Fuzzy Techniques Are . . . Versions of Fuzzy . . . Linear Interpolation Is . . . Explaining Trapezoid . . . Simple Linear Interpolation Explains All Explaining f & ( a, b ) = . . . Usual Choices in Fuzzy Techniques: Linear Interpolation . . . Membership Functions, t-Norms, What If We . . . t-Conorms, and Defuzzification Simple Linear . . . Simple Linear . . . Vladik Kreinovich 1 , Jonathan Quijas 1 , Home Page Esthela Gallardo 1 , Caio De Sa Lopes 1 , Title Page Olga Kosheleva 2 , and Shahnaz Shahbazova 3 ◭◭ ◮◮ Departments of 1 Computer Science and 2 Teacher Education ◭ ◮ University of Texas at El Paso, El Paso, Texas 79968, USA vladik@utep.edu, jkquijas@miners.utep.edu, Page 1 of 20 egallardo5@miners.utep.edu, cdesalopes@miners.utep.edu, olgak@utep.edu Go Back 3 Azerbaijan Technical University, Baku, Azerbaijan Full Screen shahbazova@gmail.com Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 1. Fuzzy Techniques Are Needed Linear Interpolation Is . . . • In many application areas, we have experts whose ex- Explaining Trapezoid . . . perience we would like to capture. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • Often, experts’ rules use imprecise (“fuzzy”) words What If We . . . from natural language, like “small”, “large”, etc. Simple Linear . . . • To formalize these rules, L. Zadeh proposed special Simple Linear . . . fuzzy techniques . Home Page • A usual application of fuzzy techniques consists of the Title Page following three stages: ◭◭ ◮◮ 1) reformulate expert knowledge in computer- ◭ ◮ understandable terms – i.e., as numbers; Page 2 of 20 2) process these numbers to come up with the degrees Go Back to which different actions are reasonable; 3) if needed, “defuzzify” this “fuzzy” recommendation Full Screen into an exact strategy. Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 2. First Stage of Fuzzy Technique Linear Interpolation Is . . . • In the first stage, we formalize the imprecise terms used Explaining Trapezoid . . . by the experts, such as “small”, “hot”, and “fast”. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • Each such term is described by assigning, What If We . . . – to different possible values x , Simple Linear . . . – a degree µ ( x ) to which x satisfies this term Simple Linear . . . (e.g., to which x is small). Home Page • Some values µ ( x ) are obtained by asking the expert. Title Page ◭◭ ◮◮ • However, there are infinitely many real numbers x , and we can only ask a finite number of questions, ◭ ◮ • Thus, we need to perform interpolation to estimate the Page 3 of 20 degrees µ ( x ) for intermediate values x . Go Back • The result µ ( x ) is called the membership function. Full Screen Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 3. Interval-Valued Fuzzy Degrees Linear Interpolation Is . . . • Often, an expert is unable to describe his or her degree Explaining Trapezoid . . . of confidence by an exact number. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • A more natural approach is to use intervals of possible What If We . . . values. Simple Linear . . . • In this case, for each x , we get an interval [ µ ( x ) , µ ( x )] Simple Linear . . . of possible values. Home Page • Determining such interval-valued membership function Title Page is equivalent to determining µ ( x ) and µ ( x ). ◭◭ ◮◮ • These functions are known as the lower and the upper ◭ ◮ membership functions. Page 4 of 20 Go Back Full Screen Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 4. Second Stage of Fuzzy Techniques: “And”- and Linear Interpolation Is . . . “Or”-Operations Explaining Trapezoid . . . • Many expert rules involve several conditions. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • Example: a doctor will prescribe a certain medicine if What If We . . . the fever is high and blood pressure is normal. Simple Linear . . . • To handle such rules, we need to be able to transform: Simple Linear . . . Home Page – the degrees a = d ( A ) and b = d ( B ) of individual conditions A and B Title Page – into a degree of confidence in the composite state- ◭◭ ◮◮ ment A & B . ◭ ◮ • The corresponding estimate f & ( a, b ) is known as an Page 5 of 20 “and”-operation , or, alternatively, as a t-norm . Go Back • Similarly, we need an “or”-operation f ∨ ( a, b ) Full Screen ( t-conorm ) and a negation operation f ¬ ( a ). Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 5. Third Stage of Fuzzy Techniques: Linear Interpolation Is . . . Defuzzification Explaining Trapezoid . . . • After performing the first two stages, Explaining f & ( a, b ) = . . . Linear Interpolation . . . – for the given input x and for all possible control What If We . . . values u , Simple Linear . . . – we get a degree µ ( u ) to which this control value is Simple Linear . . . reasonable to apply. Home Page • Sometimes, we want to use this expert knowledge in Title Page an automated system. ◭◭ ◮◮ • In this case, we need to transform this membership ◭ ◮ function µ ( u ) into a single value u . Page 6 of 20 Go Back Full Screen Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 6. Versions of Fuzzy Techniques Linear Interpolation Is . . . • There are many different membership functions µ ( x ), Explaining Trapezoid . . . “and”- and “or”-operations, and defuzzifications. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • In practice, a few choices are the most efficient: What If We . . . – trapezoid µ ( x ): start with 0, linearly got to 1, stay Simple Linear . . . at 1, then linearly decrease to 0; Simple Linear . . . – f & ( a, b ) = min( a, b ) or f & ( a, b ) = a · b ; Home Page – f ∨ ( a, b ) = max( a, b ) or f ∨ ( a, b ) = a + b − a · b ; Title Page – negation operation f ¬ ( a ) = 1 − a ; and ◭◭ ◮◮ � u · µ ( u ) du – centroid defuzzification u = µ ( u ) du . ◭ ◮ � Page 7 of 20 • Similarly, for interval-valued case, both lower and up- per membership functions are usually trapezoidal. Go Back • We show that all these choices can be explained by the Full Screen use of the simplest (linear) interpolation. Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 7. Linear Interpolation Is the Simplest Linear Interpolation Is . . . • Interpolation means that we find a function that at- Explaining Trapezoid . . . tains known values at given points. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • The simplest possible non-constant functions are linear What If We . . . functions. Simple Linear . . . • Thus, linear interpolation is the simplest possible in- Simple Linear . . . terpolation. Home Page • If we know that y 1 = f ( x 1 ) and y 2 = f ( x 2 ), then these Title Page two values uniquely determine a linear function: ◭◭ ◮◮ f ( x ) = f ( x 1 ) + y 2 − y 1 · ( x − x 1 ) . ◭ ◮ x 2 − x 1 Page 8 of 20 • In this talk, we show that this simplest (linear) inter- Go Back polation explains all usual choices of fuzzy techniques. Full Screen Close Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 8. Explaining Trapezoid Membership Functions Linear Interpolation Is . . . • For each property like “small”: Explaining Trapezoid . . . Explaining f & ( a, b ) = . . . – first, there are some values which are definitely not Linear Interpolation . . . small (e.g., negative ones), What If We . . . – then some values which are small to some extent; Simple Linear . . . – then, we have an interval of values which are defi- Simple Linear . . . nitely small; Home Page – this is followed by values which are somewhat small; Title Page – finally, we get values which are absolutely not small. ◭◭ ◮◮ • Let us denote the values (“thresholds”) that separate ◭ ◮ these regions by t 1 , t 2 , t 3 , and t 4 . Page 9 of 20 • Then: µ ( x ) = 0 for x ≤ t 1 ; µ ( x ) = 1 for t 2 ≤ x ≤ t 3 ; Go Back and µ ( x ) = 0 for x ≥ t 4 . Full Screen • Linear interpolation indeed leads to trapezoid func- Close tions. Quit
Fuzzy Techniques Are . . . Versions of Fuzzy . . . 9. Explaining f & ( a, b ) = a · b Linear Interpolation Is . . . • If one of the component statements A is false, then the Explaining Trapezoid . . . composite statement A & B is also false: f & (0 , b ) = 0. Explaining f & ( a, b ) = . . . Linear Interpolation . . . • If A is absolutely true, then our belief in A & B is equiv- What If We . . . alent to our degree of belief in B : f & (1 , b ) = b . Simple Linear . . . def • Let us fix b and consider a function F b ( a ) = f & ( a, b ) Simple Linear . . . that maps a into the value f & ( a, b ). Home Page • We know that F b (0) = 0 and F b (1) = b . Title Page • Linear interpolation leads to F b ( a ) = a · b , i.e., to the ◭◭ ◮◮ algebraic product f & ( a, b ) = a · b . ◭ ◮ • Please note that: Page 10 of 20 – while the resulting operation is commutative and Go Back associative, Full Screen – we did not require commutativity or associativity; – all we required was linear interpolation. Close Quit
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