Interval Part: Outline Straightforward . . . Discussion Reason for Excess Width From Interval Computations Constraint-Based Set . . . to Constraint-Related Set From Main Idea to . . . Implementing . . . Computations: Towards Limitations of This . . . Estimating Variance . . . Faster Estimation Other Statistical . . . of Statistics and ODEs Dynamical Systems . . . Possibility to Take . . . under Interval, p-Box, p-Boxes and Classes of . . . and Fuzzy Uncertainty Set Computations for . . . Acknowledgments Title Page Martine Ceberio, Vladik Kreinovich, Andrzej Pownuk, and Barnab´ as Bede ◭◭ ◮◮ University of Texas, El Paso, Texas 79968, USA email vladik@utep.edu ◭ ◮ Page 1 of 21
Interval Part: Outline Straightforward . . . 1. Outline Discussion • Interval computations: at each intermediate stage of Reason for Excess Width the computation, we have intervals of possible values Constraint-Based Set . . . of the corresponding quantities. From Main Idea to . . . Implementing . . . • In our previous papers, we proposed an extension of this technique to set computations . Limitations of This . . . Estimating Variance . . . • Set computations: on each stage, in addition to inter- Other Statistical . . . vals of possible values of the quantities, we also keep Dynamical Systems . . . sets of possible values of pairs (triples, etc.). Possibility to Take . . . • In this paper, we consider several practical problems: p-Boxes and Classes of . . . – estimating statistics (variance, correlation, etc.), Set Computations for . . . Acknowledgments – solving ordinary differential equations (ODEs). Title Page • For these problems, the new formalism enables us to ◭◭ ◮◮ find estimates in feasible (polynomial) time. ◭ ◮ Page 2 of 21
Interval Part: Outline Straightforward . . . 2. Need for Data Processing Discussion • Problem: in many real-life situations, we are interested Reason for Excess Width in the value of a physical quantity y that is difficult or Constraint-Based Set . . . impossible to measure directly. From Main Idea to . . . Implementing . . . • Examples: distance to a star, amount of oil in a well. Limitations of This . . . • Solution: Estimating Variance . . . – find easier-to-measure quantities x 1 , . . . , x n which Other Statistical . . . are related to y by a known relation y = f ( x 1 , . . . , x n ); Dynamical Systems . . . – measure or estimate the values of the quantities Possibility to Take . . . x 1 , . . . , x n ; results are � x i ≈ x i ; p-Boxes and Classes of . . . – estimate y as � y = f ( � x 1 , . . . , � x n ). Set Computations for . . . Acknowledgments • Computing � y is called data processing. Title Page • Comment: algorithm f can be complex, e.g., solving ◭◭ ◮◮ ODEs. ◭ ◮ Page 3 of 21
Interval Part: Outline Straightforward . . . 3. Measurement Uncertainty Discussion • Measurement errors: measurement are never 100% ac- Reason for Excess Width def curate, so ∆ x i = � x i − x i � = 0. Constraint-Based Set . . . From Main Idea to . . . • Result: the estimate � y = f ( � x 1 , . . . , � x n ) is, in general, Implementing . . . different from the actual value y = f ( x 1 , . . . , x n ). Limitations of This . . . • Problem: based on the information about ∆ x i , esti- Estimating Variance . . . def mate the error ∆ y = � y − y . Other Statistical . . . • What do we know about ∆ x i : the manufacturer of the Dynamical Systems . . . measuring instrument (MI) supplies an upper bound ∆ i : Possibility to Take . . . p-Boxes and Classes of . . . | ∆ x i | ≤ ∆ i . Set Computations for . . . • Interval uncertainty: x i ∈ [ � x i − ∆ i , � x i + ∆ i ]. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 21
Interval Part: Outline Straightforward . . . 4. Measurement Uncertainty: from Probabilities to Intervals Discussion Reason for Excess Width • Reminder: we know that ∆ x i ∈ [ − ∆ i , ∆ i ]. Constraint-Based Set . . . From Main Idea to . . . • Probabilistic uncertainty: often, we also know the prob- Implementing . . . ability of different values ∆ x i ∈ [∆ i , ∆ i ]. Limitations of This . . . • We can determine these probabilities by using standard Estimating Variance . . . measuring instruments. Other Statistical . . . • Two cases when this is not done: Dynamical Systems . . . Possibility to Take . . . – cutting edge measurements (e.g., Hubble telescope); p-Boxes and Classes of . . . – manufacturing. Set Computations for . . . • In these cases, we have a purely interval uncertainty. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 21
