Membership . . . Triangular and . . . Need for a Type-2 . . . Expert Rules Type-2 Fuzzy Analysis Definitions Explains Ubiquity of Main Results How Robust Are . . . Triangular and Trapezoid What If We Use a . . . Statistics Approach . . . Membership Functions Home Page Title Page Vladik Kreinovich 1 , Olga Kosheleva 1 , and Shahnaz Shahbazova 2 ◭◭ ◮◮ ◭ ◮ 1 University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 31 olgak@utep.edu, vladik@utep.edu 2 Azerbaijan Technical University Go Back Baku, Azerbaijan Full Screen shahbazova@gmail.com Close Quit
Membership . . . Triangular and . . . 1. Membership Functions: Brief Reminder Need for a Type-2 . . . • One of the main ideas behind fuzzy logic is: Expert Rules Definitions – to represent an imprecise (“fuzzy”) natural- Main Results language property P like “small” How Robust Are . . . – by its membership function µ ( x ). What If We Use a . . . • Such a function assigns: Statistics Approach . . . Home Page – to each possible value x of the corresponding prop- erty, Title Page – the degree µ ( x ) ∈ [0 , 1] this value satisfies the prop- ◭◭ ◮◮ erty P (e.g., to what extent x is small). ◭ ◮ Page 2 of 31 Go Back Full Screen Close Quit
Membership . . . Triangular and . . . 2. Triangular and Trapezoid Membership Func- Need for a Type-2 . . . tions Are Ubiquitous: Why? Expert Rules • According to this definition, we can have many differ- Definitions ent membership functions; however: Main Results How Robust Are . . . – in many applications of fuzzy techniques, What If We Use a . . . – the simplest piece-wise linear µ ( x ) – e.g., triangular Statistics Approach . . . and trapezoid ones – works very well. Home Page • Why? In this talk, we use fuzzy techniques to analyze Title Page this question. ◭◭ ◮◮ ◭ ◮ Page 3 of 31 Go Back Full Screen Close Quit
Membership . . . Triangular and . . . 3. How Can We Analyze the Problem: Need for Need for a Type-2 . . . a Type-2 Approach Expert Rules • Traditionally – e.g., in control – fuzzy logic is used to Definitions select a value of a quantity, e.g., a control u . Main Results How Robust Are . . . • To come up with such a value, first, we use the experts’ What If We Use a . . . rules to come up, Statistics Approach . . . – for each possible control value u , Home Page – with a degree d ( u ) to which this control value is Title Page reasonable. ◭◭ ◮◮ • Then, we select a value u – e.g., the one for which the ◭ ◮ degree of reasonableness is the largest: Page 4 of 31 d ( u ) → max . u Go Back • In our problem, instead of selecting a single value u , Full Screen we select the whole membership function µ ( x ). Close Quit
Membership . . . Triangular and . . . 4. Need for a Type-2 Approach (cont-d) Need for a Type-2 . . . • To use fuzzy techniques for selecting µ , we thus need Expert Rules to do the following. Definitions Main Results • First, we need to use experts’ rules to assign, How Robust Are . . . – to each possible membership function µ ( x ), What If We Use a . . . – a degree d ( µ ) to which this membership function is Statistics Approach . . . reasonable. Home Page • Then, out of all possible members functions, Title Page ◭◭ ◮◮ – we select the one which is the most reasonable, – i.e., the one for which the degree of reasonableness ◭ ◮ d ( µ ) is the largest: d ( µ ) → max . Page 5 of 31 µ Go Back • Let us follow this path. Full Screen Close Quit
Membership . . . Triangular and . . . 5. Comment Need for a Type-2 . . . • Traditionally: Expert Rules Definitions – situations in which we use fuzzy to reason about Main Results real values is known as type-1 fuzzy; while How Robust Are . . . – situations in which we use fuzzy to reason about What If We Use a . . . fuzzy is known as type-2 fuzzy approach. Statistics Approach . . . • From this viewpoint, what we plan to use is an example Home Page of the type-2 fuzzy approach. Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 31 Go Back Full Screen Close Quit
Membership . . . Triangular and . . . 6. Expert Rules Need for a Type-2 . . . • First, we need to select expert rules. Expert Rules Definitions • We consider the problem in its utmost generality. Main Results • We want rules that will be applicable to all possible How Robust Are . . . fuzzy properties. What If We Use a . . . • In this case, the only appropriate rule that comes to Statistics Approach . . . Home Page mind is the following natural natural-language rule: – if x and x ′ are close, Title Page – then µ ( x ) and µ ( x ′ ) should be close. ◭◭ ◮◮ ◭ ◮ • This rule exemplifies the whole idea of fuzziness: Page 7 of 31 – for crisp properties (like x ≥ 0), the degree of con- fidence abruptly changes the from 0 to 1; Go Back – instead, we have a smooth transition from 0 to 1. Full Screen Close Quit
