Which Value x Analysis of the Problem Towards an Algorithm Best - PowerPoint PPT Presentation

Need to Combine . . . Utility-Based Approach Case of Interval . . . Which Value � x Analysis of the Problem Towards an Algorithm Best Represents Resulting Algorithm What Is the . . . a Sample x 1 , . . . , x n : Acknowledgments Utility-Based Approach Home Page Title Page Under Interval Uncertainty ◭◭ ◮◮ Andrzej Pownuk and Vladik Kreinovich ◭ ◮ Page 1 of 15 Computational Science Program University of Texas at El Paso Go Back 500 W. University El Paso, Texas 79968, USA Full Screen ampownuk@utep.edu, vladik@utep.edu Close Quit

Need to Combine . . . 1. Need to Combine Several Estimates Utility-Based Approach Case of Interval . . . • In many practical situations, we have several estimates Analysis of the Problem x 1 , . . . , x n of the same quantity x . Towards an Algorithm • In such situations, it is often desirable to combine this Resulting Algorithm information into a single estimate � x . What Is the . . . Acknowledgments • Sometimes, we know the probability distribution of the Home Page corresponding estimation errors x i − x . Title Page • Then, we can use known statistical techniques to find � x . ◭◭ ◮◮ • E.g., we can use the Maximum Likelihood Method. ◭ ◮ • In many cases, however, we do not have any informa- Page 2 of 15 tion about the corresponding probability distribution. Go Back • How can we then find � x ? Full Screen Close Quit

Need to Combine . . . 2. Utility-Based Approach Utility-Based Approach Case of Interval . . . • According to the decision theory, decisions of a rational Analysis of the Problem person are ⇔ maximizing utility value u . Towards an Algorithm • Let us thus find the estimate � x for which the utility Resulting Algorithm u ( � x ) is the largest. What Is the . . . • We use a single value � x instead of all n values x i ; for Acknowledgments Home Page each i : Title Page – if the actual estimate is x i and we use a different value � x � = x i instead, ◭◭ ◮◮ – then we are not doing an optimal thing. ◭ ◮ • For example: Page 3 of 15 – if the optimal speed at which the car needs the least Go Back amount of fuel is x i , Full Screen – and we instead run it at a speed � x � = x i , we thus Close waste some fuel. Quit

Need to Combine . . . 3. Utility-Based Approach (cont-d) Utility-Based Approach Case of Interval . . . • For each i , the disutility d comes from the fact that the Analysis of the Problem difference � x − x i is different from 0. Towards an Algorithm • There is no disutility if we use the actual value, so Resulting Algorithm d = d ( � x − x i ) for some function d ( y ). What Is the . . . • Here d (0) = 0 and d ( y ) > 0 for y � = 0. Acknowledgments Home Page • The estimates are usually reasonably accurate, so the difference x i − � x is small. Title Page • So, we can expand the function d ( y ) in Taylor series ◭◭ ◮◮ and keep only the first few terms in this expansion: ◭ ◮ d ( y ) = d 0 + d 1 · y + d 2 · y 2 + . . . Page 4 of 15 • From d (0) = 0 we conclude that d 0 = 0. Go Back • From d ( y ) > 0 for y � = 0 we conclude that d 1 = 0 (else Full Screen we would have d ( y ) < 0 for small y ) and d 2 > 0, so Close d ( y ) = d 2 · y 2 = d 2 · ( � x − x i ) 2 . Quit

Need to Combine . . . 4. Utility-Based Approach (final) Utility-Based Approach Case of Interval . . . • The overall disutility d ( � x ) of using � x instead of each of Analysis of the Problem the values x 1 , . . . , x n can be computed as the sum: Towards an Algorithm n n � � Resulting Algorithm x − x i ) 2 = d 2 · x − x i ) 2 . d ( � x ) = d ( � ( � What Is the . . . i =1 i =1 Acknowledgments def Home Page • u ( � x ) = − d ( � x ) → max ⇔ d ( � x ) → min. Title Page • Since d 2 > 0, minimizing disutility is equivalent to min- imizing re-scaled disutility: ◭◭ ◮◮ n � ◭ ◮ = d ( � x ) def x − x i ) 2 . D ( � x ) = ( � d 2 Page 5 of 15 i =1 Go Back • Equating the derivative to 0, we get the well-known � n Full Screen x = 1 sample mean: � n · x i . Close i =1 Quit

