Traditional Approach: . . . Need to Go Beyond . . . Sugeno λ -Measures Problem Why Sugeno λ -Measures Sugeno λ -Measure is . . . Processing Sugeno . . . Saiful Abu 1 , Vladik Kreinovich 1 How to Explain the . . . Joe Lorkowski 1 , and Hung T. Nguyen 2 , 3 Sugeno Measure in . . . Which Fuzzy . . . 1 Department of Computer Science Home Page University of Texas at El Paso 500 W. University Title Page El Paso, Texas 79968 sabu@miners.utep.edu, vladik@utep.edu ◭◭ ◮◮ lorkowski@computer.org ◭ ◮ 2 Department of Mathematical Sciences Page 1 of 17 New Mexico State University Las Cruces, NM 88003 Go Back 3 Faculty of Economics Chiang Mai University, Thailand Full Screen hunguyen@nmsu.edu Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 1. Traditional Approach: Probability Measures Sugeno λ -Measures • Traditionally, uncertainty has been described by prob- Problem abilities. Sugeno λ -Measure is . . . Processing Sugeno . . . • The probability p ( A ) of a set A is usually interpreted as How to Explain the . . . the frequency with which events from the set A occur. Sugeno Measure in . . . • In this interpretation: Which Fuzzy . . . – if we have two disjoint sets A and B with A ∩ B = ∅ , Home Page – then the frequency p ( A ∪ B ) with which the events Title Page from A or B happen ◭◭ ◮◮ – is equal to the sum of the frequencies p ( A ) and p ( B ) ◭ ◮ corresponding to each of these sets. Page 2 of 17 • This property of probabilities measures is known as Go Back additivity : if A ∩ B = ∅ , then Full Screen p ( A ∪ B ) = p ( A ) + p ( B ) . Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 2. Need to Go Beyond Probability Measures Sugeno λ -Measures • To adequately describe expert knowledge, we often Problem need to go beyond probabilities. Sugeno λ -Measure is . . . Processing Sugeno . . . • In general, instead of probabilities, we have the ex- How to Explain the . . . pert’s degree of confidence g ( A ) in A . Sugeno Measure in . . . • Clearly, g ( ∅ ) = 0 and g ( X ) = 1. Which Fuzzy . . . • Also, clearly, the larger the set, the more confident we Home Page are that an event from this set will occur: Title Page A ⊆ B implies g ( A ) ≤ g ( B ) . ◭◭ ◮◮ ◭ ◮ • Functions g ( A ) that satisfy these properties are known Page 3 of 17 as fuzzy measures. Go Back Full Screen Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 3. Sugeno λ -Measures Sugeno λ -Measures • M. Sugeno introduced a specific class of fuzzy measures Problem which are now known as Sugeno λ -measures . Sugeno λ -Measure is . . . Processing Sugeno . . . • If we know g ( A ) and g ( B ) for two disjoint sets, we can How to Explain the . . . still reconstruct the degree g ( A ∪ B ). Sugeno Measure in . . . • For Sugeno measure, Which Fuzzy . . . Home Page g ( A ∪ B ) = g ( A ) + g ( B ) + λ · g ( A ) · g ( B ) . Title Page • When λ = 0, this formula transforms into additivity. ◭◭ ◮◮ • Sugeno λ -measures are among the most widely used ◭ ◮ and most successful fuzzy measures. Page 4 of 17 Go Back Full Screen Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 4. Problem Sugeno λ -Measures • The success of Sugeno measures is somewhat paradox- Problem ical: Sugeno λ -Measure is . . . Processing Sugeno . . . – The main point of using fuzzy measures is to go How to Explain the . . . beyond probability measures. Sugeno Measure in . . . – On the other hand, Sugeno λ -measures are, in some Which Fuzzy . . . reasonable sense, equivalent to probabilities. Home Page • In this talk, we explain this seeming paradox: from the Title Page computational viewpoint, ◭◭ ◮◮ – processing Sugeno measure directly is much more ◭ ◮ computationally efficient Page 5 of 17 – than using a reduction to a probability measure. Go Back • We also analyze which other probability-equivalent fuzzy measures have this property. Full Screen Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 5. Sugeno λ -Measure is Mathematically Equiva- Sugeno λ -Measures lent to a Probability Measure Problem • In Sugeno measure, if we know a = g ( A ) and b = g ( B ) Sugeno λ -Measure is . . . for A ∩ B = ∅ , then we can compute c = g ( A ∪ B ) as Processing Sugeno . . . How to Explain the . . . c = a + b + λ · a · b. Sugeno Measure in . . . • We would like to find a 1-1 function f ( x ) for which Which Fuzzy . . . def = f − 1 ( g ( A )) is a probability measure. p ( A ) Home Page • This means that if c = a + b + λ · a · b , then c ′ = a ′ + b ′ , Title Page where a ′ = f − 1 ( a ) , b ′ = f − 1 ( b ) , and c ′ = f − 1 ( c ) . ◭◭ ◮◮ • For λ > 0, this holds for f ( x ′ ) = 1 ◭ ◮ λ · (exp( x ′ ) − 1) . Page 6 of 17 • For λ < 0, this holds for f ( x ′ ) = 1 | λ | · (1 − exp( − x ′ )) . Go Back Full Screen • So, a Sugeno λ -measure is indeed equivalent to a prob- ability measure. Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 6. Processing Sugeno Measures Is More Compu- Sugeno λ -Measures tationally Efficient than Using Probabilities Problem • If we know g ( A ) and g ( B ), then we can compute Sugeno λ -Measure is . . . Processing Sugeno . . . g ( A ∪ B ) = g ( A ) + g ( B ) + λ · g ( A ) · g ( B ) . How to Explain the . . . • This computation uses only hardware supported (thus, Sugeno Measure in . . . fast) + and · . Alternative is to: Which Fuzzy . . . – compute p ( A ) = f − 1 ( g ( A )) and p ( B ) = f − 1 ( g ( B )); Home Page – add these probabilities p ( A ∪ B ) = p ( A ) + p ( B ); Title Page – finally, re-scale this resulting probability back into ◭◭ ◮◮ degree-of-confidence: g ( A ∪ B ) = f ( p ( A ∪ B )) . ◭ ◮ • In this approach, we compute logarithm (to compute Page 7 of 17 f − 1 ( x )) and exponential function (to compute f ( x )). Go Back • These computations are much slower than + and · . Full Screen • Thus, the direct use of Sugeno measure is definitely much more computationally efficient. Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 7. How to Explain the Use of Sugeno Measure in Sugeno λ -Measures Probabilistic Terms Problem • We are interested in expert estimates of probabilities Sugeno λ -Measure is . . . of different sets of events. Processing Sugeno . . . How to Explain the . . . • It is known that expert estimates of the probabilities Sugeno Measure in . . . are biased: Which Fuzzy . . . – the expert’s subjective estimates g ( A ) of the corre- Home Page sponding probabilities p ( A ) Title Page – are equal to g ( A ) = f ( p ( A )) for an appropriate re- ◭◭ ◮◮ scaling function f ( A ). ◭ ◮ • In this case, a natural ideas seems to be: Page 8 of 17 – to re-scale all the estimates back into the probabil- ities: p ( A ) = f − 1 ( g ( A )) , and Go Back – to use the usual algorithms to process these prob- Full Screen abilities. Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 8. Sugeno Measure in Prob. Terms (cont-d) Sugeno λ -Measures • If we know the expert’s estimates g ( A ) and g ( B ) for Problem A ∩ B = ∅ , to predict the g ( A ∪ B ), we: Sugeno λ -Measure is . . . Processing Sugeno . . . – re-scale g ( A ) and g ( B ) into probabilities: How to Explain the . . . p ( A ) = f − 1 ( g ( A )) and p ( B ) = f − 1 ( g ( B )); Sugeno Measure in . . . Which Fuzzy . . . – compute p ( A ∪ B ) = p ( A ) + p ( B ); and Home Page – estimate g ( A ∪ B ) as g ( A ∪ B ) = f ( p ( A ∪ B )) . Title Page • For some biasing functions f ( x ), it is computationally ◭◭ ◮◮ more efficient ◭ ◮ – not to re-scale into probabilities, Page 9 of 17 – but to store and process the original values g ( A ). Go Back • This is, in effect, the essence of applications of a Sugeno Full Screen λ -measure are about. Close Quit
Traditional Approach: . . . Need to Go Beyond . . . 9. Which Fuzzy Measures Have This Property Sugeno λ -Measures • If we know the expert’s estimates a = g ( A ) and b = Problem g ( B ) for A ∩ B = ∅ , to predict the g ( A ∪ B ), we: Sugeno λ -Measure is . . . Processing Sugeno . . . – re-scale a and b into probabilities: How to Explain the . . . p ( A ) = f − 1 ( a ) and p ( B ) = f − 1 ( b ); Sugeno Measure in . . . Which Fuzzy . . . – compute p ( A ∪ B ) = f − 1 ( a ) + f − 1 ( b ); and Home Page – estimate g ( A ∪ B ) as F ( a, b ) = f ( f − 1 ( a ) + f − 1 ( b )) . Title Page • One can check that F ( a, b ) is commutative, associative, ◭◭ ◮◮ and F (0 , a ) = a . ◭ ◮ • We want to find all such F ( a, b ) for which direct com- Page 10 of 17 putation is faster than this 3-stage approach. Go Back • Computation is fast it consists of a sequence of hard- ware supported elementary operations: +, − , · , /. Full Screen Close Quit
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