Main Objective of . . . It Is important to . . . Imprecise (“Fuzzy”) Rules Fuzzy Logic Fuzzy Systems Are Universal . . . Universal Approximators Often, We Can Only . . . Main Idea: What Is . . . for Random Dependencies: In These Terms, What . . . This Leads to a . . . A Simplified Proof Home Page Title Page Mahdokht Afravi and Vladik Kreinovich Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA mafravi@miners.utep.edu, vladik@utep.edu Page 1 of 19 Go Back Full Screen Close Quit
Main Objective of . . . It Is important to . . . 1. Main Objective of Science in General Imprecise (“Fuzzy”) Rules • One of the main objectives of science is: Fuzzy Logic Universal . . . – to find the state of the world and Often, We Can Only . . . – to predict the future state of the world. Main Idea: What Is . . . • We need to do it both: In These Terms, What . . . This Leads to a . . . – in situations when we do not interfere and Home Page – in situations when we perform a certain action. Title Page • The state of the world is usually characterized by the ◭◭ ◮◮ values of appropriate physical quantities. ◭ ◮ • For example: Page 2 of 19 – we would like to know the distance y to a distant Go Back star, – we would like to predict tomorrow’s temperature y Full Screen at a given location, etc. Close Quit
Main Objective of . . . It Is important to . . . 2. Direct Measurement Is Not Always Possible Imprecise (“Fuzzy”) Rules • In some cases, we can directly measure the current Fuzzy Logic value of the quantity y of interest. Universal . . . Often, We Can Only . . . • However, in many practical cases, such a direct mea- Main Idea: What Is . . . surement is not possible – e.g.: In These Terms, What . . . – while it is possible to measure a distance to a This Leads to a . . . nearby town by just driving there, Home Page – it is not yet possible to directly travel to a faraway Title Page star. ◭◭ ◮◮ • And it is definitely not possible to measure tomorrow’s ◭ ◮ temperature y today. Page 3 of 19 Go Back Full Screen Close Quit
Main Objective of . . . It Is important to . . . 3. Need for Indirect Measurements Imprecise (“Fuzzy”) Rules • In such situations, since we cannot directly measure Fuzzy Logic the value of the desired quantity y , a natural idea is: Universal . . . Often, We Can Only . . . – to measure related easier-to-measure quantities Main Idea: What Is . . . x 1 , . . . , x n , and then In These Terms, What . . . – to use the known dependence y = f ( x 1 , . . . , x n ) be- This Leads to a . . . tween these quantities to estimate y . Home Page • For example, to predict tomorrow’s temperature at a Title Page given location, we can: ◭◭ ◮◮ – measure today’s values of temperature, wind veloc- ◭ ◮ ity, humidity, etc. in nearby locations, and then Page 4 of 19 – use the known equations of atmospheric physics to predict tomorrow’s temperature y . Go Back Full Screen Close Quit
Main Objective of . . . It Is important to . . . 4. It Is important to Determine Dependencies Imprecise (“Fuzzy”) Rules • In some cases we know the exact form of the depen- Fuzzy Logic dence y = f ( x 1 , . . . , x n ). Universal . . . Often, We Can Only . . . • However, in many other practical situations, we do not Main Idea: What Is . . . have this information. In These Terms, What . . . • Instead, we have to rely on experts. This Leads to a . . . Home Page • Experts often formulate their rules in terms of impre- cise (“fuzzy”) words from natural language. Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 19 Go Back Full Screen Close Quit
Main Objective of . . . It Is important to . . . 5. Imprecise (“Fuzzy”) Rules Imprecise (“Fuzzy”) Rules • What kind of imprecise rules can we have? Fuzzy Logic Universal . . . • In some cases, the experts formulating the rule are im- Often, We Can Only . . . precise both about x i and about y . Main Idea: What Is . . . • In such situations, we may have rules like this: In These Terms, What . . . • if today’s temperature is very low and the Northern This Leads to a . . . Home Page wind is strong, • the temperature will remain very low tomorrow. Title Page ◭◭ ◮◮ • In this case: ◭ ◮ • x 1 is temperature today, Page 6 of 19 • x 2 is the speed of the Northern wind, • y is tomorrow’s temperature, and Go Back • the properties “very low” and “strong” are impre- Full Screen cise. Close Quit