Membership . . . Triangular and . . . 7. How Can We Formalize This Expert Rule? Need for a Type-2 . . . • There are infinitely many possible values of x and x ′ . Expert Rules Definitions • Thus, the above rule consists of infinitely many impli- cations – one implication for each pair ( x, x ′ ). Main Results How Robust Are . . . • Dealing with infinitely many rules is difficult. What If We Use a . . . • It is therefore desirable to try to limit ourselves to finite Statistics Approach . . . number of rules. Home Page • Such a limitation is indeed possible. Title Page • Indeed, theoretically, we can consider all infinitely ◭◭ ◮◮ many possible values x . ◭ ◮ • However, in practice, the values of any physical quan- Page 8 of 31 tity are bounded: e.g., Go Back – locations on the Earth are bounded by the Earth’s Full Screen diameter, – speeds are limited by the speed of light, etc. Close Quit
Membership . . . Triangular and . . . 8. Formalizing the Expert Rule (cont-d) Need for a Type-2 . . . • Thus, it is reasonable to assume that all possible values Expert Rules x are within some interval [ x, x ]. Definitions • Second, we only know x and x ′ with a certain accuracy Main Results How Robust Are . . . ε > 0. What If We Use a . . . • From this viewpoint, there is no need to consider all Statistics Approach . . . infinitely many values. Home Page • It is sufficient to consider only values on the grid of Title Page width ε , i.e., values ◭◭ ◮◮ x 0 = x, x 1 = x + ε, x 2 = x + 2 ε, . . . , x n = x + n · ε = x. ◭ ◮ def Page 9 of 31 • So, it is sufficient to describe the values µ i = µ ( x i ) of the membership function for x 0 , x 1 , . . . , x n . Go Back • We call these values discrete (d-)membership function . Full Screen Close Quit
Membership . . . Triangular and . . . 9. Formalizing the Expert Rule (cont-d) Need for a Type-2 . . . • For these values, it is sufficient to formulate the “close- Expert Rules ness” rule only for neighboring values µ i and µ i +1 : Definitions Main Results For all i, µ i is close to µ i +1 , i.e. How Robust Are . . . ( µ 1 is close to µ 2 ) and . . . and (( µ n − 1 is close to µ n ) . What If We Use a . . . Statistics Approach . . . • This formula can be formalized according to the usual Home Page fuzzy methodology. Title Page • Intuitively, closeness of x and x ′ is means that d = | x − x ′ | is small. ◭◭ ◮◮ ◭ ◮ • Thus, to express closeness, we need to select a mem- bership function s ( d ) describing “small”. Page 10 of 31 • The larger the difference, the less small it is. Go Back • So it is reasonable to require that the membership func- Full Screen tion s ( d ) be strictly decreasing. Close Quit
Membership . . . Triangular and . . . 10. Formalizing the Expert Rule (cont-d) Need for a Type-2 . . . • At least, s ( d ) decreases until it reaches value 0 for the Expert Rules differences d which are clearly not small. Definitions Main Results • n is usually large, thus, 1 /n is small. How Robust Are . . . • So, without losing generality, we can safely assume that What If We Use a . . . the distance 1 /n is small, i.e., that s (1 /n ) > 0. Statistics Approach . . . Home Page • In terms of s ( d ), for each i , the degree to which µ i is close to µ i +1 is s ( | µ i − µ i +1 | ) . Title Page • To find the degree d ( µ ) to which a given d-membership ◭◭ ◮◮ function µ = ( µ 1 , . . . , µ n ) is reasonable: ◭ ◮ – we need to apply some “and”-operation (t-norm) Page 11 of 31 f & ( a, b ) to these degrees, Go Back – then, d ( µ ) = f & ( s ( | µ 0 − µ 1 | ) , . . . , s ( | µ n − 1 − µ n | )) . Full Screen Close Quit
Membership . . . Triangular and . . . 11. Formalizing the Expert Rule (cont-d) Need for a Type-2 . . . • It is reasonable to consider the simplest “and”- Expert Rules operation f & ( a, b ) = min( a, b ), then we get Definitions Main Results d ( µ ) = min( s ( | µ 0 − µ 1 | ) , . . . , s ( | µ n − 1 − µ n | )) . How Robust Are . . . What If We Use a . . . • Now, we are ready to formulate the problem in precise terms. Statistics Approach . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 31 Go Back Full Screen Close Quit
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