strict monotonicity): a c and b d i(a, b) i(c, d). p. 8

12 Functions for i and u The following are several possibilities for calculation ω A S B and ω A T B, which, if combined with an appropriate function for determining r A S B and r A T B, satisfy the axiomatic requirements. p. 9

13 Functions for i and u The following are several possibilities for calculation ω A S B and ω A T B, which, if combined with an appropriate function for determining r A S B and r A T B, satisfy the axiomatic requirements. Sum: ω A + ω B Max: max(ω A, ω B ) Min: min(ω A, ω B ) { Winner Take All : ω A ω B r A > r B r B > r A p. 9

14 Functions for i and u The following are several possibilities for calculation ω A S B and ω A T B, which, if combined with an appropriate function for determining r A S B and r A T B, satisfy the axiomatic requirements. Sum: ω A + ω B Max: max(ω A, ω B ) Min: min(ω A, ω B ) { Winner Take All : ω A ω B r A > r B r B > r A In thoery i and u can simultaniously be defined by the same function but that is unlikely to be useful. p. 9

15 Complex Fuzzy Aggregation Let A 1, A 2,..., A n be complex fuzzy sets defined on the universe of discourse U, then the vector aggregation on A 1,..., A n is defined by a funcion v: v : {a a C, a 1} n {b b C, b 1} p. 10

16 Complex Fuzzy Aggregation Let A 1, A 2,..., A n be complex fuzzy sets defined on the universe of discourse U, then the vector aggregation on A 1,..., A n is defined by a funcion v: v : {a a C, a 1} n {b b C, b 1} This function produces an aggregate fuzzy set A operating on memberships grades of A i for each x U: p. 10

17 Complex Fuzzy Aggregation Let A 1, A 2,..., A n be complex fuzzy sets defined on the universe of discourse U, then the vector aggregation on A 1,..., A n is defined by a funcion v: v : {a a C, a 1} n {b b C, b 1} This function produces an aggregate fuzzy set A operating on memberships grades of A i for each x U: µ A (x) = v(µ A1 (x),..., µ An (x)) = w i {a a C, a 1}, n i=1 n i=1 w i µ Ai (x) w i = 1 p. 10

18 Complex Fuzzy Aggregation Usually w i is selected to be n 1 since no method for choosing complex valued w i has been developed(2003) yielding: p. 11

19 Complex Fuzzy Aggregation Usually w i is selected to be n 1 since no method for choosing complex valued w i has been developed(2003) yielding: µ A (x) = 1 n n i=1 µ Ai (x) p. 11

20 Complex Fuzzy Aggregation Usually w i is selected to be n 1 since no method for choosing complex valued w i has been developed(2003) yielding: µ A (x) = 1 n n i=1 µ Ai (x) The significance of vector aggregations is that all µ Ai are complex valued. p. 11

21 Complex Fuzzy Aggregation Usually w i is selected to be n 1 since no method for choosing complex valued w i has been developed(2003) yielding: µ A (x) = 1 n n i=1 µ Ai (x) The significance of vector aggregations is that all µ Ai are complex valued. If the phases amongst A i are equal for all i then the sum is maximized. p. 11

22 Complex Fuzzy Aggregation Usually w i is selected to be n 1 since no method for choosing complex valued w i has been developed(2003) yielding: µ A (x) = 1 n n i=1 µ Ai (x) The significance of vector aggregations is that all µ Ai are complex valued. If the phases amongst A i are equal for all i then the sum is maximized. If they are not "aligned" or coherent "destructive interference" may occur, yielding a smaller amplitude for µ A than the individual amplitudes of the arguments µ Ai. p. 11

23 Complex Fuzzy Relations CFR represent both the degree of presence or absence of association, interaction or interconnectedness, and the phase of association, interaction or interconnectedness between the elements of two or more crisp sets. For crisp sets U and V the complex fuzzy relation(x U and y V ): p. 12

24 Complex Fuzzy Relations CFR represent both the degree of presence or absence of association, interaction or interconnectedness, and the phase of association, interaction or interconnectedness between the elements of two or more crisp sets. For crisp sets U and V the complex fuzzy relation(x U and y V ): R(U, V ) = {((x, y), µ R (x, y)) (x, y) U V } µ R (x) = r R (x)e jω R(x) p. 12

25 Complex Fuzzy Relations CFR represent both the degree of presence or absence of association, interaction or interconnectedness, and the phase of association, interaction or interconnectedness between the elements of two or more crisp sets. For crisp sets U and V the complex fuzzy relation(x U and y V ): R(U, V ) = {((x, y), µ R (x, y)) (x, y) U V } µ R (x) = r R (x)e jω R(x) r R (x) represents the traditional fuzzy relation ω R (x) represents the phase of fuzzy relation p. 12

26 Complex Fuzzy Relations CFR represent both the degree of presence or absence of association, interaction or interconnectedness, and the phase of association, interaction or interconnectedness between the elements of two or more crisp sets. For crisp sets U and V the complex fuzzy relation(x U and y V ): R(U, V ) = {((x, y), µ R (x, y)) (x, y) U V } µ R (x) = r R (x)e jω R(x) r R (x) represents the traditional fuzzy relation ω R (x) represents the phase of fuzzy relation Again the phase is application specific but a few good examples are give and referred to in the article these notes are based on. p. 12

27 Complex Fuzzy Logic The if then else rules using complex fuzzy sets like: X A Y B. This expression can be examined using membership functions and is denoted(a and B are complex fuzzy sets): µ A B (x, y) = µ A (x)µ B (y) = r A (x)r B (x)e j(ω A(x)+ω B (x)) p. 13

28 Complex Fuzzy Logic The if then else rules using complex fuzzy sets like: X A Y B. This expression can be examined using membership functions and is denoted(a and B are complex fuzzy sets): µ A B (x, y) = µ A (x)µ B (y) = r A (x)r B (x)e j(ω A(x)+ω B (x)) or: r A B (x, y) = r A (x)r B (y) ω A B (x, y) = ω A (x) + ω B (y) p. 13

29 Conclusion Rules based on CFL are strongly related which is manifested in the phase term associated with complex fuzzy implication. p. 14

30 Conclusion Rules based on CFL are strongly related which is manifested in the phase term associated with complex fuzzy implication. This relation leads to a unique interaction or dependence between rules which is enhanced by the use of vector aggregation. p. 14

31 Conclusion Rules based on CFL are strongly related which is manifested in the phase term associated with complex fuzzy implication. This relation leads to a unique interaction or dependence between rules which is enhanced by the use of vector aggregation. Thus CFLs provides a framework which can be utilized to solve problems in which the rules are related, with the natrue of the relation varying as a function of the input to the system which can be very difficult or impossible to solve using traditional methods of fuzzy logic. p. 14