Measuring the average . . . Measuring the average . . . Measuring the actual . . . Approximation inaccuracy Trade-Off Between Trade-off problems for . . . Sample Size and Accuracy: Solutions Case of Dynamic Measurements Case of non-smooth . . . under Interval Uncertainty Case of non-smooth . . . Case of more accurate . . . Hung T. Nguyen 1 , Olga Kosheleva 2 , Conclusions Vladik Kreinovich 2 , and Scot Ferson 3 Acknowledgments Title Page 1 New Mexico State University Las Cruces, NM 88003, USA ◭◭ ◮◮ 2 University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA Page 1 of 13 vladik@utep.edu Go Back 3 Applied Biomathematics Setauket, NY 11733, USA Full Screen Close Quit
Measuring the average . . . Measuring the average . . . 1. Formulation of the problem Measuring the actual . . . • For dynamic quantities, we may have two different ob- Approximation inaccuracy jectives: Trade-off problems for . . . Solutions – We may be interested in knowing the average value Case of non-smooth . . . of the measured quantity. Case of non-smooth . . . – We may want to know the actual dependence of the Case of more accurate . . . measured quantity on space location and/or time. Conclusions • Meteorological example: Acknowledgments – to study general weather patterns, we need average Title Page wind speed; ◭◭ ◮◮ – to guide airplanes, we need to know the exact wind ◭ ◮ speed at different locations. Page 2 of 13 Go Back Full Screen Close Quit
Measuring the average . . . Measuring the average . . . 2. Measuring the average value: case of ideal mea- suring instrument Measuring the actual . . . Approximation inaccuracy • Case description: measurement errors are negligible. Trade-off problems for . . . • Details: we measure the values x 1 , . . . , x n at n different Solutions locations, and estimate Case of non-smooth . . . = x 1 + . . . + x n Case of non-smooth . . . def E . n Case of more accurate . . . • Notation: let σ 0 be the standard deviation of the dif- Conclusions ference x i − x 0 . Acknowledgments • Conclusion: for the average, standard deviation is σ 0 / √ n , Title Page so error bound is ∆ = k 0 · σ 0 / √ n , where: ◭◭ ◮◮ ◭ ◮ – 95% confidence corresponds to k 0 = 2, – 99.9% corresponds to k 0 = 3. Page 3 of 13 • Recommendation: to get error ≤ ∆ 0 , we need Go Back n ≈ k 2 0 · σ 2 Full Screen 0 . ∆ 2 0 Close Quit
Measuring the average . . . Measuring the average . . . 3. Measuring the average value: case of realistic mea- suring instrument Measuring the actual . . . Approximation inaccuracy • Case description: we have random error with st. dev. σ Trade-off problems for . . . and a systematic error ∆ s bounded by ∆: | ∆ s | ≤ ∆. Solutions • Resulting random error component in measuring aver- Case of non-smooth . . . def σ 2 + σ 2 � Case of non-smooth . . . age: st. dev. σ t = 0 . Case of more accurate . . . • Resulting overall error bound: Conclusions ∆ 0 = ∆ + k 0 · σ t √ n. Acknowledgments Title Page • Observation: situation is similar to static measure- ◭◭ ◮◮ ments. ◭ ◮ • Conclusion: we have use the recommendations from Page 4 of 13 the static case. Go Back • Example: to achieve accuracy ∆ 0 with the minimal Full Screen cost, take ∆ ≈ (1 / 3) · ∆ 0 . Close Quit
Measuring the average . . . Measuring the average . . . 4. Measuring the actual dependence: formulation of the problem. Measuring the actual . . . Approximation inaccuracy • We are often interested in the actual dependence of a Trade-off problems for . . . quantity on space and/or time. Solutions • Possible situations: Case of non-smooth . . . Case of non-smooth . . . – A quantity that only depends on time t . Case of more accurate . . . – Example: temperature at a given location. Conclusions – A quantity that only depends on the spatial loca- Acknowledgments tion t = ( t 1 , t 2 ) or t = ( t 1 , t 2 , t 3 ). Title Page – Example: density inside the Earth. ◭◭ ◮◮ – A quantity that depends both on time t 1 and on ◭ ◮ the spatial location ( t 2 , . . . ). – Example: temperature in the atmosphere. Page 5 of 13 • General description: we measure x ( t ) for different Go Back t = ( t 1 , . . . , t d ) . Full Screen Close Quit
