Introduction How to Get More . . . How to Speed Up . . . Combining Interval and Towards a Better . . . How General Can We . . . Probabilistic Uncertainty in Proofs Engineering Applications Home Page Title Page Andrew Pownuk ◭◭ ◮◮ Computational Science Program ◭ ◮ University of Texas at El Paso El Paso, Texas 79968, USA Page 1 of 72 ampownuk@utep.edu Go Back Full Screen Close Quit
Part I Introduction How to Get More . . . Introduction How to Speed Up . . . Towards a Better . . . How General Can We . . . Proofs Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 72 Go Back Full Screen Close Quit
1. Need for Data Processing Introduction How to Get More . . . • One of the main objectives of science is to predict fu- How to Speed Up . . . ture values y of physical quantities: Towards a Better . . . – in meteorology, we need to predict future weather; How General Can We . . . – in airplane control, we need to predict the location Proofs and the velocity of the plane under current control. Home Page • To make this prediction: Title Page – we need to know the relation y = f ( x 1 , . . . , x n ) be- ◭◭ ◮◮ tween y and related quantities x 1 , . . . , x n ; ◭ ◮ – then, we measure or estimate x 1 , . . . , x n ; Page 3 of 72 – finally, we use the results � x i of measurement (or estimation) to compute an estimate Go Back y = f ( � x 1 , . . . , � x n ) . � Full Screen Close • This computation of � y is an important case of data processing . Quit
2. Need to Take Uncertainty Into Account When Introduction Processing Data How to Get More . . . How to Speed Up . . . • Measurement are never absolutely accurate: in general, Towards a Better . . . def How General Can We . . . ∆ x i = � x i − x i � = 0 . Proofs • As a result, the estimate � y = f ( � x 1 , . . . , � x n ) is, in gen- Home Page eral, different from the ideal value y = f ( x 1 , . . . , x n ). Title Page def • To estimate the accuracy ∆ y = � y − y , we need to have ◭◭ ◮◮ some information about the measurement errors ∆ x i . ◭ ◮ • Traditional engineering approach assumes that we Page 4 of 72 know the probability distribution of each ∆ x i . Go Back • Often, ∆ x i ∼ N (0 , σ i ), and different ∆ x i are assumed to be independent. Full Screen • In such situations, our goal is to find the probability Close distribution for ∆ y . Quit
3. Cases of Interval and Fuzzy Uncertainty Introduction How to Get More . . . • Often, we only know the upper bound ∆ i : | ∆ x i | ≤ ∆ i . How to Speed Up . . . Towards a Better . . . • Then, the only information about the x i is that How General Can We . . . def x i ∈ x i = [ � x i − ∆ i , � x i + ∆ i ] . Proofs • Different x i ∈ x i lead, in general, to different Home Page y = f ( x 1 , . . . , x n ) . Title Page ◭◭ ◮◮ • We want to find the range y of possible values of y : ◭ ◮ y = { f ( x 1 , . . . , x n ) : x 1 ∈ x 1 , . . . , x n ∈ x n } . Page 5 of 72 • To gauge the accuracy of expert estimates, it is reason- Go Back able to use fuzzy techniques, i.e., to describe: Full Screen – for each possible value x i , Close – the degree µ i ( x i ) to which x i is possible. Quit
4. Measurement and Estimation Inaccuracies Are Introduction Usually Small How to Get More . . . How to Speed Up . . . • In many practical situations, the measurement and es- Towards a Better . . . timation inaccuracies ∆ x i are relatively small. How General Can We . . . • Then, we can safely ignore terms which are quadratic Proofs (or of higher order) in terms of ∆ x i : Home Page ∆ y = � y − y = f ( � x 1 , . . . , � x n ) − f ( � x 1 − ∆ x 1 , . . . , � x n − ∆ x n ) = Title Page n � c i · ∆ x i , where c i = ∂f ◭◭ ◮◮ . ∂x i ◭ ◮ i =1 • If needed, the derivative can be estimated by numerical Page 6 of 72 differentiation Go Back c i ≈ f ( � x 1 , . . . , � x i − 1 , � x i + h, � x i +1 , . . . , � x n ) − � y . Full Screen h Close Quit
5. Case of Interval Uncertainty Introduction How to Get More . . . � n • Let us consider the case when ∆ y = c i · ∆ x i . How to Speed Up . . . i =1 Towards a Better . . . � n How General Can We . . . • In this case, y = [ � y − ∆ , � y + ∆], where ∆ = | c i | · ∆ i . i =1 Proofs • Sometimes, we have explicit expressions or efficient al- Home Page gorithms for the partial derivatives c i . Title Page • Often, however, we use proprietary software in our ◭◭ ◮◮ computations. ◭ ◮ • Then, we cannot use differentiation formulas, but we Page 7 of 72 can use numerical differentiation. Go Back • Problem: We need n + 1 calls to f , to compute � y and n values c i . Full Screen • When f is time-consuming and n is large, this takes Close too long. Quit
