Outline Need for Uncertainty . . . Case Study: Seismic . . . Propagation and Provenance Need to go Beyond . . . Model Fusion: We . . . of Uncertainty in Optimal Stationary . . . Dynamic Sensors: . . . Cyberinfrastructure-Related Home Page Data Processing Title Page ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ Cyber-ShARE Center Page 1 of 24 University of Texas at El Paso El Paso, TX 79968, USA Go Back contact email vladik@utep.edu Full Screen Close Quit
Outline 1. Outline Need for Uncertainty . . . • Need for uncertainty estimation in cyberinfrastructure- Case Study: Seismic . . . related data processing. Need to go Beyond . . . Model Fusion: We . . . • Geophysical case study: need to go beyond traditional Optimal Stationary . . . techniques. Dynamic Sensors: . . . • Estimating uncertainty and spatial resolution. Home Page • Combining different types of uncertainty: model fu- Title Page sion, with additional continuous vs. discrete problem. ◭◭ ◮◮ • This is all based on known measurement results, how ◭ ◮ can be better plan the measurements? Page 2 of 24 • Optimal location of a sensor, on the example of a me- teorological tower. Go Back • Optimal placement of stationary sensors. Full Screen • Optimal trajectories of mobile sensors, on the example Close of UAV-based sensors. Quit
Outline 2. Need for Uncertainty Estimation in Need for Uncertainty . . . Cyberinfrastructure-Related Data Processing Case Study: Seismic . . . • In the past: communications were much slower. Need to go Beyond . . . Model Fusion: We . . . • Conclusion: use centralization. Optimal Stationary . . . • At present: communications are much faster. Dynamic Sensors: . . . • Conclusion: use cyberinfrastructure. Home Page Title Page • Related problems: ◭◭ ◮◮ – gauge the the uncertainty of the results obtained by using cyberinfrastructure; ◭ ◮ – which data points contributed most to uncertainty; Page 3 of 24 – how an improved accuracy of these data points will Go Back improve the accuracy of the result. Full Screen • We need: algorithms for solving these problems. Close Quit
Outline 3. Case Study: Seismic Inverse Problem in the Need for Uncertainty . . . Geosciences Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 24 Go Back Full Screen Close Quit
Outline Need for Uncertainty . . . Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 24 Go Back Full Screen Close Quit
Outline 4. Need to go Beyond Traditional Probabilistic Need for Uncertainty . . . Techniques Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 24 Go Back Full Screen Close Quit
Outline 5. Towards Interval Approach Need for Uncertainty . . . • Manufacturer of the measuring instrument (MI) sup- Case Study: Seismic . . . def Need to go Beyond . . . plies ∆ i s.t. | ∆ x i | ≤ ∆ i , where ∆ x i = � x i − x i . Model Fusion: We . . . • The actual (unknown) value x i of the measured quan- Optimal Stationary . . . tity is in the interval x i = [ � x i − ∆ i , � x i + ∆ i ]. Dynamic Sensors: . . . • Probabilistic uncertainty: often, we know the probabil- Home Page ities of different values ∆ x i ∈ [ − ∆ i , ∆ i ]. 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 Page 7 of 24 the probabilities: Go Back – cutting-edge measurements; Full Screen – measurements on the shop floor. Close • In both cases, we only know that x i ∈ [ � x i − ∆ i , � x i +∆ i ]. Quit
Outline 6. Estimating Uncertainty, Second Try: Interval Need for Uncertainty . . . Approach Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 24 Go Back Full Screen Close Quit
Outline 7. Towards a Better Estimate Need for Uncertainty . . . � n = ∂f def Case Study: Seismic . . . • Linearization: ∆ y = c i · ∆ x i , where c i . ∂x i i =1 Need to go Beyond . . . � n � n Model Fusion: We . . . • Formulas: σ 2 = c 2 i · σ 2 i , ∆ = | c i | · ∆ i . Optimal Stationary . . . i =1 i =1 Dynamic Sensors: . . . • Numerical differentiation: n iterations, too long. Home Page • Monte-Carlo approach: if ∆ x i are Gaussian w/ σ i , then � n Title Page ∆ y = c i · ∆ x i is also Gaussian, w/desired σ . ◭◭ ◮◮ i =1 • Advantage: # of iterations does not grow with n . ◭ ◮ ∆ i Page 9 of 24 • Interval estimates: if ∆ x i are Cauchy, w/ ρ i ( x ) = i + x 2 , ∆ 2 Go Back � n then ∆ y = c i · ∆ x i is also Cauchy, w/desired ∆. Full Screen i =1 Close Quit