Need to Combine . . . 5. Case of Interval Uncertainty Utility-Based Approach Case of Interval . . . • In many practical situations, we only know the inter- Analysis of the Problem vals [ x i , x i ] that contain the unknown values x i . Towards an Algorithm • For different x i ∈ [ x i , x i ], we get, in general, different Resulting Algorithm values of utility What Is the . . . U ( � x, x 1 , . . . , x n ) = − D ( � x, x 1 , . . . , x n ) , where Acknowledgments n Home Page � x − x i ) 2 . D ( � x, x 1 , . . . , x n ) = ( � Title Page i =1 ◭◭ ◮◮ • Thus, all we know is that the actual (unknown) value of the utility belongs to the interval ◭ ◮ [ U ( � x ) , U ( � x )] = [ − D ( � x ) , − D ( � x )] , where Page 6 of 15 D ( � x ) = min D ( � x, x 1 , . . . , x n ) , D ( � x ) = max D ( � x, x 1 , . . . , x n ) . Go Back • In such situations, decision theory recommends using Full Screen Hurwicz optimism-pessimism criterion, i.e., maximize: Close def U ( � x ) = α · U ( � x ) + (1 − α ) · U ( � x ) . Quit

Need to Combine . . . 6. Case of Interval Uncertainty (cont-d) Utility-Based Approach Case of Interval . . . • According to Hurwicz criterion, we maximize: Analysis of the Problem def Towards an Algorithm U ( � x ) = α · U ( � x ) + (1 − α ) · U ( � x ) . Resulting Algorithm • The parameter α ∈ [0 , 1] describes the decision maker’s What Is the . . . degree of optimism. Acknowledgments Home Page • For U = − D , this is equivalent to minimizing the ex- pression Title Page ◭◭ ◮◮ D ( � x ) = − U ( � x ) = α · D ( � x ) + (1 − α ) · D ( � x ) . ◭ ◮ • In this paper, we describe an efficient algorithm for Page 7 of 15 computing such � x . Go Back Full Screen Close Quit

Need to Combine . . . 7. Analysis of the Problem Utility-Based Approach Case of Interval . . . x − x i ) 2 in the sum D ( � • Each term ( � x, x 1 , . . . , x n ) depends Analysis of the Problem only on its own variable x i . Thus, with respect to x i : Towards an Algorithm – the sum is the smallest when each term is min, and Resulting Algorithm – the sum is the largest when each term is the largest. What Is the . . . x − x i ) 2 is attained: Acknowledgments • When x i ∈ [ x i , x i ], the max of ( � Home Page = x i + x i def – at x i = x i when � x ≥ � x i and Title Page 2 – at x i = x i when � x i . x < � ◭◭ ◮◮ x ) = � x − x i ) 2 + � x − x i ) 2 . • Thus, D ( � ( � ( � ◭ ◮ i : � x< � i : � x ≥ � x i x i Page 8 of 15 x − x i ) 2 is attained: • Similarly, the minimum of the term ( � Go Back – for x i = � x when � x ∈ [ x i , x i ] (in this case, min = 0); Full Screen – for x i = x i when � x < x i ; and Close – for x i = x i when � x > x i . Quit

Need to Combine . . . 8. Analysis of the Problem (cont-d) Utility-Based Approach x ) = � x − x i ) 2 + � Case of Interval . . . x − x i ) 2 . • Thus, D ( � ( � ( � Analysis of the Problem i : � x>x i i : � x<x i Towards an Algorithm • So, for D ( � x ) = α · D ( � x ) + (1 − α ) · D ( � x ), we get Resulting Algorithm � � x − x i ) 2 + α · x − x i ) 2 + What Is the . . . D ( � x ) = α · ( � ( � Acknowledgments i : � i : � x>x i x<x i � � Home Page x − x i ) 2 + (1 − α ) · x − x i ) 2 . (1 − α ) · ( � ( � Title Page x ≥ � i : � x< � x i i : � x i ◭◭ ◮◮ ◭ ◮ Page 9 of 15 Go Back Full Screen Close Quit

Need to Combine . . . 9. Towards an Algorithm Utility-Based Approach Case of Interval . . . • The terms depends on the relation between � x and the Analysis of the Problem values Towards an Algorithm x i , x i , and � x i . Resulting Algorithm • Let us sort these 3 n values into a sequence What Is the . . . Acknowledgments s 1 ≤ s 2 ≤ . . . ≤ s 3 n . Home Page • Then on each interval [ s j , s j +1 ], the function D ( � x ) is Title Page simply a quadratic function of � x . ◭◭ ◮◮ • A quadratic function attains min on an interval: ◭ ◮ – either at one of its midpoints, Page 10 of 15 – or at a point when the derivative is equal to 0 (if Go Back this point is inside the given interval). Full Screen Close Quit