6. A Faster Method: Cauchy-Based Monte-Carlo Introduction How to Get More . . . • Idea: use Cauchy distribution ρ ∆ ( x ) = ∆ 1 π · 1 + x 2 / ∆ 2 . How to Speed Up . . . Towards a Better . . . • Why: when ∆ x i ∼ ρ ∆ i ( x ) are indep., then How General Can We . . . � � n n ∆ y = c i · ∆ x i ∼ ρ ∆ ( x ), with ∆ = | c i | · ∆ i . Proofs i =1 i =1 Home Page • Thus, we simulate ∆ x ( k ) ∼ ρ ∆ i ( x ); then, i Title Page ∆ y ( k ) def x 1 − ∆ x ( k ) = � y − f ( � 1 , . . . ) ∼ ρ ∆ ( x ). ◭◭ ◮◮ • Maximum Likelihood method can estimate ∆: � � N N 1 + (∆ y ( k ) ) 2 / ∆ 2 = N 1 ◭ ◮ ρ ∆ (∆ y ( k ) ) → max, so 2 . Page 8 of 72 k =1 k =1 • To find ∆ from this equation, we can use, e.g., the Go Back 1 ≤ k ≤ N | ∆ y ( k ) | . bisection method for ∆ = 0 and ∆ = max Full Screen Close Quit
7. Monte-Carlo: Successes and Limitations Introduction How to Get More . . . √ • Fact: for Monte-Carlo, accuracy is ε ∼ 1 / N . How to Speed Up . . . Towards a Better . . . • Good news: the number N of calls to f depends only the desired accuracy ε . How General Can We . . . Proofs • Example: to find ∆ with accuracy 20% and certainty 95%, we need N = 200 iterations. Home Page Title Page • Limitation: this method is not realistic ; indeed: ◭◭ ◮◮ – we know that ∆ x i is inside [ − ∆ i , ∆ i ], but – Cauchy-distributed variable has a high probability ◭ ◮ to be outside this interval. Page 9 of 72 • Natural question: is it a limitation of our method, or Go Back of a problem itself? Full Screen Close Quit
8. Fuzzy Case: A Problem Introduction How to Get More . . . µ 1 ( x 1 ) How to Speed Up . . . ✲ µ 2 ( x 2 ) Towards a Better . . . µ = f ( µ 1 , . . . , µ n ) f ✲ How General Can We . . . ✲ · · · µ n ( x n ) Proofs ✲ Home Page • Given: an algorithm y = f ( x 1 , . . . , x n ) and n fuzzy Title Page numbers µ i ( x i ). ◭◭ ◮◮ • Compute: µ ( y ) = x 1 ,...,x n : f ( x 1 ,...,x n )= y min( µ 1 ( x 1 ) , . . . , µ n ( x n )) . max ◭ ◮ • Motivation: y is a possible value of Y ↔ ∃ x 1 , . . . , x n s.t. Page 10 of 72 each x i is a possible value of X i and f ( x 1 , . . . , x n ) = y . Go Back • Details: “and” is min, ∃ (“or”) is max, hence Full Screen µ ( y ) = max x 1 ,...,x n min( µ 1 ( x 1 ) , . . . , µ n ( x n ) , t ( f ( x 1 , . . . , x n ) = y )) , Close where t (true) = 1 and t (false) = 0 . Quit
9. Fuzzy Case: Reduction to Interval Computa- Introduction tions How to Get More . . . How to Speed Up . . . • Given: an algorithm y = f ( x 1 , . . . , x n ) and n fuzzy Towards a Better . . . numbers X i described by membership functions µ i ( x i ). How General Can We . . . • Compute: Y = f ( X 1 , . . . , X n ), where Y is defined by Proofs Zadeh’s extension principle: Home Page µ ( y ) = x 1 ,...,x n : f ( x 1 ,...,x n )= y min( µ 1 ( x 1 ) , . . . , µ n ( x n )) . max Title Page ◭◭ ◮◮ • Idea: represent each X i by its α -cuts ◭ ◮ X i ( α ) = { x i : µ i ( x i ) ≥ α } . Page 11 of 72 • Advantage: for continuous f , for every α , we have Go Back Y ( α ) = f ( X 1 ( α ) , . . . , X n ( α )) . Full Screen • Resulting algorithm: for α = 0 , 0 . 1 , 0 . 2 , . . . , 1 apply in- Close terval computations techniques to compute Y ( α ). Quit
10. Open Problems Introduction How to Get More . . . • In engineering applications, we want methods for esti- How to Speed Up . . . mating uncertainty which are: Towards a Better . . . – accurate – this is most important in most engineer- How General Can We . . . ing applications; Proofs – fast : this is important in some engineering applica- Home Page tions where we need real-time computations, Title Page – understandable to engineers – otherwise, engineers ◭◭ ◮◮ will be reluctant to use them, and ◭ ◮ – sufficiently general – so that they can be applied in all kinds of situations. Page 12 of 72 • It is thus desirable to design more accurate, faster, Go Back more understandable, and/or more general methods. Full Screen Close Quit
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