Measuring the average . . . Measuring the average . . . 5. Approximation inaccuracy Measuring the actual . . . • Fact: we only measure x ( t ) at n different values t (1) , . . . , t ( n ) . Approximation inaccuracy • To get values x ( t ) for t � = t ( i ) , we use interpolation. Trade-off problems for . . . Solutions • Main assumptions: we assume that: x ( t ) is smooth, Case of non-smooth . . . and we know the bound g on the rate of change: Case of non-smooth . . . | x ( t ) − x ( t ′ ) | ≤ g · � t − t ′ � . Case of more accurate . . . • Conclusion: to minimize the approximation error Conclusions | x ( t ) − x ( t ( i ) ) | , we must minimize the distance � t − t ( i ) � . Acknowledgments Title Page • If each value t is within distance ρ from one of t ( i ) , then n balls centered in t ( i ) cover the domain of volume V . ◭◭ ◮◮ ◭ ◮ • Hence, V ≤ n · c · ρ d , and ρ ≥ c · ( V/n ) 1 /d . • For a grid: ρ ≈ c 1 · ( V/n ) 1 /d . Page 6 of 13 Go Back • Conclusion: approximation error is d · c 1 · ( V/n ) 1 /d . Full Screen • Overall error: ∆ + d · c 1 · ( V/n ) 1 /d . Close Quit
Measuring the average . . . Measuring the average . . . 6. Trade-off problems for engineering and science: for- mulation Measuring the actual . . . Approximation inaccuracy • In engineering applications: Trade-off problems for . . . Solutions – we know the overall accuracy ∆ 0 , and Case of non-smooth . . . – we want to minimize the cost of the resulting mea- Case of non-smooth . . . surement: Case of more accurate . . . ∆ ,n under the constraint ∆+ g 0 Minimize n · F (∆) → min n 1 /d = ∆ 0 . Conclusions Acknowledgments • In scientific applications: Title Page – we are given the cost F 0 , and ◭◭ ◮◮ – the problem is to achieve the highest possible ac- ◭ ◮ curacy within this cost: Page 7 of 13 Minimize ∆+ g 0 n 1 /d → min ∆ ,n under the constraint n · F (∆) = F 0 . Go Back Full Screen Close Quit
Measuring the average . . . Measuring the average . . . 7. Solutions Measuring the actual . . . • Reminder: basic cost model F (∆) = c/ ∆. Approximation inaccuracy ∆ ,n under the constraint ∆+ g 0 Trade-off problems for . . . Minimize n · F (∆) → min n 1 /d = ∆ 0 . Solutions • Solution for engineering situations: Case of non-smooth . . . � g 0 � d Case of non-smooth . . . 1 · d + 1 ∆ opt = d + 1 · ∆ 0 ; n opt = . Case of more accurate . . . ∆ 0 d Conclusions Minimize ∆+ g 0 n 1 /d → min ∆ ,n under the constraint n · F (∆) = F 0 . Acknowledgments Title Page • Solution for science situations: ◭◭ ◮◮ � d/ ( d +1) � F 0 c · g 0 ∆ opt = n opt · c ◭ ◮ n opt = ; . d F 0 Page 8 of 13 • Observation: the optimal trade-off is when both error Go Back components are of approximately the same size. Full Screen • Comment: this is similar to the static case. Close Quit
Measuring the average . . . Measuring the average . . . 8. Case of non-smooth processes Measuring the actual . . . • Example of a non-smooth process: Brownian motion: Approximation inaccuracy | x ( t ) − x ( t ′ ) | ≤ � t − t ′ � 1 / 2 . Trade-off problems for . . . Solutions • General (fractal) case: | x ( t ) − x ( t ′ ) | ≤ � t − t ′ � β . Case of non-smooth . . . • For n measurements, distance ρ is ≈ ( V/n ) 1 /d , so ap- Case of non-smooth . . . proximation error is ∼ ( V/n ) β/d . Case of more accurate . . . • Overall error: ∆+ g β = C · d β/ 2 · 1 def 2 β/d · V β/d . Conclusions n β/d , where g β Acknowledgments • Trade-off problems for engineering: Title Page Min n · F (∆) under the constraint ∆ + g β n β/d = ∆ 0 . ◭◭ ◮◮ ◭ ◮ • Trade-off problems for science: Min ∆ + g β Page 9 of 13 n β/d under the constraint n · F (∆) = F 0 . Go Back • Observation: formulas same as in the smooth case, with d ′ def Full Screen = d/β instead of d . Close Quit
Measuring the average . . . Measuring the average . . . 9. Case of non-smooth processes: solutions Measuring the actual . . . • Case: basic cost model F (∆) = c/ ∆. Approximation inaccuracy Trade-off problems for . . . • Engineering problem – reminder: Min n · F (∆) under the constraint ∆ + g β Solutions n β/d = ∆ 0 . Case of non-smooth . . . Case of non-smooth . . . • Engineering problem – solution: � g β Case of more accurate . . . � d β · d + β ∆ opt = d + β · ∆ 0 ; n opt = . Conclusions ∆ 0 d Acknowledgments • Min ∆ + g β Title Page n β/d under the constraint n · F (∆) = F 0 . ◭◭ ◮◮ • Science problem – solution: ◭ ◮ � d/ ( d + β ) � F 0 c · g β ∆ opt = n opt · c n opt = ; . Page 10 of 13 d F 0 Go Back • Comment: in the optimal trade-off, both error compo- Full Screen nents are of approximately the same value. Close Quit
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