Bilinear Models For System Dynamics, . . . The Fact that We Are . . - PowerPoint PPT Presentation

System Approach: In . . . Systems Approach in . . . Approximations of . . . First and Second . . . Bilinear Models For System Dynamics, . . . The Fact that We Are . . . from System Approach Explanation of Bilinear . . . Discussion: A Similar . . . Justified for Classification, Auxiliary Question: . . . Auxiliary Question: . . . Acknowledgments with Potential Applications Title Page to Bioinformatics ◭◭ ◮◮ o 1 , Fran¸ cois Modave 2 Richard Al´ ◭ ◮ Vladik Kreinovich 2 , David Herrera 2 , Xiaojing Wang 2 Page 1 of 12 1 Center for Computational Sciences & Advanced Distributed Simulation, Go Back University of Houston-Downtown, One Main Street, Houston, TX 77002, USA, RAlo@uh.edu Full Screen 2 Department of Computer Science, University of Texas at El Paso, Close El Paso, TX 79968, USA, vladik@utep.edu Quit

System Approach: In . . . Systems Approach in . . . 1. System Approach: In Brief Approximations of . . . First and Second . . . • Problem: often, we do not know the exact dynamics For System Dynamics, . . . x i = f i ( x 1 , . . . , x n ) . ˙ The Fact that We Are . . . • Solution: build the expressions for f i based on common Explanation of Bilinear . . . sense (von Bertalanffy, Forrester, Meadows). Discussion: A Similar . . . • Case 1: increase in x i slows down the growth of x j . Auxiliary Question: . . . Auxiliary Question: . . . • Suggestion: f j includes a term − k · x i with k > 0. Acknowledgments • Case 2: x j and x k , when combined, enhance the growth Title Page of x i . ◭◭ ◮◮ • Suggestion: f i includes + k · x j · x k . ◭ ◮ • Comment: values of k are determined empirically. Page 2 of 12 • Common sense rarely goes beyond simple interaction, Go Back so we have bilinear f i . Full Screen • Successes: good qualitative predictions. Close Quit

System Approach: In . . . Systems Approach in . . . 2. Systems Approach in Classification Approximations of . . . First and Second . . . • Examples: For System Dynamics, . . . – separate stocks with a good growth potential from The Fact that We Are . . . the risky ones, or Explanation of Bilinear . . . – separate cancerous cells from the normal ones. Discussion: A Similar . . . Auxiliary Question: . . . • Idea: use a discrimination function f ( x 1 , . . . , x n ): Auxiliary Question: . . . – objects with f > 0 belong to the first class, and Acknowledgments – objects with f < 0 belong to the second class. Title Page • Common sense: leads to bilinear f . ◭◭ ◮◮ • Unexpected phenomenon: ◭ ◮ Page 3 of 12 – for system dynamics, we had qualitative predictions; – for classification, we have good quantitative fit. Go Back Full Screen • Our objective: explain this mystery. Close Quit

System Approach: In . . . Systems Approach in . . . 3. Approximations of Different Order of Accuracy: Approximations of . . . General Idea First and Second . . . For System Dynamics, . . . • Assumption: the actual (unknown) discrimination func- The Fact that We Are . . . tion f ( x 1 , . . . , x n ) is smooth. Explanation of Bilinear . . . • Conclusion: in the neighborhood U of a point � x = Discussion: A Similar . . . ( � x 1 , . . . , � x n ), we keep only lower order Taylor terms. Auxiliary Question: . . . • Different points � x : Auxiliary Question: . . . – if � x is in class 1, then U is in class 1; Acknowledgments Title Page – if � x is in class 2, then U is in class 2; ◭◭ ◮◮ – conclusion: the problem is interesting only when � x is on the border, i.e., f ( � x n ) = 0. x 1 , . . . , � ◭ ◮ • Simplification: Page 4 of 12 – Idea: take new coordinates x i → x i − � x i in which Go Back the starting point is (0 , . . . , 0). Full Screen – Result: starting point is 0, and f (0 , . . . , 0) = 0. Close Quit

System Approach: In . . . Systems Approach in . . . 4. First and Second Approximations Approximations of . . . First and Second . . . • General idea: f (0 , . . . , 0) = 0. For System Dynamics, . . . • First approximation: linear discrimination function The Fact that We Are . . . n � Explanation of Bilinear . . . f ( x 1 , . . . , x n ) = a i · x i . Discussion: A Similar . . . i =1 Auxiliary Question: . . . • Second approximation: quadratic function Auxiliary Question: . . . n n n � � � Acknowledgments f ( x 1 , . . . , x n ) = a i · x i + a ij · x i · x j . Title Page i =1 i =1 j =1 ◭◭ ◮◮ • Beyond bilinear: this general expression: ◭ ◮ – has linear terms a i · x i ; Page 5 of 12 – has bilinear terms a ij · x i · x j for i � = j ; Go Back – also has purely quadratic (not bilinear) terms a ii · x 2 i Full Screen (corresponding to i = j ). Close Quit