Main Objective of . . . It Is important to . . . 6. Fuzzy Rules: General Case Imprecise (“Fuzzy”) Rules • In general, we have rules of the following type, where Fuzzy Logic A ki and A k are imprecise properties: Universal . . . Often, We Can Only . . . “if x 1 is A k 1 , . . . , and x n is A kn , then y is A k ”, Main Idea: What Is . . . In These Terms, What . . . • It is worth mentioning that in some cases, This Leads to a . . . – the information about x i is imprecise, but Home Page – the conclusion about y is described by a precise Title Page expression. ◭◭ ◮◮ • For example, in non-linear mechanics, we can say that: ◭ ◮ – when the stress x 1 is small, the strain y is deter- Page 7 of 19 mined by a formula y = k · x 1 , with known k , but Go Back – when the stress is high, we need to use a nonlinear expression y = k · x 1 − a · x 2 2 with known k and a . Full Screen Close Quit
Main Objective of . . . It Is important to . . . 7. Fuzzy Logic Imprecise (“Fuzzy”) Rules • To transform such expert rules into a precise expres- Fuzzy Logic sion, Zadeh invented fuzzy logic. Universal . . . Often, We Can Only . . . • In fuzzy logic, to describe each imprecise property P , Main Idea: What Is . . . we ask the expert to assign, In These Terms, What . . . – to each possible value x of the corresponding quan- This Leads to a . . . tity, Home Page – a degree µ P ( x ) to which the value x satisfies this Title Page property – e.g., to what extent the value x is small. ◭◭ ◮◮ • We can do this, e.g., by asking the expert to mark, on ◭ ◮ a 0-to-10 scale, to what extent the given x is small. Page 8 of 19 • If the expert marks 7, we take µ P ( x ) = 7 / 10. Go Back • The function µ P ( x ) that assigns this degree is known Full Screen as the membership function corr. to P . Close Quit
Main Objective of . . . It Is important to . . . 8. Fuzzy Logic (cont-d) Imprecise (“Fuzzy”) Rules • For given inputs x 1 , . . . , x n , a value y is possible if it Fuzzy Logic fits within one of the rules, i.e., if: Universal . . . Often, We Can Only . . . – either the first rule is satisfied, i.e., x 1 is A 11 , . . . , Main Idea: What Is . . . x n is A 1 n , and y is A 1 , In These Terms, What . . . – or the second rule is satisfied, i.e., x 1 is A 21 , . . . , x n This Leads to a . . . is A 2 n , and y is A 2 , etc. Home Page • We assumed that we know the membership functions Title Page µ ki ( x i ) and µ k ( y ) corresponding to A ki and A k . ◭◭ ◮◮ • We can thus find the degrees µ ki ( x i ) and µ k ( y ) to which ◭ ◮ each corresponding property is satisfied. Page 9 of 19 • To estimate the degree to which y is possible, we must Go Back be able to deal with “or” and “and”. Full Screen Close Quit
Main Objective of . . . It Is important to . . . 9. Need for “And”- and “Or”-Operations Imprecise (“Fuzzy”) Rules • In other words, we need to come up with a way Fuzzy Logic Universal . . . – to estimate our degrees of confidence in statements Often, We Can Only . . . A ∨ B and A & B Main Idea: What Is . . . – based on the known degrees of confidence a and b In These Terms, What . . . of the elementary statements A and B . This Leads to a . . . • Such estimation algorithms are known as t-conorms Home Page (“or”-operations) and t-norms (“and”-operations). Title Page • We will denote them by f ∨ ( a, b ) and f & ( a, b ). ◭◭ ◮◮ • In these terms, the degree µ ( y ) to which y is possible ◭ ◮ can be estimated as µ ( y ) = f ∨ ( r 1 , r 2 , . . . ), where Page 10 of 19 def r k = f & ( µ k 1 ( x 1 ) , . . . , µ kn ( x n ) , µ k ( y )) . Go Back • We can then transform these degrees into a numerical Full Screen estimate y . Close Quit
Main Objective of . . . It Is important to . . . 10. Fuzzy Technique: Final Step Imprecise (“Fuzzy”) Rules • We can then transform the degrees µ ( y ) into a numer- Fuzzy Logic ical estimate y . Universal . . . Often, We Can Only . . . • This can be done, e.g., by minimizing the weighted � µ ( y ) · ( y − y ) 2 dy . Main Idea: What Is . . . mean square difference In These Terms, What . . . • This results in This Leads to a . . . � y · µ ( y ) dy Home Page � y = µ ( y ) dy . Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 19 Go Back Full Screen Close Quit
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