Growth Rates What Is Known New Algorithm under Interval - PowerPoint PPT Presentation

General Systems: . . . The Notion of a . . . Growth under Interval . . . Growth Rates What Is Known New Algorithm under Interval Uncertainty Justification of the . . . Case of Fuzzy Uncertainty Acknowledgments Janos Hajagos 1 and Vladik Kreinovich 2 Title Page 1 Applied Biomathematics, 100 North Country Road ◭◭ ◮◮ Setauket, New York 11733, USA logistic@sdf-eu.org ◭ ◮ Page 1 of 9 2 Department of Computer Science, University of Texas at El Paso, Go Back El Paso, TX 79968, USA vladik@utep.edu Full Screen http://www.cs.utep.edu/vladik Close Quit

General Systems: . . . 1. General Systems: Linear Approximation The Notion of a . . . • State: is described by the values of the parameters Growth under Interval . . . x 1 , . . . , x n . What Is Known New Algorithm • Dynamic for continuous time: ˙ x i = f i ( x 1 , . . . , x n ). Justification of the . . . • Dynamic for discrete time: x i ( t +1) = f i ( x 1 ( t ) , . . . , x n ( t )). Case of Fuzzy Uncertainty • Linearization: f i ( x 1 , . . . , x n ) is smooth, so in a small Acknowledgments � n region, f i ( x 1 , . . . , x n ) ≈ b i + a ij · x j . Title Page j =1 ◭◭ ◮◮ • Further simplification: an appropriate shift x i → x i − s i � n ◭ ◮ leads to f i ( x 1 , . . . , x n ) = a ij · x j . j =1 Page 2 of 9 � n Go Back • Resulting dynamic: ˙ x i = a ij · x j ; j =1 Full Screen n � Close x i ( t + 1) = a ij · x j ( t ) . Quit j =1

General Systems: . . . 2. The Notion of a Growth Rate The Notion of a . . . • New variables: coefficients y i in the expansion of x i ( t ) Growth under Interval . . . in eigenvectors of A = ( a ij ). What Is Known New Algorithm • Generic case (all eigenvalues different): Justification of the . . . y i = λ i · y i , or y i ( t + 1) = λ i · y i ( t ) . ˙ Case of Fuzzy Uncertainty • General solution: a linear combination of Acknowledgments y i ( t ) = y i (0) · exp( λ i · t ) or y i ( t ) = y i (0) · λ t Title Page i . ◭◭ ◮◮ • Asymptotically: x i ( t ) ∼ exp( λ · t ) or x i ( t ) ∼ λ t , where λ – growth rate – is the largest of the eigenvalues. ◭ ◮ • Applications: Page 3 of 9 – population growth, Go Back – growth in animals and plants, Full Screen – spread of an epidemic, Close – financial growth. Quit

General Systems: . . . 3. Growth under Interval Uncertainty The Notion of a . . . • Idealized case – exactly known coefficients: find the Growth under Interval . . . eigenvalues and find the largest of them. What Is Known New Algorithm • In practice: a ij are only known with uncertainty. Justification of the . . . • Case of interval uncertainty: we know Case of Fuzzy Uncertainty – approximate values � a ij , and Acknowledgments – upper bounds ∆ ij on the approx. error Title Page | � a ij − a ij | . ◭◭ ◮◮ def ◭ ◮ • Hence a ij ∈ a ij = [ a ij , a ij ] = [ � a ij − ∆ ij , � a ij + ∆ ij ] . Page 4 of 9 • Problem: find the interval [ λ, λ ] of possible values of λ when a ij ∈ a ij . Go Back • Alternative: at least find an interval I ⊇ [ λ, λ ]. Full Screen • Comment: upper bound λ is of special interest: it de- Close termines, e.g., how fast a disease can spread. Quit

