Uncertainty Typical Situation: . . . Case of Data Processing in - PowerPoint PPT Presentation

Cyberinfrastructure: . . . Data Processing vs. . . . Need for Uncertainty . . . Uncertainty of the . . . Uncertainty Typical Situation: . . . Case of Data Processing in Cyberinfrastructure: Beyond Probabilistic . . . Case Study: Seismic . . . Results, Algorithms, Conclusions Request for Collaboration Acknowledgments Challenges, and Request Title Page for Collaboration ◭◭ ◮◮ Ann Gates, Vladik Kreinovich, Paulo Pinheiro da Silva, ◭ ◮ Craig Tweedie, Leonardo Salayandia, and Christian Servin Page 1 of 26 Center of Excellence for Sharing resources for Go Back the Advancement of Research and Education through Cyberinfrastructure Cyber-ShARE, Full Screen University of Texas at El Paso (UTEP) Close http://trust.cs.utep.edu/cybershare/ contact email vladik@utep.edu Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 1. Cyberinfrastructure: A Brief Overview Need for Uncertainty . . . Uncertainty of the . . . • Practical problem: need to combine geographically sep- Typical Situation: . . . arate computational resources. Case of Data Processing • Centralization of computational resources – traditional Beyond Probabilistic . . . approach to combining computational resources. Case Study: Seismic . . . • Limitations of centralization: Conclusions – need to reformat all the data; Request for Collaboration Acknowledgments – need to rewrite data processing programs: make Title Page compatible w/selected formats and w/each other ◭◭ ◮◮ • Cyberinfrastructure – a more efficient approach to com- bining computational resources: ◭ ◮ – keep resources at their current locations, and Page 2 of 26 – in their current formats. Go Back • Technical advantages of cyberinfrastructure: a brief Full Screen summary. Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 2. Data Processing vs. Data Fusion Need for Uncertainty . . . Uncertainty of the . . . • Practically important situation: difficult to measure Typical Situation: . . . the desired quantity y with a given accuracy. Case of Data Processing • Data processing : Beyond Probabilistic . . . – measure related easier-to-measure quantities x 1 , . . . , x n ; Case Study: Seismic . . . – estimate y from the results � x i of measuring x i as Conclusions y = f ( � � x 1 , . . . , � x n ). Request for Collaboration Acknowledgments • Example: seismic inverse problem. Title Page • Data fusion: ◭◭ ◮◮ – measure the quantity y several times; ◭ ◮ – combine the results � y 1 , . . . � y n of these measurements. Page 3 of 26 • Specifics of cyberinfrastructure: first looks for stored Go Back results � x i (corr., � y i ), measure only if necessary. Full Screen • Combination of data processing and data fusion. Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 3. Need for Uncertainty Propagation, and for Prove- Need for Uncertainty . . . nance of Uncertainty Uncertainty of the . . . Typical Situation: . . . • Need for uncertainty propagation. Case of Data Processing – main reasons for data processing and data fusion: Beyond Probabilistic . . . accuracy is not high enough; Case Study: Seismic . . . – we must make sure that after the data processing Conclusions (data fusion), we get the desired accuracy. Request for Collaboration • In cyberinfrastructure this is especially important: Acknowledgments Title Page – accuracy varies greatly, and ◭◭ ◮◮ – we do not have much control over these accuracies. ◭ ◮ • Need for the provenance of uncertainty: Page 4 of 26 – sometimes, the resulting accuracy is still too low; Go Back – it is desirable to find out which data points con- Full Screen tributed most to the inaccuracy. Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 4. Uncertainty of the Results of Direct Measurements: Need for Uncertainty . . . Probabilistic and Interval Approaches Uncertainty of the . . . Typical Situation: . . . • Manufacturer of the measuring instrument (MI) sup- Case of Data Processing def plies ∆ i s.t. | ∆ x i | ≤ ∆ i , where ∆ x i = � x i − x i . Beyond Probabilistic . . . • The actual (unknown) value x i of the measured quan- Case Study: Seismic . . . tity is in the interval x i = [ � x i − ∆ i , � x i + ∆ i ]. Conclusions • Probabilistic uncertainty: often, we know the probabil- Request for Collaboration ities of different values ∆ x i ∈ [ − ∆ i , ∆ i ]. Acknowledgments Title Page • How probabilities are determined: by comparing our MI with a much more accurate (standard) MI. ◭◭ ◮◮ • Interval uncertainty: in two cases, we do not determine ◭ ◮ the probabilities: Page 5 of 26 – cutting-edge measurements; Go Back – measurements on the shop floor. Full Screen • In both cases, we only know that x i ∈ [ � x i − ∆ i , � x i +∆ i ]. Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 5. Typical Situation: Measurement Errors are Rea- Need for Uncertainty . . . sonably Small Uncertainty of the . . . Typical Situation: . . . • Typical situation: Case of Data Processing – direct measurements are accurate enough; Beyond Probabilistic . . . – the resulting approximation errors ∆ x i are small; Case Study: Seismic . . . – terms which are quadratic (or of higher order) in Conclusions ∆ x i can be safely neglected. Request for Collaboration • Example: for an error of 1%, its square is a negligible Acknowledgments 0.01%. Title Page ◭◭ ◮◮ • Linearization: ◭ ◮ – expand f in Taylor series around the point ( � x 1 , . . . , � x n ); – restrict ourselves only to linear terms: Page 6 of 26 ∆ y = c 1 · ∆ x 1 + . . . + c n · ∆ x n , Go Back = ∂f def Full Screen where c i . ∂x i Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 6. Case of Data Processing Need for Uncertainty . . . Uncertainty of the . . . • Propagation (probabilistic case): if ∆ x i are indepen- Typical Situation: . . . dent with st. dev. σ i (and 0 mean), then ∆ y has st. dev. Case of Data Processing σ 2 = c 2 1 · σ 2 1 + . . . + c 2 n · σ 2 n . Beyond Probabilistic . . . Case Study: Seismic . . . • Provenance: Conclusions – we know which component σ 2 comes from the i -th Request for Collaboration measurement; Acknowledgments – we can predict how replacing the i -th measurement Title Page with a more accurate one ( σ new ≪ σ i ) will affect σ 2 . ◭◭ ◮◮ i • Propagation of interval uncertainty: ◭ ◮ ∆ = | c 1 | · ∆ 1 + . . . + | c n | · ∆ n . Page 7 of 26 Go Back • We can predict how replacing the i -th measurement with a more accurate one (∆ new Full Screen ≪ ∆ i ) will affect ∆. i Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 7. Beyond Probabilistic and Interval Uncertainty Need for Uncertainty . . . Uncertainty of the . . . • Up to now: we considered two extreme situations: Typical Situation: . . . – probabilistic uncertainty, when we know all the prob- Case of Data Processing abilities; Beyond Probabilistic . . . – interval uncertainty, when we have no information Case Study: Seismic . . . about the probabilities. Conclusions • Fact: probabilistic situation is a particular case of the Request for Collaboration interval situation. Acknowledgments Title Page • Conclusion: interval bounds are wider. ◭◭ ◮◮ • In practice: often, we have partial information about probabilities. ◭ ◮ • As a result: Page 8 of 26 – probabilistic bounds are too narrow, Go Back – interval bounds are too wide. Full Screen • We need: intermediate bounds. Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 8. Case Study: Seismic Inverse Problem in the Geo- Need for Uncertainty . . . sciences Uncertainty of the . . . Typical Situation: . . . Case of Data Processing Beyond Probabilistic . . . Case Study: Seismic . . . Conclusions Request for Collaboration Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 26 Go Back Full Screen Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . Need for Uncertainty . . . Uncertainty of the . . . Typical Situation: . . . Case of Data Processing Beyond Probabilistic . . . Case Study: Seismic . . . Conclusions Request for Collaboration Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 26 Go Back Full Screen Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 9. Estimating Uncertainty, First Try: Probabilistic Ap- Need for Uncertainty . . . proach Uncertainty of the . . . Typical Situation: . . . Case of Data Processing Beyond Probabilistic . . . Case Study: Seismic . . . Conclusions Request for Collaboration Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 26 Go Back Full Screen Close Quit

Cyberinfrastructure: . . . Data Processing vs. . . . 10. Estimating Uncertainty, Second Try: Interval Ap- Need for Uncertainty . . . proach Uncertainty of the . . . Typical Situation: . . . Case of Data Processing Beyond Probabilistic . . . Case Study: Seismic . . . Conclusions Request for Collaboration Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 26 Go Back Full Screen Close Quit