Single-Quantity Data . . . Limitation of Fuzzy . . . Formulation of the . . . How to Relate Fuzzy and Fuzzy Fusion for . . . Resulting . . . OWA Estimates A Similar Problem Is . . . Relation between M- . . . Resulting Solution: . . . Tanja Magoc and Vladik Kreinovich From OWA to Fuzzy Department of Computer Science Title Page University of Texas at El Paso 500 W. University ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ contact email vladik@utep.edu Page 1 of 15 http://www.cs.utep.edu/vladik Go Back Full Screen Close Quit
Single-Quantity Data . . . 1. Single-Quantity Data Fusion: A Problem Limitation of Fuzzy . . . Formulation of the . . . • In many practical situations, we have several estimates Fuzzy Fusion for . . . x 1 , . . . , x n of the same quantity x : Resulting . . . x 1 ≈ x, x 2 ≈ x, . . . , x n ≈ x. A Similar Problem Is . . . • It is desirable to combine (fuse) these estimates into a Relation between M- . . . single estimate for x . Resulting Solution: . . . From OWA to Fuzzy • From the fuzzy viewpoint, a natural way to combine Title Page these estimates is as follows: ◭◭ ◮◮ – to describe, for each x and for each i , the degree µ ≈ ( x i − x ) to which x is close to x i ; ◭ ◮ – to use a t-norm (“and”-operation) t & ( a, b ) to com- Page 2 of 15 bine these degrees into a single degree Go Back d ( x ) = t & ( µ ≈ ( x 1 − x ) , . . . , µ ≈ ( x n − x )); Full Screen – and find the estimate x for which the degree d ( x ) Close – that x is close to all x i – is the largest. Quit
Single-Quantity Data . . . 2. Limitation of Fuzzy and Emergence of OWA Limitation of Fuzzy . . . Formulation of the . . . • Reminder: find x that maximizes Fuzzy Fusion for . . . d ( x ) = t & ( µ ≈ ( x 1 − x ) , . . . , µ ≈ ( x n − x )) . Resulting . . . A Similar Problem Is . . . • Main problem: the corresponding procedure is compu- tationally complex, esp. for generic µ ≈ ( x ) and t & ( a, b ). Relation between M- . . . Resulting Solution: . . . • Solution: OWA (Ordered Weighted Average) approach: From OWA to Fuzzy – sort the values x 1 , . . . , x n into an increasing sequence Title Page x (1) ≤ x (2) ≤ . . . ≤ x ( n ) ; ◭◭ ◮◮ – select the weights w 1 , . . . , w n ≥ 0 for which ◭ ◮ n Page 3 of 15 � w i = 1; Go Back i =1 n Full Screen � – use the weighted average x = w i · x ( i ) as the Close i =1 desired fused estimate. Quit
Single-Quantity Data . . . 3. Formulation of the Problem Limitation of Fuzzy . . . Formulation of the . . . • To get a better fusion: Fuzzy Fusion for . . . – we must appropriately select the membership func- Resulting . . . tion µ ≈ ( x ) and the t-norm (in the fuzzy case), and A Similar Problem Is . . . – we must appropriately select the weights w i (in the Relation between M- . . . OWA case). Resulting Solution: . . . From OWA to Fuzzy • Both approaches – when applied properly – lead to Title Page reasonable data fusion. ◭◭ ◮◮ • It is therefore desirable to be able to relate the corre- sponding selections: ◭ ◮ Page 4 of 15 – once we have found the appropriate µ ≈ ( x ) and t- norm, we should be able to deduce the weights; Go Back – once we have found the appropriate weights, we Full Screen should be able to deduce µ ≈ ( x ) and t-norm. Close Quit
Single-Quantity Data . . . 4. Reducing to the Case of Archimedean t-Norms Limitation of Fuzzy . . . Formulation of the . . . • Archimedean t-norms have the form Fuzzy Fusion for . . . t & ( a, b ) = f − 1 ( f ( a ) · f ( b )) . Resulting . . . A Similar Problem Is . . . • It is known that a general t-norm can be obtained: Relation between M- . . . – by setting Archimedean t-norms on several (maybe Resulting Solution: . . . infinitely many) subintervals of the interval [0 , 1], From OWA to Fuzzy Title Page – by using min( a, b ) as the value of t & ( a, b ) for the cases when a and b are not in the same interval. ◭◭ ◮◮ • Conclusion: for every t-norm and for every ε > 0, there ◭ ◮ exists an ε -close Archimedean t-norm. Page 5 of 15 • Idea of the proof: replace min with a close Archimedean Go Back t-norm, e.g., with ( a − p + b − p ) − 1 /p for a large p . Full Screen • So, from the practical viewpoint, we can always safely Close assume that the t-norm is Archimedean. Quit
