Need to Gauge . . . Such Gauging Is . . . How We Elicit Fuzzy . . . How to Gauge the Accuracy Let Us Use This Idea . . . Resulting . . . of Fuzzy-Control But What Should We . . . Resulting Algorithm Recommendations: Proof A Simple Idea Home Page Title Page Patricia Melin 1 , Oscar Castillo 1 , ◭◭ ◮◮ Andrzej Pownuk 2 , Olga Kosheleva 2 , and Vladik Kreinovich 2 ◭ ◮ 1 Department of Computer Science, Tijuana Institute of Technology, Page 1 of 15 Tijuana, Baja California, Mexico, ocastillo@tectijuana.mx, pmelin@tectijuana.mx Go Back 2 University of Texas at El Paso, El Paso, TX 79968, USA ampownuk@utep.edu, olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Need to Gauge . . . 1. Need to Gauge Accuracy of Fuzzy Recommen- Such Gauging Is . . . dations How We Elicit Fuzzy . . . Let Us Use This Idea . . . • Fuzzy logic has been successfully applied to many dif- Resulting . . . ferent application areas, e.g., in control. But What Should We . . . • A natural question is: with what accuracy do we need Resulting Algorithm to implement this recommendation? Proof Home Page • In many applications, this is an important question: Title Page – it is often much easier to implement the control value approximately, ◭◭ ◮◮ – but maybe a more accurate actuator is needed? ◭ ◮ • To answer this question, we must be able to gauge the Page 2 of 15 accuracy of the corresponding recommendations. Go Back Full Screen Close Quit
Need to Gauge . . . 2. Such Gauging Is Possible for Probabilistic Un- Such Gauging Is . . . certainty How We Elicit Fuzzy . . . Let Us Use This Idea . . . • Probabilistic uncertainty means that instead of the ex- Resulting . . . act value x , we only know a probability distribution. But What Should We . . . • This distribution can be described, e.g., by the proba- Resulting Algorithm bility density ρ ( x ). Proof Home Page • If we need to select a single value x , a natural idea is � to select, e.g., the mean value x = x · ρ ( x ) dx . Title Page • A natural measure of accuracy is the mean square devi- ◭◭ ◮◮ ation from the mean, known as the standard deviation: ◭ ◮ �� Page 3 of 15 def ( x − x) 2 dx. = σ Go Back Full Screen • We need a similar formula for the fuzzy case. Close Quit
Need to Gauge . . . 3. How We Elicit Fuzzy Degrees: A Brief Re- Such Gauging Is . . . minder How We Elicit Fuzzy . . . Let Us Use This Idea . . . • For each possible value x of the corresponding quantity, Resulting . . . we ask the expert to mark: But What Should We . . . – on a scale from 0 to 1, Resulting Algorithm – his/her degree of confidence that x satisfies the Proof given property. Home Page Title Page • For example, we ask the expert to specify the degree to which the value x is small. ◭◭ ◮◮ • In some cases, this is all we need. ◭ ◮ • However, in many other cases, we get a non-normalized Page 4 of 15 membership function, for which max µ ( x ) < 1. Go Back x • Most fuzzy techniques assume that the membership Full Screen function is normalized. Close Quit
Need to Gauge . . . 4. How We Elicit Fuzzy Degrees (cont-d) Such Gauging Is . . . How We Elicit Fuzzy . . . • So, we sometimes need to perform an additional step Let Us Use This Idea . . . to get an easy-to-process membership function. Resulting . . . • Namely, we normalize the original values µ ( x ) by di- But What Should We . . . viding them by the largest of the values µ ( y ): Resulting Algorithm µ ( x ) Proof def µ ′ ( x ) = µ ( y ) . Home Page max y Title Page • Sometimes, the experts have some subjective probabil- ◭◭ ◮◮ ities ρ ( x ) assigned to different values x . ◭ ◮ • In this case, when asked to indicate their degree of Page 5 of 15 certainty, they list µ ( x ) = ρ ( x ). Go Back • After normalizing this µ ( x ), we get the membership ρ ( x ) Full Screen function µ ( x ) = ρ ( y ) . max Close y Quit