Outline 8. A New (Heuristic) Approach Need for Uncertainty . . . Case Study: Seismic . . . • Problem: guaranteed (interval) bounds are too high. Need to go Beyond . . . • Gaussian case: we only have bounds guaranteed with Model Fusion: We . . . confidence, say, 90%. Optimal Stationary . . . • How: cut top 5% and low 5% off a normal distribution. Dynamic Sensors: . . . • New idea: to get similarly estimates for intervals, we Home Page “cut off” top 5% and low 5% of Cauchy distribution. Title Page • How: ◭◭ ◮◮ – find the threshold value x 0 for which the probability ◭ ◮ of exceeding this value is, say, 5%; Page 10 of 24 – replace values x for which x > x 0 with x 0 ; Go Back – replace values x for which x < − x 0 with − x 0 ; Full Screen – use this “cut-off” Cauchy in error estimation. Close • Example: for 95% confidence level, we need x 0 = 12 . 706. Quit
Outline 9. Heuristic Approach: Results with 95% Confi- Need for Uncertainty . . . dence Level Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 24 Go Back Full Screen Close Quit
Outline 10. Heuristic Approach: Results with 90% Confi- Need for Uncertainty . . . dence Level Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 24 Go Back Full Screen Close Quit
Outline 11. Model Fusion: We Also Have Different Spatial Need for Uncertainty . . . Resolution Case Study: Seismic . . . • In many situations, different models have not only dif- Need to go Beyond . . . ferent accuracy, but also different spatial resolution. Model Fusion: We . . . Optimal Stationary . . . • Example: Dynamic Sensors: . . . – seismic data leads to higher spatial resolution esti- Home Page mates of the density at different locations, while Title Page – gravity data leads to lower-spatial resolution esti- ◭◭ ◮◮ mates of the same densities. ◭ ◮ • Towards precise formulation of the problem: Page 13 of 24 – High spatial resolution estimates correspond to small spatial cells. Go Back – A low spatial resolution estimate is affected by sev- Full Screen eral neighboring spatial cells. Close Quit
Outline 12. Estimates of High and Low Spatial Resolu- Need for Uncertainty . . . tion: Illustration Case Study: Seismic . . . Need to go Beyond . . . Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . x 1 = 2 . 0 � x 2 = 3 . 0 � Home Page � X 1 = 3 . 7 Title Page ◭◭ ◮◮ x 3 = 5 . 0 x 4 = 6 . 0 � � ◭ ◮ Page 14 of 24 Go Back Full Screen Close Quit
Outline 13. Numerical Example: Discussion Need for Uncertainty . . . • We assume that the low spatial resolution estimate is Case Study: Seismic . . . accurate ( σ l ≈ 0). Need to go Beyond . . . Model Fusion: We . . . • So, the average of the four cell values is equal to the Optimal Stationary . . . result � X 1 = 3 . 7 of this estimate: Dynamic Sensors: . . . x 1 + x 2 + x 3 + x 4 ≈ 3 . 7 . Home Page 4 Title Page • For the high spatial resolution estimates � x i , the average ◭◭ ◮◮ is slightly different: ◭ ◮ x 1 + � � x 2 + � x 3 + � = 2 . 0 + 3 . 0 + 5 . 0 + 6 . 0 x 4 = 4 . 0 � = 3 . 7 . 4 4 Page 15 of 24 • Reason: high spatial resolution estimates are much less Go Back accurate: σ h = 0 . 5. Full Screen • We use the low spatial resolution estimate to “correct” Close the high spatial resolution estimate. Quit
Outline 14. The Result of Model Fusion Need for Uncertainty . . . Case Study: Seismic . . . Need to go Beyond . . . � x 1 ≈ 1 . 89 x 2 ≈ 2 . 79 � Model Fusion: We . . . Optimal Stationary . . . Dynamic Sensors: . . . Home Page � x 3 ≈ 4 . 62 x 4 ≈ 5 . 53 � Title Page ◭◭ ◮◮ ◭ ◮ • The arithmetic average of these four values is equal to Page 16 of 24 x 1 + x 2 + x 3 + x 4 ≈ 1 . 89 + 2 . 79 + 4 . 62 + 5 . 53 ≈ 3 . 71 . Go Back 4 4 Full Screen • So, within our computation accuracy, it coincides with the low spatial resolution estimate � X 1 = 3 . 7. Close Quit
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