Fuzzy Techniques: A . . . Need to Select Proper . . . Re-Scaling Scale-Invariance: Idea Natural Invariance Explains Empirical Let Us Apply This Idea . . . Success of Specific Membership So, What Are . . . Functions, Hedge Operations, and What Membership . . . Negation Operations Definitions and the . . . Which Hedge . . . Julio C. Urenda 1 , Orsolya Csisz´ ar 2 , 3 , G´ ar 4 , abor Csisz´ Home Page ozsef Dombi 5 , Gy¨ orgy Eigner 3 , and Vladik Kreinovich 1 J´ Title Page 1 University of Texas at El Paso, El Paso, TX 79968, USA ◭◭ ◮◮ jcurenda@utep.edu, vladik@utep.edu 2 Faculty of Basic Sciences, University of Applied Sciences Esslingen ◭ ◮ Esslingen, Germany Page 1 of 40 3 Institute of Applied Mathematics, ´ Obuda University, Budapest, Hungary eigner.gyorgy@nik.uni-obuda.hu, orsolya.csiszar@nik.uni-obuda.hu Go Back 4 Institute of Materials Physics, University of Stuttgart Stuttgart, Germany, gabor.csiszar@mp.imw.uni-stuttgart.de Full Screen 5 Institute of Informatics, University of Szeged, Szeged, Hungary Close dombi@inf.u-szeged.hu Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 1. Fuzzy Techniques: A Brief Reminder Re-Scaling • In many applications, we have knowledge formulated: Scale-Invariance: Idea Let Us Apply This Idea . . . – in terms of imprecise (“fuzzy”) terms from natural So, What Are . . . language, What Membership . . . – like “small”, “somewhat small”, etc. Definitions and the . . . • To translate this knowledge into computer-understandable Which Hedge . . . form, Lotfi Zadeh proposes fuzzy techniques . Home Page • According to these techniques, each imprecise property Title Page like “small” can be described by assigning: ◭◭ ◮◮ – to each value x of the corresponding quantity, ◭ ◮ – a degree µ ( x ) to which, according to the expert, Page 2 of 40 this property is true. Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 2. Fuzzy Techniques (cont-d) Re-Scaling • These degrees are usually selected from the interval Scale-Invariance: Idea [0 , 1], so that: Let Us Apply This Idea . . . So, What Are . . . – 1 corresponds to full confidence, What Membership . . . – 0 to complete lack of confidence, and Definitions and the . . . – values between 0 and 1 describe intermediate de- Which Hedge . . . grees of confidence. Home Page • The resulting function µ ( x ) is known as a membership Title Page function . ◭◭ ◮◮ • In practice, we can only ask finitely many questions to ◭ ◮ the expert. Page 3 of 40 • So we only elicit a few values µ ( x 1 ), µ ( x 2 ), etc. Go Back • Based on these values, we need to estimate the values Full Screen µ ( x ) for all other values x . Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 3. Fuzzy Techniques (cont-d) Re-Scaling • For this purpose, usually: Scale-Invariance: Idea Let Us Apply This Idea . . . – we select a family of membership functions – e.g., So, What Are . . . triangular, trapezoidal, etc. – and What Membership . . . – we select a function from this family which best fits Definitions and the . . . the known values. Which Hedge . . . • For terms like “somewhat small”, “very small”, the Home Page situation is more complicated. Title Page • We can add different “hedges” like “somewhat”, “very”, ◭◭ ◮◮ etc., to each property. ◭ ◮ • As a result, we get a large number of possible terms. Page 4 of 40 Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 4. Fuzzy Techniques (cont-d) Re-Scaling • It is not realistically possible to ask the expert about Scale-Invariance: Idea each such term; instead: Let Us Apply This Idea . . . So, What Are . . . – practitioners estimate the degree to which, e.g., What Membership . . . “somewhat small” is true Definitions and the . . . – based on the degree to which “small” is true. Which Hedge . . . • In other words, with each linguistic hedge, we associate Home Page a function h from [0 , 1] to [0 , 1] that: Title Page – transforms the degree to which a property is true ◭◭ ◮◮ – into an estimate for the degree to which the hedged ◭ ◮ property is true. Page 5 of 40 Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 5. Fuzzy Techniques (cont-d) Re-Scaling • Similarly to the membership functions: Scale-Invariance: Idea Let Us Apply This Idea . . . – we can elicit a few values h ( x i ) of the hedge oper- So, What Are . . . ation from the experts, and What Membership . . . – then we extrapolate and/or interpolate to get all Definitions and the . . . the other values of h ( x ). Which Hedge . . . • Usually, a family of hedge operations is pre-selected. Home Page • Then we select a specific operation from this family Title Page which best fits the elicited values h ( x i ). ◭◭ ◮◮ ◭ ◮ Page 6 of 40 Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 6. Fuzzy Techniques (cont-d) Re-Scaling • Similarly: Scale-Invariance: Idea Let Us Apply This Idea . . . – instead of asking experts for their degrees of confi- So, What Are . . . dence in statements like “not small”, What Membership . . . – we estimate these degrees based on their degrees of Definitions and the . . . confidence in the positive statements. Which Hedge . . . • The corresponding operation n ( x ) is known as the nega- Home Page tion operation. Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 40 Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 7. Need to Select Proper Membership Functions, Re-Scaling Hedge Operations, And Negation Operations Scale-Invariance: Idea • Fuzzy techniques have been successfully applied to many Let Us Apply This Idea . . . application areas. So, What Are . . . What Membership . . . • However, this does not necessarily mean that every Definitions and the . . . time we try to use fuzzy techniques, we get a success. Which Hedge . . . • The success (or not) often depends on which member- Home Page ship functions etc. we select: Title Page – for some selections, we get good results (e.g., good ◭◭ ◮◮ control), ◭ ◮ – for other selections, the results are not so good. Page 8 of 40 • There is a lot of empirical data about which selections work better. Go Back Full Screen • In this talk, we provide a general explanation for sev- eral of these empirically best selections. Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 8. Need to Select Proper Functions (cont-d) Re-Scaling • This explanation is based on the natural concepts of Scale-Invariance: Idea invariance. Let Us Apply This Idea . . . So, What Are . . . • For symmetric membership functions that describe prop- What Membership . . . erties like “small”, Definitions and the . . . – for which µ ( x ) = µ ( − x ) and the degree µ ( | x | ) de- Which Hedge . . . creases with | x | , Home Page – in many practical situations, the most empirically Title Page successful are so-called distending functions: ◭◭ ◮◮ 1 µ ( x ) = 1 + a · | x | b . ◭ ◮ Page 9 of 40 • Among hedge and negation operations, often, the most Go Back efficient are fractional linear functions: h ( x ) = a + b · x Full Screen 1 + c · x. Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 9. Re-Scaling Re-Scaling • The variable x describes the value of some physical Scale-Invariance: Idea quantity, such a distance, height, etc. Let Us Apply This Idea . . . So, What Are . . . • When we process these values, we deal with numbers. What Membership . . . • Numbers depend on the selection of the measuring Definitions and the . . . unit: Which Hedge . . . Home Page – if we replace the original measuring unit with a new one which is λ times smaller, Title Page – then all the numerical values will be multiplied by ◭◭ ◮◮ λ : x → X = λ · x . ◭ ◮ • For example, 2 meters become 2 · 100 = 200 cm. Page 10 of 40 • This transformation from one measuring scale to an- Go Back other is known as re-scaling . Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 10. Scale-Invariance: Idea Re-Scaling • In many physical situations, the choice of a measuring Scale-Invariance: Idea unit is rather arbitrary. Let Us Apply This Idea . . . So, What Are . . . • In such situations, all the formulas remain the same no What Membership . . . matter what unit we use. Definitions and the . . . • For example, the formula y = x 2 for the area of the Which Hedge . . . square with side x remains valid: Home Page – if we replace the unit for measuring sides from me- Title Page ters with centimeters, ◭◭ ◮◮ – of course, we then need to appropriately change the unit for y , from m 2 to cm 2 . ◭ ◮ Page 11 of 40 Go Back Full Screen Close Quit
Fuzzy Techniques: A . . . Need to Select Proper . . . 11. Scale-Invariance (cont-d) Re-Scaling • In general, invariance of the formula y = f ( x ) means Scale-Invariance: Idea that: Let Us Apply This Idea . . . So, What Are . . . – for each re-scaling x → X = λ · x , there exists an What Membership . . . appropriate re-scaling y → Y Definitions and the . . . – for which the same formula Y = f ( X ) will be true Which Hedge . . . for the re-scaled variables X and Y . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 40 Go Back Full Screen Close Quit
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