Need to Gauge . . . 5. Let Us Use This Idea to Gauge the Accuracy Such Gauging Is . . . of Fuzzy Recommendations How We Elicit Fuzzy . . . Let Us Use This Idea . . . • We assign, to each probability density function ρ ( x ), a Resulting . . . ρ ( x ) membership function µ ( x ) = ρ ( y ) . But What Should We . . . max y Resulting Algorithm • Vice versa, if we know that µ ( x ) was obtained by nor- Proof malizing some ρ ( x ), we can uniquely reconstruct ρ ( x ): Home Page µ ( x ) Title Page ρ ( x ) = µ ( y ) dy. � ◭◭ ◮◮ • Our idea is then to use the probabilistic formulas cor- ◭ ◮ responding to this artificial distribution. Page 6 of 15 • At first glance, this does not make sense. Go Back • The probabilistic measure of accuracy is based on the Full Screen assumption that we use the mean. Close • But don’t we use something else in fuzzy? Quit
Need to Gauge . . . 6. Let Us Use This Idea to Gauge the Accuracy Such Gauging Is . . . of Fuzzy Recommendations (cont-d) How We Elicit Fuzzy . . . Let Us Use This Idea . . . • Don’t we use something else in fuzzy? Resulting . . . • Actually, not really. But What Should We . . . µ ( x ) Resulting Algorithm • The mean of the distribution ρ ( x ) = µ ( y ) dy is � Proof Home Page � x · µ ( x ) dx � x = x · ρ ( x ) dx = µ ( x ) dx . Title Page � ◭◭ ◮◮ • This is the centroid defuzzification – one of the main ◭ ◮ ways to transform µ ( x ) into a control recommendation. Page 7 of 15 • Since the above idea makes sense, let us use it to gauge Go Back the accuracy of the fuzzy control recommendation. Full Screen Close Quit
Need to Gauge . . . 7. Resulting Recommendation Such Gauging Is . . . How We Elicit Fuzzy . . . • For a given membership function µ ( x ), we usually gen- Let Us Use This Idea . . . erate the result x of its centroid defuzzification. Resulting . . . • We should also generate, as a measure of the accuracy But What Should We . . . of this recommendation, the following value σ : Resulting Algorithm ( x − x) 2 · µ ( x ) dx � Proof � σ 2 = ( x − x) 2 · ρ ( x ) dx = = Home Page � µ ( x ) dx Title Page x 2 · µ ( x ) dx � 2 � �� x · µ ( x ) dx . − ◭◭ ◮◮ � � µ ( x ) dx µ ( x ) dx ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit
Need to Gauge . . . 8. But What Should We Do in the Interval-Valued Such Gauging Is . . . Fuzzy Case? How We Elicit Fuzzy . . . Let Us Use This Idea . . . • Often, experts cannot tell us the exact values µ ( x ). Resulting . . . • Instead, for each x , they tell us the interval [ µ ( x ) , µ ( x )] But What Should We . . . of possible value of degree of confidence µ ( x ). Resulting Algorithm Proof • For different functions µ ( x ) ∈ [ µ ( x ) , µ ( x )], we get dif- Home Page ferent values σ 2 . Title Page • It is desirable to find the range of possible values σ 2 when µ ( x ) ∈ [ µ ( x ) , µ ( x )]. ◭◭ ◮◮ ◭ ◮ Page 9 of 15 Go Back Full Screen Close Quit
Need to Gauge . . . 9. Resulting Algorithm Such Gauging Is . . . How We Elicit Fuzzy . . . • For all possible pairs x < x , we compute σ 2 ( µ − ) and Let Us Use This Idea . . . σ 2 ( µ + ), where: Resulting . . . • µ + ( x ) = µ ( x ) when x < x or x > x , and But What Should We . . . µ + ( x ) = µ ( x ) when x < x < x ; Resulting Algorithm • µ − ( x ) = µ ( x ) when x < x or x > x , and Proof µ − ( x ) = µ ( x ) when x < x < x . Home Page • As the upper bound for σ 2 , we take the maximum of the Title Page values σ 2 ( µ + ) corresponding to different pairs x < x . ◭◭ ◮◮ • As the lower bound for σ 2 , we take the minimum of the ◭ ◮ values σ 2 ( µ − ) corresponding to different pairs x < x . Page 10 of 15 Go Back Full Screen Close Quit
Need to Gauge . . . 10. Proof Such Gauging Is . . . How We Elicit Fuzzy . . . • According to calculus, when f ( z ) attains max on [ z, z ] Let Us Use This Idea . . . at z 0 ∈ [ z, z ], then we have one of the three cases: Resulting . . . – we can have z 0 ∈ ( z, z ), in which case d f dz ( z 0 ) = 0; But What Should We . . . Resulting Algorithm – we can have z 0 = z , in which case d f dz ( z 0 ) ≤ 0, or Proof Home Page – we can have z 0 = z , in which case d f dz ( z 0 ) ≥ 0. Title Page • Similarly, when f ( z ) attains min on [ z, z ] at z 0 ∈ [ z, z ], ◭◭ ◮◮ then we have one of the three cases: ◭ ◮ – we can have z 0 ∈ ( z, z ), in which case d f dz ( z 0 ) = 0; Page 11 of 15 – we can have z 0 = z , in this case, in which case Go Back d f dz ( z 0 ) ≥ 0, or Full Screen – we can have z 0 = z , in which case d f dz ( z 0 ) ≤ 0. Close Quit
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