Common Sense Addition Towards Precise . . . Hurwicz Optimism- . . . Common Sense Addition Computing ∆ Computing ∆ Explained by Hurwicz Let Us Apply Hurwicz . . . Optimism-Pessimism Acknowledgments Home Page Criterion Title Page Bibek Aryal, Laxman Bokati, Karla Godinez, ◭◭ ◮◮ Shammir Ibarra, Heyi Liu, Bofei Wang, and Vladik Kreinovich ◭ ◮ University of Texas at El Paso, El Paso, TX 79968, USA Page 1 of 11 baryal@miners.utep.edu, lbokati@miners.utep.edu, kpgodinezma@miners.utep.edu, saibarra@miners.utep.edu Go Back hliu2@miners.utep.edu, bwang2@miners.utep.edu, vladik@utep.edu Full Screen Close Quit
Common Sense Addition 1. Common Sense Addition Towards Precise . . . • Suppose that we have two factors that affect the accu- Hurwicz Optimism- . . . racy of a measuring instrument. Computing ∆ Computing ∆ • One factor leads to errors ± 10%. Let Us Apply Hurwicz . . . • This means that the resulting error component can Acknowledgments take any value from − 10% to +10%. Home Page • The second factor leads to errors of ± 0 . 1%. Title Page • What is the overall error? ◭◭ ◮◮ • From the purely mathematical viewpoint, the largest ◭ ◮ possible error is 10.1%. Page 2 of 11 • However, from the common sense viewpoint, an engi- Go Back neer would say: 10%. Full Screen Close Quit
Common Sense Addition 2. Common Sense Addition (cont-d) Towards Precise . . . Hurwicz Optimism- . . . • A similar common sense addition occurs in other situ- ations as well. Computing ∆ Computing ∆ • For example: Let Us Apply Hurwicz . . . – if we have a car that weight 1 ton = 1000 kg, Acknowledgments – and we place a coke can that weighs 0.35 kg in the Home Page car, Title Page – what will be now the weight of the car? ◭◭ ◮◮ • Mathematics says 1000.35 kg, but common sense clearly ◭ ◮ says: still 1 ton. Page 3 of 11 • How can we explain this common sense addition? Go Back Full Screen Close Quit
Common Sense Addition 3. Towards Precise Formulation of the Problem Towards Precise . . . • We know that the overall measurement error ∆ x is Hurwicz Optimism- . . . equal to ∆ x 1 + ∆ x 2 , where: Computing ∆ Computing ∆ – the value ∆ x 1 can take all possible values from the Let Us Apply Hurwicz . . . interval [ − ∆ 1 , ∆ 1 ], and Acknowledgments – the value ∆ x 2 can take all possible values from the Home Page interval [ − ∆ 2 , ∆ 2 ]. Title Page • What can we say about the largest possible value ∆ of ◭◭ ◮◮ the absolute value | ∆ x | of the sum ∆ x = ∆ x 1 + ∆ x 2 ? ◭ ◮ • Let us describe this problem in precise terms. Page 4 of 11 • For every pair ( x 1 , x 2 ), let π 1 ( x 1 , x 2 ) denote x 1 and let π 2 ( x 1 , x 2 ) stand for x 2 . Go Back Full Screen • Let ∆ 1 > 0 and ∆ 2 > 0 be two numbers. Close • Without losing generality, we can assume ∆ 1 ≥ ∆ 2 . Quit
Common Sense Addition 4. Towards Precise Formulation (cont-d) Towards Precise . . . Hurwicz Optimism- . . . • By S , let us denote the class of all possible sets S ⊆ [ − ∆ 1 , ∆ 1 ] × [ − ∆ 2 , ∆ 2 ] for which Computing ∆ Computing ∆ π 1 ( S ) = [ − ∆ 1 , ∆ 1 ] and π 2 ( S ) = [ − ∆ 2 , ∆ 2 ] . Let Us Apply Hurwicz . . . • We are interested in the value Acknowledgments ∆( S ) = max {| ∆ x 1 + ∆ x 2 | : (∆ x 1 , ∆ 2 ) ∈ S } . Home Page • Here, S is the actual (unknown) set. Title Page • We do not know what is the actual set S , we only know ◭◭ ◮◮ that S ∈ S . ◭ ◮ • For different sets S ∈ S , we may get different ∆( S ). Page 5 of 11 • The only thing we know about ∆( S ) is that Go Back ∆( S ) ∈ [∆ , ∆], where: Full Screen ∆ = min S ∈S ∆( S ) , ∆ = max S ∈S ∆( S ) . Close • Which value ∆ from this interval should we choose? Quit