Interval Part: Outline Straightforward . . . 5. Case of Fuzzy Uncertainty and its Reduction to Interval Uncertainty Discussion Reason for Excess Width • Situation: an expert uses natural language, e.g., “most Constraint-Based Set . . . probably, the value of the quantity is between 3 and 4”. From Main Idea to . . . • Natural formalization: fuzzy set theory, as fuzzy num- Implementing . . . bers µ i ( x i ). Limitations of This . . . Estimating Variance . . . • Equivalent reformulation: in terms of α -cuts Other Statistical . . . def x i ( α ) = { x i | µ i ( x i ) > α } . Dynamical Systems . . . Possibility to Take . . . • Zadeh’s extension principle transforms fuzzy numbers p-Boxes and Classes of . . . for x i into a fuzzy number for y = f ( x 1 , . . . , x n ). Set Computations for . . . • Known result: y ( α ) = f ( x 1 ( α ) , . . . , x n ( α )). Acknowledgments • Reduction: fuzzy data processing can be implemented Title Page as layer-by-layer interval computations. ◭◭ ◮◮ • In view of this reduction, in the following, we will ◭ ◮ mainly concentrate on interval computations. Page 6 of 21
Interval Part: Outline Straightforward . . . 6. Interval Part: Outline Discussion • We start by recalling the basic techniques of interval Reason for Excess Width computations and their drawbacks. Constraint-Based Set . . . From Main Idea to . . . • Then we will describe the new set computation tech- Implementing . . . niques. Limitations of This . . . • We describe a class of problems for which these tech- Estimating Variance . . . niques are efficient. Other Statistical . . . • Finally, we talk about how we can extend these tech- Dynamical Systems . . . niques to other types of uncertainty (e.g., classes of Possibility to Take . . . probability distributions). p-Boxes and Classes of . . . Set Computations for . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 21
Interval Part: Outline Straightforward . . . 7. Straightforward Interval Computations: Main Idea Discussion • Parsing: inside the computer, every algorithm consists Reason for Excess Width of elementary operations (arithmetic operations, min, Constraint-Based Set . . . max, etc.). From Main Idea to . . . • Interval arithmetic: for each elementary operation f ( a, b ), Implementing . . . if we know the intervals a and b , we can compute the Limitations of This . . . exact range f ( a , b ): Estimating Variance . . . Other Statistical . . . [ a, a ]+[ b, b ] = [ a + b, a + b ]; [ a, a ] − [ b, b ] = [ a − b, a − b ]; Dynamical Systems . . . [ a, a ] · [ b, b ] = [min( a · b, a · b, a · b, a · b ) , max( a · b, a · b, a · b, a · b )]; Possibility to Take . . . � 1 � 1 a, 1 [ a, a ] 1 p-Boxes and Classes of . . . [ a, a ] = if 0 �∈ [ a, a ]; [ b, b ] = [ a, a ] · [ b, b ] . a Set Computations for . . . Acknowledgments • Main idea: replace each elementary operation in f by Title Page the corresponding operation of interval arithmetic. ◭◭ ◮◮ • Known: we get an enclosure Y ⊇ y for the desired range. ◭ ◮ Page 8 of 21
Interval Part: Outline Straightforward . . . 8. Discussion Discussion • Fact: not every real number can be exactly imple- Reason for Excess Width mented in a computer. Constraint-Based Set . . . From Main Idea to . . . • Conclusion: after implementing an operation of inter- val arithmetic, we must enclose the result [ r − , r + ] in a Implementing . . . computer-representable interval: Limitations of This . . . Estimating Variance . . . – round-off r − to a smaller computer-representable Other Statistical . . . value r , and Dynamical Systems . . . – round-off r + to a larger computer-representable value r . Possibility to Take . . . • Computation time: increase by a factor of ≤ 4. p-Boxes and Classes of . . . Set Computations for . . . • Computing exact range: NP-hard. Acknowledgments • Conclusion: excess width is inevitable. Title Page • More accurate techniques exist: centered form, bisec- ◭◭ ◮◮ tion, etc. ◭ ◮ Page 9 of 21
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