Need for Fuzzy Extensions Need for Fuzzy . . . Need for Robustness, . . . Most Robust Fuzzy Need for Robustness . . . What Is Known and . . . Extensions of Binary Logical Results Operations Results (cont-d) Home Page Irvin L. Bosquez and Vladik Kreinovich Title Page Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA, ilbosquez@miners.utep.edu, vladik@utep.edu Page 1 of 8 Go Back Full Screen Close Quit
Need for Fuzzy Extensions 1. Need for Fuzzy Extensions Need for Fuzzy . . . • In many practical situations when we solicit expert Need for Robustness, . . . opinions, we are interested in a property: Need for Robustness . . . What Is Known and . . . – which experts cannot easily directly estimate, Results – but we know that this property is a boolean com- Results (cont-d) bination of easier-to-estimate properties. Home Page • Example: a medicine is efficient when the blood pres- Title Page sure is high ( H ) but not very high ( ¬ V ): ◭◭ ◮◮ H & ¬ V. ◭ ◮ • Only a few experts can estimate the medicine’s effi- Page 2 of 8 ciency. Go Back • However, doctors usually have a good understanding Full Screen of when the blood pressure is high or very high. Close Quit
Need for Fuzzy Extensions 2. Need for Fuzzy Extensions (cont-d) Need for Fuzzy . . . Need for Robustness, . . . • The difficulty is that in many situations, an expert is not 100% confident in his/her statement. Need for Robustness . . . What Is Known and . . . • At best, the expert can mark, on a scale from 0 to 1, Results to what extent he/she believes in this statement. Results (cont-d) • We can thus get expert’s degrees of belief a and b in Home Page statements A and B . Title Page • We need to estimate the resulting degree of belief in ◭◭ ◮◮ the corr. boolean combination A ∗ B (such as A & ¬ B ). ◭ ◮ • When experts are absolutely sure, i.e., when each a and b is 0 or 1, we should get the usual boolean results. Page 3 of 8 • Thus, what we need is to extend the usual boolean Go Back operations a ∗ b : Full Screen – from the 2-valued set { 0 , 1 } Close – to the function f ∗ ( a, b ) defined for all a, b ∈ [0 , 1]. Quit
Need for Fuzzy Extensions 3. Need for Robustness, and Resulting Extension Need for Fuzzy . . . Procedure Need for Robustness, . . . • The values a and b are not very accurate. Need for Robustness . . . What Is Known and . . . • If we ask the same expert twice, he/she may give slightly Results different values. Results (cont-d) • It is desirable to make sure that the resulting estimate Home Page f ∗ ( a, b ) be minimally affected by this difference. Title Page • For example, if we fix a and for some n , take values ◭◭ ◮◮ b i = i/n , then for c i = f ∗ ( a, b i ) we should have ◭ ◮ c i ≈ c i +1 , i.e., c i − c i +1 ≈ 0 . Page 4 of 8 • In other words, a multi-D point ( c 1 − c 2 , c 2 − c 3 , . . . ) Go Back should be close to (0 , 0 , . . . ). Full Screen • The distance between these points is the smallest when ( c i − c i +1 ) 2 is the smallest. Close its square � i Quit
Need for Fuzzy Extensions 4. Need for Robustness (cont-d) Need for Fuzzy . . . ( c i − c i +1 ) 2 . Need for Robustness, . . . • Reminder: we minimize the expression � i Need for Robustness . . . • Equating the derivative of this expression to 0, we con- What Is Known and . . . clude that c i − c i +1 = c i − 1 − c i for all i . Results Results (cont-d) • Thus, the expression f ∗ ( a, b ) is linear in b . Home Page • So, if we know f ∗ ( a, 0) and f ∗ ( a, 1), we can get the Title Page values f ∗ ( a, b ) for all b by linear interpolation. ◭◭ ◮◮ • Similarly, f ∗ ( a, b ) should be linear in a . ◭ ◮ • Thus, based on the values f ∗ (0 , 0) and f ∗ (0 , 1), we can use linear interpolation to find the values f ∗ (0 , b ). Page 5 of 8 Go Back • Similarly, we get f ∗ (1 , b ). Full Screen • For each b , based on the values f ∗ (0 , b ) and f ∗ (1 , b ), we can similarly find the values f ∗ ( a, b ). Close Quit
Need for Fuzzy Extensions 5. What Is Known and What We Do Need for Fuzzy . . . • The results of using this procedure are known for & Need for Robustness, . . . and ∨ . Need for Robustness . . . What Is Known and . . . • We are extending it to all possible binary boolean op- Results erations. Results (cont-d) Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 8 Go Back Full Screen Close Quit
Need for Fuzzy Extensions 6. Results Need for Fuzzy . . . • There are four pairs of boolean values: Need for Robustness, . . . Need for Robustness . . . 00 , 01 , 10 , and 11 . What Is Known and . . . • For each such pair, the operation can give 0 or 1. Results • Thus, there are 2 4 = 16 possible binary boolean oper- Results (cont-d) ations. Home Page Title Page • We can describe each such operation by listing the 0-1 values corresponding to inputs 00, 01, 10, and 11. ◭◭ ◮◮ • The sequence 0000 corresponds to a ∗ b = 0 which ◭ ◮ interpolates to f ∗ ( a, b ) = 0. Page 7 of 8 • For 0001, we get a ∗ b = a & b , and f ∗ ( a, b ) = a · b . Go Back • For 0010, we get a ∗ b = a & ¬ b and f ∗ ( a, b ) = a · (1 − b ). Full Screen • For 0011, we get a ∗ b = a and f ∗ ( a, b ) = a . Close • For 0100, we get a ∗ b = ¬ a & b and f ∗ ( a, b ) = (1 − a ) · b . Quit
7. Results (cont-d) Need for Fuzzy Extensions Need for Fuzzy . . . • For 0101, we get a ∗ b = b and f ∗ ( a, b ) = b . Need for Robustness, . . . Need for Robustness . . . • For 0110, we get exclusive “or”, with What Is Known and . . . Results f ∗ ( a, b ) = a + b − 2 a · b. Results (cont-d) • For 0111, we get a ∗ b = a ∨ b and f ∗ ( a, b ) = a + b − a · b . Home Page • For ∗ = 1 ε 1 ε 2 ε 3 , we get a ∗ b = ¬ ( a ∗ ′ b ), where ∗ ′ = Title Page 0(1 − ε 1 )(1 − ε 2 )(1 − ε 3 ), and f ∗ ( a, b ) = 1 − f ∗ ′ ( a, b ). ◭◭ ◮◮ • Comment: ◭ ◮ – all boolean operations can be described in terms of Page 8 of 8 &, ∨ , and ¬ ; Go Back – however, fuzzy exclusive “or” cannot be described Full Screen in terms of fuzzy &, ∨ , and ¬ ; Close – so, we need to add exclusive “or” to basic opera- tions. Quit
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