Common Sense Addition 5. Hurwicz Optimism-Pessimism Criterion Towards Precise . . . • Often, we do not know the value of a quantity, we only Hurwicz Optimism- . . . know the interval of its possible values. Computing ∆ Computing ∆ • In such situations, decision theory recommends using Let Us Apply Hurwicz . . . Hurwicz optimism-pessimism criterion . Acknowledgments • Namely, we select the value α · ∆+(1 − α ) · ∆ for some Home Page α ∈ [0 , 1]. Title Page • A usual recommendation is to use α = 0 . 5. ◭◭ ◮◮ • Let us see what will be the result of applying this cri- ◭ ◮ terion to our problem. Page 6 of 11 Go Back Full Screen Close Quit
Common Sense Addition 6. Computing ∆ Towards Precise . . . Hurwicz Optimism- . . . • For every set S ∈ S , from | ∆ x 1 | ≤ ∆ 1 and | ∆ x 2 | ≤ ∆ 2 , Computing ∆ we conclude that | ∆ x 1 + ∆ x 1 | ≤ ∆ 1 + ∆ 2 . Computing ∆ • Thus always ∆( S ) ≤ ∆ 1 + ∆ 2 and hence, Let Us Apply Hurwicz . . . ∆ = max ∆( S ) ≤ ∆ 1 + ∆ 2 . Acknowledgments Home Page • Let us take S 0 = { ( v, (∆ 2 / ∆ 1 ) · v ) : v ∈ [ − ∆ 1 , ∆ 1 ] } ∈ S . Title Page • For S 0 , we have ∆ x 1 + ∆ x 2 = ∆ x 1 · (1 + ∆ 2 / ∆ 1 ) . ◭◭ ◮◮ • Thus in this case, the largest possible value ∆( S 0 ) of ◭ ◮ ∆ x 1 + ∆ x 2 is equal to Page 7 of 11 ∆( S 0 ) = ∆ 1 · (1 + ∆ 2 / ∆ 1 ) = ∆ 1 + ∆ 2 . Go Back • So, ∆ = max ∆( S ) ≥ ∆( S 0 ) = ∆ 1 + ∆ 2 . Full Screen • Hence, ∆ = ∆ 1 + ∆ 2 . Close Quit
Common Sense Addition 7. Computing ∆ Towards Precise . . . • For every S ∈ S , since π 1 ( S ) = [ − ∆ 1 , ∆ 1 ] , we have Hurwicz Optimism- . . . ∆ 1 ∈ π 1 ( S ) . Computing ∆ Computing ∆ • Thus, there exists a pair (∆ 1 , ∆ x 2 ) ∈ S corresponding Let Us Apply Hurwicz . . . to ∆ x 1 = ∆ 1 . Acknowledgments • For this pair, we have Home Page | ∆ x 1 + ∆ x 2 | ≥ | ∆ x 1 | − | ∆ x 2 | = ∆ 1 − | ∆ x 2 | . Title Page ◭◭ ◮◮ • Here, | ∆ x 2 | ≤ ∆ 2 , so | ∆ x 1 + ∆ x 2 | ≥ ∆ 1 − ∆ 2 . ◭ ◮ • Thus, for each S ∈ S , the largest possible value ∆( S ) of | ∆ x 1 + ∆ x 2 | cannot be smaller than ∆ 1 − ∆ 2 : Page 8 of 11 Go Back ∆( S ) ≥ ∆ 1 − ∆ 2 . Full Screen • Hence, ∆ = min S ∈S ∆( S ) ≥ ∆ 1 − ∆ 2 . Close Quit
Common Sense Addition 8. Computing ∆ (cont-d) Towards Precise . . . Hurwicz Optimism- . . . • Take S 0 = { ( v, − (∆ 2 / ∆ 1 ) · v ) : v ∈ [ − ∆ 1 , ∆ 1 ] } ∈ S . Computing ∆ • For S 0 , we have ∆ x 1 + ∆ x 2 = ∆ x 1 · (1 − ∆ 2 / ∆ 1 ) . Computing ∆ • Thus in this case, the largest possible value ∆( S 0 ) of Let Us Apply Hurwicz . . . ∆ x 1 + ∆ x 2 is equal to Acknowledgments Home Page ∆( S 0 ) = ∆ 1 · (1 − ∆ 2 / ∆ 1 ) = ∆ 1 − ∆ 2 . Title Page • So, ∆ = min S ∈S ∆( S ) ≥ ∆( S 0 ) = ∆ 1 − ∆ 2 . ◭◭ ◮◮ • Thus, ∆ ≤ ∆ 1 − ∆ 2 . ◭ ◮ • Hence, ∆ = ∆ 1 − ∆ 2 . Page 9 of 11 Go Back Full Screen Close Quit
Common Sense Addition 9. Let Us Apply Hurwicz Criterion Towards Precise . . . • Let us apply Hurwicz criterion with α = 0 . 5 to the Hurwicz Optimism- . . . interval [∆ , ∆] = [∆ 1 − ∆ 2 , ∆ 1 + ∆ 2 ] . Computing ∆ Computing ∆ • Then, we get ∆ = 0 . 5 · ∆ + 0 . 5 · ∆ = ∆ 1 . Let Us Apply Hurwicz . . . • For example, for ∆ 1 = 10% and ∆ 2 = 0 . 1%, we get Acknowledgments ∆ = 10%, in full accordance with common sense. Home Page • In other words, Hurwicz criterion explains the above- Title Page described common-sense addition. ◭◭ ◮◮ ◭ ◮ Page 10 of 11 Go Back Full Screen Close Quit
10. Acknowledgments Common Sense Addition Towards Precise . . . This work was supported in part by the National Science Hurwicz Optimism- . . . Computing ∆ Foundation grant HRD-1242122 (Cyber-ShARE Center). Computing ∆ Let Us Apply Hurwicz . . . Acknowledgments Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 11 Go Back Full Screen Close Quit
Recommend
More recommend