General Systems: . . . 4. What Is Known The Notion of a . . . def • Case of small uncertainty: ∆ a ij = � a ij − a ij ≪ � a ij . Growth under Interval . . . What Is Known • Solution: linearize the dependence λ k ( a ij ), and get a New Algorithm range for λ . Justification of the . . . • General case: the problem is computationally intractable Case of Fuzzy Uncertainty (NP-hard). Acknowledgments • Important case: non-negative matrix a ij ≥ 0. Title Page • Examples: ◭◭ ◮◮ – population growth, ◭ ◮ – spread of disease, Page 5 of 9 – financial situations. Go Back • What we propose: a new efficient algorithm that com- Full Screen putes λ for non-negative matrices. Close Quit

General Systems: . . . 5. New Algorithm The Notion of a . . . • We use: a known algorithm A that computing λ ( A ) Growth under Interval . . . for exact matrices A . What Is Known New Algorithm • Input: interval-valued matrix A = � a ij � = � [ a ij , a ij ] � . Justification of the . . . • Algorithm: Case of Fuzzy Uncertainty – First, we apply A to A = � a ij � ; the resulting value Acknowledgments is returned as λ . Title Page – Then, we apply A to A = � a ij � ; the resulting value ◭◭ ◮◮ is returned as λ . ◭ ◮ • Comment: this idea only works for non-negative ma- Page 6 of 9 trices. Go Back Full Screen Close Quit

General Systems: . . . 6. Justification of the New Algorithm The Notion of a . . . • Notation: A ≤ B means a ij ≤ b ij for all i, j . Growth under Interval . . . What Is Known • Notation: x ≥ 0 means x i ≥ 0 for all i . � New Algorithm � n � Ax � 2 def x 2 • Known fact: λ ( A ) = max , where � x � 2 = Justification of the . . . i � x � 2 x � =0 i =1 Case of Fuzzy Uncertainty is the length of x . Acknowledgments • Perron-Frobenius Theorem: for A ≥ 0, one of the Title Page eigenvectors x corresponding to the largest λ ( A ) is non- ◭◭ ◮◮ negative: x ≥ 0. ◭ ◮ � Ax � 2 • Conclusion: λ ( A ) = max . � x � 2 Page 7 of 9 x ≥ 0 & x � =0 • If 0 ≤ A ≤ B and x ≥ 0, then 0 ≤ Ax ≤ Bx hence Go Back � Ax � 2 ≤ � Bx � 2 , � Ax � 2 ≤ � Bx � 2 , and λ ( A ) ≤ λ ( B ). Full Screen � x � 2 � x � 2 Close • Interval matrix: for all A ∈ [ A, A ], we have Quit A ≤ A ≤ A , hence λ ( A ) ≤ λ ( A ) ≤ λ ( A ) .

General Systems: . . . 7. Case of Fuzzy Uncertainty The Notion of a . . . • Situation: often, we have uncertain expert estimates. Growth under Interval . . . What Is Known • Fuzzy set description: for each x , describe the degree New Algorithm µ ( x ) ∈ [0 , 1] to which this value is possible. Justification of the . . . • Reducing fuzzy uncertainty to interval uncertainty: Case of Fuzzy Uncertainty – for each degree of certainty α , values possible with Acknowledgments uncertainty ≥ α form an interval ( α -cut) Title Page x ( α ) = { x | µ ( x ) ≥ α } ; ◭◭ ◮◮ – if we know α -cuts for all α , we can reconstruct µ ( x ). ◭ ◮ • Computation under fuzzy uncertainty: for each level α , Page 8 of 9 we apply the interval algorithm to the α -cuts. Go Back • Case of growth rate: for α = 0 , 0 . 1 , . . . , 1, we apply our Full Screen algorithm to α -cuts a ij ( α ). Close • Result: intervals [ λ, λ ]( α ) form the fuzzy set for λ . Quit

8. Acknowledgments General Systems: . . . The Notion of a . . . This work was supported in part by: Growth under Interval . . . What Is Known • NASA under cooperative agreement NCC5-209, New Algorithm Justification of the . . . • NSF grants EAR-0225670 and DMS-0532645, Case of Fuzzy Uncertainty Acknowledgments • Star Award from the University of Texas System, and • Texas Department of Transportation grant No. 0-5453. Title Page The authors are thankful to Arnold Neumaier for his valu- ◭◭ ◮◮ able advise. ◭ ◮ Page 9 of 9 Go Back Full Screen Close Quit