Data Fusion: Main . . . Data Fusion for Fuzzy . . . Problem: Intersection . . . Towards Improved Usual Solution to the . . . How to Improve the . . . Trapezoidal Approximation Least Squares Method to Intersection (Fusion) of Problem with the New . . . Properties of the . . . Trapezoidal Fuzzy Numbers: General Non- . . . Title Page Specific Procedure and ◭◭ ◮◮ General Non-Associativity ◭ ◮ Theorem Page 1 of 17 Go Back Gang Xiang 1 and Vladik Kreinovich 2 Full Screen 1 Philips Healthcare, El Paso, TX 79902, USA Close 2 University of Texas, El Paso, TX 79968, USA, vladik@utep.edu Quit
Data Fusion: Main . . . 1. Data Fusion: Main Idea and Case of Interval Un- Data Fusion for Fuzzy . . . certainty Problem: Intersection . . . Usual Solution to the . . . • Situation: often, we have several pieces of information How to Improve the . . . about the same quantity x . Least Squares Method • Data fusion: we combine (“fuse”) this information. Problem with the New . . . • Frequent case: we have estimates � x i with upper bounds Properties of the . . . ∆ i on their uncertainty: | � x i − x | ≤ ∆ i . General Non- . . . Title Page def • Interval uncertainty: x i ∈ X i = [ x i , x i ], where ◭◭ ◮◮ x i = � x i − ∆ i and x i = � x i + ∆ i . ◭ ◮ • Fusion of intervals: if we know that x ∈ X 1 , . . . , and x ∈ X n , then x ∈ X = [ x, x ] = X 1 ∩ . . . ∩ X n , where: Page 2 of 17 x = max( x 1 , . . . , x n ) and x = min( x 1 , . . . , x n ) . Go Back Full Screen • Natural properties: X 1 ∩ X 2 = X 2 ∩ X 1 and associativity Close X 1 ∩ ( X 2 ∩ X 3 ) = ( X 1 ∩ X 2 ) ∩ X 3 . Quit
Data Fusion: Main . . . 2. Data Fusion for Fuzzy Numbers Data Fusion for Fuzzy . . . Problem: Intersection . . . • Frequent situation: we have several piece of imprecise Usual Solution to the . . . (“fuzzy”) expert knowledge X i about a quantity x . How to Improve the . . . • Natural description: in terms of fuzzy numbers µ X i ( x ). Least Squares Method • Natural fusion operation: intersection Problem with the New . . . Properties of the . . . X = X 1 ∩ . . . ∩ X n , General Non- . . . defined as µ X ( x ) = min( µ X 1 ( x ) , . . . , µ X n ( x )) . Title Page • Towards an algorithm: a fuzzy set X can be described ◭◭ ◮◮ def by its α -cuts X ( α ) = { x : µ ( x ) ≥ α } . ◭ ◮ • Useful result: the α -cut of the intersection X is equal Page 3 of 17 to the intersection of α -cuts: Go Back X ( α ) = X 1 ( α ) ∩ . . . ∩ X n ( α ) . Full Screen • Resulting algorithm: use the above formula to find the Close α -cut of the intersection for α = 0 , 0 . 1 , . . . , 1 . 0. Quit
Data Fusion: Main . . . 3. Computer Representation of Fuzzy numbers: Data Fusion for Fuzzy . . . Trapezoidal Numbers Problem: Intersection . . . Usual Solution to the . . . • Ideally: to adequately describe a fuzzy number, we How to Improve the . . . need to store α -cuts for all α ∈ [0 , 1]. Least Squares Method • In practice: we can only store finitely many α -cuts. Problem with the New . . . • Usually: we store the lower ( α = 0) and upper ( α = 1) Properties of the . . . α -cuts and use linear interpolation. General Non- . . . Title Page ✻ ◭◭ ◮◮ 1 � ❅ � ❅ ◭ ◮ � ❅ � ❅ Page 4 of 17 � ❅ ✲ a a 0 Go Back Full Screen • Resulting (approximate) fuzzy numbers are called Close trapezoidal . Quit
Data Fusion: Main . . . 4. Intersection of Two Trapezoidal Fuzzy Numbers Data Fusion for Fuzzy . . . Problem: Intersection . . . Desirable: find the intersection µ i ( x ) of the trapezoidal Usual Solution to the . . . membership functions How to Improve the . . . ✻ Least Squares Method ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ Problem with the New . . . Properties of the . . . ✲ General Non- . . . a a Title Page ◭◭ ◮◮ and ◭ ◮ ✻ Page 5 of 17 � � Go Back � � Full Screen � ✲ b b Close Quit
Data Fusion: Main . . . 5. Problem: Intersection of Two Trapezoidal Fuzzy Data Fusion for Fuzzy . . . Numbers Is, In General, Not Trapezoidal Problem: Intersection . . . Usual Solution to the . . . • We are interested in: intersection (fusion) of member- How to Improve the . . . ship functions. Least Squares Method • We limit ourselves to: trapezoidal membership func- Problem with the New . . . tions. Properties of the . . . • Problem: the intersection of two trapezoidal member- General Non- . . . ship functions, in general, has the non-trapezoidal form: Title Page ◭◭ ◮◮ ✻ ◭ ◮ ✏ ✏✏✏✏✏✏✏ � Page 6 of 17 ✏ ✏ ✏ ✏ ✏ � � � Go Back � ✲ a a b b Full Screen Close Quit