Single-Quantity Data . . . 5. Fuzzy Fusion for Archimedean t-Norms Limitation of Fuzzy . . . Formulation of the . . . • Reminder: we maximize Fuzzy Fusion for . . . d ( x ) = t & ( µ ≈ ( x 1 − x ) , . . . , µ ≈ ( x n − x )) . Resulting . . . A Similar Problem Is . . . • Archimedean t-norm: t & ( a, b ) = f − 1 ( f ( a ) · f ( b )), so Relation between M- . . . d ( x ) = f − 1 ( f ( µ ≈ ( x 1 − x )) · . . . · f ( µ ≈ ( x n − x ))) . Resulting Solution: . . . From OWA to Fuzzy def • Fact: d ( x ) → max ⇔ D ( x ) = f ( d ( x )) → max, where Title Page ◭◭ ◮◮ D ( x ) = f ( µ ≈ ( x 1 − x )) · . . . · f ( µ ≈ ( x n − x )) . ◭ ◮ • Alternative description: Page 6 of 15 n � D ( x ) = ρ ( x i − x ) , Go Back i =1 Full Screen def where ρ ( x ) = f ( µ ≈ ( x )) . Close Quit
Single-Quantity Data . . . 6. Resulting Reformulation of the Problem Limitation of Fuzzy . . . Formulation of the . . . • We have two ways to fuse estimates x 1 , . . . , x n into a Fuzzy Fusion for . . . single estimate x : Resulting . . . n – find x for which the value � ρ ( x i − x ) is the largest A Similar Problem Is . . . i =1 Relation between M- . . . possible ( fuzzy approach ), and Resulting Solution: . . . n � – find x as w i · x ( i ) ( OWA approach ). From OWA to Fuzzy i =1 Title Page • The problem is: ◭◭ ◮◮ – given ρ ( x ), find w i for which the OWA estimate is ◭ ◮ close to the original fuzzy estimate; and Page 7 of 15 – given w i , find ρ ( x ) for which the fuzzy estimate is close to the original OWA estimate. Go Back Full Screen Close Quit
Single-Quantity Data . . . 7. A Similar Problem Is Already Solved In Robust Limitation of Fuzzy . . . Statistics Formulation of the . . . Fuzzy Fusion for . . . • Robust statistics: making estimates under partial in- Resulting . . . formation about the probability distribution f ( x ). A Similar Problem Is . . . • Typical techniques: use statistical techniques correspond- Relation between M- . . . ing to some pdf f 0 ( x ). Resulting Solution: . . . • M-methods: Max Likelihood From OWA to Fuzzy n Title Page � f 0 ( x i − a ) → max . a ◭◭ ◮◮ i =1 � i n ◭ ◮ • L-estimates: a L = 1 � � · x ( i ) for some m ( p ). m n · Page 8 of 15 n i =1 Go Back • Observation: these are exactly our formulas for fuzzy and OWA estimates, with Full Screen � i ρ ( x ) = f 0 ( x ) and w i = 1 � Close n · m . n Quit
Single-Quantity Data . . . 8. Relation between M-methods and L-Estimates Limitation of Fuzzy . . . Formulation of the . . . • Reminder: we have estimates: Fuzzy Fusion for . . . n � • a m s.t. f 0 ( x i − a M ) → max a , and Resulting . . . i =1 � i A Similar Problem Is . . . n • a L = 1 � � m · x ( i ) . n · Relation between M- . . . n i =1 Resulting Solution: . . . • Fact: in robust statistics, it is known how, given f 0 ( x ), From OWA to Fuzzy to find m ( p ) for which a M and a L are asympt. close: Title Page – we compute the cumulative distribution function ◭◭ ◮◮ � x F 0 ( x ) as F 0 ( x ) = −∞ f 0 ( t ) dt ; ◭ ◮ – we find the auxiliary function M ( p ) = z ( F − 1 0 ( p )), Page 9 of 15 def = − (ln( f 0 ( x )) ′′ ; where z ( x ) Go Back M ( p ) – we normalize m ( p ) = . � 1 Full Screen 0 M ( q ) dq • Our idea: use this relation to compare fuzzy and OWA Close estimates. Quit
Single-Quantity Data . . . 9. M-methods vs. L-Estimates: Example Limitation of Fuzzy . . . Formulation of the . . . • Reminder: Fuzzy Fusion for . . . � x – we compute cdf F 0 ( x ) = −∞ f 0 ( t ) dt ; Resulting . . . def – we find M ( p ) = z ( F − 1 = − (ln( f 0 ( x )) ′′ ; 0 ( p )), where z ( x ) A Similar Problem Is . . . M ( p ) Relation between M- . . . – we compute m ( p ) = . � 1 0 M ( q ) dq Resulting Solution: . . . From OWA to Fuzzy � − 1 � 2 · x 2 • The Gaussian function f 0 ( x ) = exp is pro- Title Page portional to the pdf of the normal distribution. ◭◭ ◮◮ � x • Hence, F 0 ( x ) = −∞ f 0 ( t ) dt is proportional to the cdf ◭ ◮ of a normal distribution. Page 10 of 15 • Here, ln( f 0 ( x )) = − 1 2 · x 2 , hence Go Back z ( x ) = − ln( f 0 ( x )) ′′ = 1 . Full Screen • So, M ( p ) = z ( F − 1 0 ( p )) = 1; the integral of M ( p ) = 1 Close over the interval [0 , 1] is 1, hence m ( p ) = M ( p ) = 1 . Quit
Single-Quantity Data . . . 10. Relation Between Fuzzy and OWA Estimates: Our Limitation of Fuzzy . . . Main Idea Formulation of the . . . Fuzzy Fusion for . . . • We have seen that, mathematically, Resulting . . . – M-estimates correspond to fuzzy estimates, and A Similar Problem Is . . . – L-estimates correspond to OWA estimates. Relation between M- . . . Resulting Solution: . . . • We can therefore From OWA to Fuzzy – use the solution provided by robust statistics Title Page – to find the desired correspondence between the util- ◭◭ ◮◮ ity function and the spectral risk measures. ◭ ◮ Page 11 of 15 Go Back Full Screen Close Quit
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