System Approach: In . . . Systems Approach in . . . 5. For System Dynamics, Bilinear Functions Provide Approximations of . . . a Rather Crude Approximation First and Second . . . For System Dynamics, . . . • Fact: in linear approximation, we ignore quadratic The Fact that We Are . . . (and higher order) terms. Explanation of Bilinear . . . • Accuracy of linear approximation: quadratic in x i . Discussion: A Similar . . . • Accuracy of quadratic approximation: cubic in x i . Auxiliary Question: . . . Auxiliary Question: . . . • Accuracy of bilinear approximation: Acknowledgments – fact: we ignore quadratic terms a ii · x 2 i ; Title Page – conclusion: accuracy is quadratic in x i . ◭◭ ◮◮ • Bad news: same asymptotic as linear. ◭ ◮ • Good news: Page 6 of 12 – linear approximation: we ignore n 2 terms a ij · x i · x j , Go Back so accuracy is n 2 · δ ; Full Screen – bilinear approximation: we ignore n terms a ii · x 2 i , so accuracy is n · δ ≪ n 2 · δ . Close Quit

System Approach: In . . . Systems Approach in . . . 6. The Fact that We Are Interested in Classification Approximations of . . . Applications Allows Further Simplifications First and Second . . . For System Dynamics, . . . • In dynamics applications: the function f ( x 1 , . . . , x n ) The Fact that We Are . . . can be determined from observations. Explanation of Bilinear . . . • In the classification applications: we only observe the Discussion: A Similar . . . signs of f ( x 1 , . . . , x n ). Auxiliary Question: . . . a new function f ′ ( x 1 , . . . , x n ) with the • Conclusion: Auxiliary Question: . . . same signs leads to the same classification: Acknowledgments Title Page f ( x 1 , . . . , x n ) > 0 if and only if f ′ ( x 1 , . . . , x n ) > 0; ◭◭ ◮◮ f ( x 1 , . . . , x n ) > 0 if and only if f ′ ( x 1 , . . . , x n ) > 0 . ◭ ◮ • What we plan to do: we prove that Page 7 of 12 – for every quadratic function f ( x 1 , . . . , x n ), Go Back – there is a bilinear function f ′ ( x 1 , . . . , x n ) with the Full Screen same signs. Close Quit

System Approach: In . . . Systems Approach in . . . 7. Explanation of Bilinear Functions Approximations of . . . � � First and Second . . . � n • Idea: f ′ ( x 1 , . . . , x n ) = f ( x 1 , . . . , x n ) · 1 + b j · x j . For System Dynamics, . . . j =1 The Fact that We Are . . . � � � � � n Explanation of Bilinear . . . � � • Good news: for x ≈ 0, b j · x j � ≪ 1, hence f ( x 1 , . . . , x n ) � � � Discussion: A Similar . . . i = j and f ′ ( x 1 , . . . , x n ) have the same sign. Auxiliary Question: . . . Auxiliary Question: . . . � n � n � n • General case: f ( x 1 , . . . , x n ) = a i · x i + a ij · x i · x j , Acknowledgments i =1 i =1 j =1 Title Page hence � n � � n � ◭◭ ◮◮ � � f ′ ( x 1 , . . . , x n ) = f ( x 1 , . . . , x n )+ a i · x i · b j · x j . ◭ ◮ i =1 j =1 Page 8 of 12 • Conclusion: for b i = − a ii , we have a bilinear f ′ . a i Go Back • Comment: this is only possible in the generic case, Full Screen when a i � = 0 for all i . Close Quit

System Approach: In . . . Systems Approach in . . . 8. Discussion: A Similar Simplification Is Not Always Approximations of . . . Possible for Higher Order Models First and Second . . . For System Dynamics, . . . • We proved: quadratic f can be reduced to bilinear f ′ . The Fact that We Are . . . • Natural question: can we reduce cubic f to trilinear Explanation of Bilinear . . . � � � f ′ = a i · x i + a ij · x i · x j + a ijk · x i · x j · x k ? Discussion: A Similar . . . Auxiliary Question: . . . i i,j i,j,k Auxiliary Question: . . . • Answer: no, even for n = 3: Acknowledgments – to describe a general trilinear function, we need 7 Title Page parameters a 1 , a 2 , a 3 , a 12 , a 13 , a 23 , and a 123 ; ◭◭ ◮◮ – a general cubic f can be described by a discrimi- ◭ ◮ nating curve x 3 = F ( x 1 , x 2 ), where a general cubic Page 9 of 12 F ( x 1 , x 2 ) = b 1 · x 1 + b 2 · x 2 + b 11 · x 2 1 + b 12 · x 1 · x 2 + b 22 · x 2 2 + Go Back b 111 · x 3 1 + b 112 · x 2 1 · x 2 + b 122 · x 1 · x 2 2 + b 222 · x 3 2 Full Screen requires 9 > 7 parameters. Close Quit