Data Fusion: Main . . . 6. Usual Solution to the Above Problem Data Fusion for Fuzzy . . . Problem: Intersection . . . • Usual solution to the above problem is simply: Usual Solution to the . . . – take an intersection of lower and alpha α -cuts, and How to Improve the . . . then Least Squares Method – as the result of the fusion, take the trapezoidal Problem with the New . . . number corresponding to this intersection: Properties of the . . . General Non- . . . ✻ Title Page ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✟✟✟✟✟✟✟✟✟✟ ◭◭ ◮◮ � � ◭ ◮ � ✲ a a Page 7 of 17 b b Go Back Full Screen • Main limitation: this approach underestimates the re- Close sulting membership function. Quit
Data Fusion: Main . . . 7. Under-Estimation Explained: Mathematical Com- Data Fusion for Fuzzy . . . ment Problem: Intersection . . . Usual Solution to the . . . • Reminder: trapezoidal membership functions are (lo- How to Improve the . . . cally) linear. Least Squares Method • Reminder: intersection is the minimum of two func- Problem with the New . . . tions. Properties of the . . . • Fact: a minimum µ X ( x ) of two linear functions is al- General Non- . . . ways concave. Title Page • Meaning: the values µ X ( x ) are always above the straight ◭◭ ◮◮ line µ t ( x ) connecting the endpoints: ◭ ◮ ✻ Page 8 of 17 µ t ✏ ✟✟✟✟✟✟✟✟✟✟ ✏✏✏✏✏✏✏ r r r r r r r r r � µ X µ X Go Back r r r r ✏ ✏ ✏ ✏ ✏ � µ t r r r r r r � µ X Full Screen � � ✲ a a b b Close Quit
Data Fusion: Main . . . 8. How to Improve the Traditional Trapezoidal Ap- Data Fusion for Fuzzy . . . proximation to Intersection: A New Procedure Problem: Intersection . . . Usual Solution to the . . . • Traditional approximation: How to Improve the . . . – starts with the α -cuts corresponding to α = 0 and Least Squares Method α = 1, and Problem with the New . . . – uses the linear interpolation to reconstruct other Properties of the . . . α -cuts. General Non- . . . • Problem: for 0 < α < 1, the reconstructed α -cuts are Title Page biased (under-estimated). ◭◭ ◮◮ • Natural idea: ◭ ◮ – First, compute the actual intersection. Page 9 of 17 – Second, approximate the non-0 and non-1 parts of Go Back this intersection by linear functions. Full Screen • Implementation: use the Least Squares Method for this Close approximation. Quit
Data Fusion: Main . . . 9. Least Squares Method Data Fusion for Fuzzy . . . Problem: Intersection . . . • Problem: find � p for which e ( a, � p ) ≈ 0 for all a . Usual Solution to the . . . � � a p ) 2 da = a ( p + q · a − µ i ( a )) 2 da. a e 2 ( a, � • Solution: min How to Improve the . . . • Resulting formulas: Least Squares Method � a p · ( a ) 2 − ( a ) 2 + q · ( a ) 3 − ( a ) 3 Problem with the New . . . = µ i ( a ) · a da. Properties of the . . . 2 3 a General Non- . . . � a p · ( a ) 2 − ( a ) 2 + q · ( a ) 3 − ( a ) 3 Title Page = µ i ( a ) · a da. 2 3 ◭◭ ◮◮ a ◭ ◮ ✻ Page 10 of 17 µ ℓ ✟✟✟✟✟✟✟✟✟✟ ✏ ✏ ✏ ✏ ✏ µ i Go Back � Full Screen � ✲ b a Close Quit
Data Fusion: Main . . . 10. Problem with the New Approximation Procedure: Data Fusion for Fuzzy . . . Non-Associativity Problem: Intersection . . . Usual Solution to the . . . • Situation: we have three different pieces of knowledge How to Improve the . . . X 1 , X 2 , and X 3 . Least Squares Method • Options: we can combine them in two different ways: Problem with the New . . . – we can first combine X 1 and X 2 into X 1 ⊗ X 2 , Properties of the . . . and then combine X 1 ⊗ X 2 with X 3 ; General Non- . . . – we can first combine X 2 and X 3 into X 2 ⊗ X 3 , Title Page and then combine X 1 with X 2 ⊗ X 3 . ◭◭ ◮◮ • Intuitively: we expect the results to be equal: ◭ ◮ ( X 1 ⊗ X 2 ) ⊗ X 3 = X 1 ⊗ ( X 2 ⊗ X 3 ) . Page 11 of 17 • In practice: the Least Squares-based operation ⊗ is Go Back not associative. Full Screen • What we prove in this paper: non-associativity is in- Close evitable. Quit
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