Comparison GUM versus GUM+1
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This document discusses the use of Monte Carlo simulations to evaluate measurement uncertainty according to Supplement 1 to the Guide to the Expression of Uncertainty in Measurement (GUM+1). It provides examples of generating random numbers and using them in Monte Carlo simulations. One example evaluates the uncertainty in measuring the volume of a cylinder based on uncertainties in diameter and height measurements. The document concludes with an example of calibrating a thermometer and determining the uncertainties in the calibration curve parameters.
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Monte-Carlo methods ((pseudo)random generator R[0,1])
Usually based on the remainder of integer divisions:
xi+1 = a * xi mod m (i = 1, 2, …) R[0,m-1]
Example: xi+1 = 75 * xi mod (231 – 1)
12 mod 3 = 0 ↔ 12 = 3.4 + 0
11 mod 3 = 2 ↔ 11 = 3.3 + 2
10 mod 3 = 1 ↔ 10 = 3.3 + 1
9 mod 3 = 0 ↔ 9 = 3.3 + 0
The GUM+1 proposes a generator (enhanced
Wichmann-Hill) that combines 4 R[0,m-1] generators.
The proposed algorithm can be easily implemented.
In the following examples, the random generator of VBA
(Excel) has been used for practical reasons.](https://image.slidesharecdn.com/comparisongum-montecarlo-151103125646-lva1-app6892/85/Comparison-GUM-versus-GUM-1-10-320.jpg)
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Monte-Carlo methods (Excel VBA generator R[0,1])
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N = 106 function calls: Rnd()](https://image.slidesharecdn.com/comparisongum-montecarlo-151103125646-lva1-app6892/85/Comparison-GUM-versus-GUM-1-11-320.jpg)
![http://economie.fgov.be
Monte-Carlo methods (Excel VBA generator T[0,2])
N = 106 function calls: Rnd() + Rnd()
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Monte-Carlo methods (Excel VBA generator N[0,1])
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N = 5.105 function calls: Box-Muller algorithm
r1 = Rnd() : r2 = Rnd(): z1 = (-2.ln(r1))1/2.cos(2.p.r2) : z2 = (-2.ln(r1))1/2.sin(2.p.r2)
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Comparison GUM versus GUM+1
- 1. http://economie.fgov.be Supplement 1 to the GUM Propagation of distributions using a Monte-Carlo method (JCGM 101:2008) Dr M. Maeck Belgian National Metrology
- 2. http://economie.fgov.be Supplement 1 to the GUM 1. Comparison GUM – GUM+1 2. Monte-Carlo methods 3. Evaluation of measurement uncertainty using a MC method 4. Calibration of a thermometer (GUM H3)
- 3. http://economie.fgov.be Comparison GUM – GUM+1 (similarities) PDF (1) PDF (2) PDF (3) PDF () Model Propagation of uncertainties PDF: probability density function
- 4. http://economie.fgov.be Comparison GUM – GUM+1 (differences) 1. Propagation of uncertainties: – GUM: series expansion – GUM+1: numerical method (Monte Carlo simulations) 2. The output PDF: – Supposed to be normal or « Student like » for the GUM ( = ? ; k = ?) – Obtained explicitly for the GUM+1 (neither nor k needed)
- 5. http://economie.fgov.be Comparison GUM – GUM+1 (advantages of GUM+1) 1. Less restrictions: – The model can be strongly nonlinear – The output PDF can be almost anything 2. Easy to program: – No computation of derivatives – The programming logic is very simple 3. Interpretation: – Based on the CDF – The confidence interval can be « personalized »
- 6. http://economie.fgov.be Supplement 1 to the GUM 1. Comparison GUM – GUM+1 2. Monte-Carlo methods 3. Evaluation of measurement uncertainty using a MC method 4. Calibration of a thermometer (GUM H3)
- 7. http://economie.fgov.be Monte-Carlo methods (principle) Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. These methods are most suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm. This method is also used to complement the theoretical derivations. (http://en.wikipedia.org/wiki/Monte_Carlo_method)
- 9. http://economie.fgov.be Monte-Carlo methods (evaluation of p) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 N Nr S S N N square circle square circle square circle 4 411 2 p pp 2.00 2.50 3.00 3.50 4.00 0 20 40 60 80 100 N = 100 p = 3,08 N = 103 p = 3,204 N = 104 p = 3,155 N = 105 p = 3,1389 N = 106 p = 3,1427 N = 107 p = 3,1418
- 10. http://economie.fgov.be Monte-Carlo methods ((pseudo)random generator R[0,1]) Usually based on the remainder of integer divisions: xi+1 = a * xi mod m (i = 1, 2, …) R[0,m-1] Example: xi+1 = 75 * xi mod (231 – 1) 12 mod 3 = 0 ↔ 12 = 3.4 + 0 11 mod 3 = 2 ↔ 11 = 3.3 + 2 10 mod 3 = 1 ↔ 10 = 3.3 + 1 9 mod 3 = 0 ↔ 9 = 3.3 + 0 The GUM+1 proposes a generator (enhanced Wichmann-Hill) that combines 4 R[0,m-1] generators. The proposed algorithm can be easily implemented. In the following examples, the random generator of VBA (Excel) has been used for practical reasons.
- 11. http://economie.fgov.be Monte-Carlo methods (Excel VBA generator R[0,1]) 0 2000 4000 6000 8000 10000 12000 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 0 10 20 30 40 50 60 70 80 90 100 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 PDF CDF (%) N = 106 function calls: Rnd()
- 12. http://economie.fgov.be Monte-Carlo methods (Excel VBA generator T[0,2]) N = 106 function calls: Rnd() + Rnd() 0 5000 10000 15000 20000 25000 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 0 10 20 30 40 50 60 70 80 90 100 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 PDF CDF (%)
- 13. http://economie.fgov.be Monte-Carlo methods (Excel VBA generator N[0,1]) 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 -5,00 -4,00 -3,00 -2,00 -1,00 0,00 1,00 2,00 3,00 4,00 5,00 N = 5.105 function calls: Box-Muller algorithm r1 = Rnd() : r2 = Rnd(): z1 = (-2.ln(r1))1/2.cos(2.p.r2) : z2 = (-2.ln(r1))1/2.sin(2.p.r2) 0 10 20 30 40 50 60 70 80 90 100 -5,0 -4,0 -3,0 -2,0 -1,0 0,0 1,0 2,0 3,0 4,0 5,0 PDF CDF
- 14. http://economie.fgov.be Enhanced Wichmann-Hill generator (C++) PDF CDF (%) N = 107 function calls
- 15. http://economie.fgov.be Supplement 1 to the GUM 1. Comparison GUM – GUM+1 2. Monte-Carlo methods 3. Evaluation of measurement uncertainty using a MC method 4. Calibration of a thermometer (GUM H3)
- 16. http://economie.fgov.be Evaluation of measurement uncertainty (1) V = p.d2.h/4 Six determinations of diameter and height: Calculation of the volume of a cylinder
- 17. http://economie.fgov.be Evaluation of measurement uncertainty (2)
- 18. http://economie.fgov.be Program (1) Dim i, N As Long: Dim Pi As Single : Dim r1, r2, d, h, V, Vmin, Vmax, Bin As Single : Dim Volumes(10000000#) As Single : Dim DeltaV, AverageV, sV, DeltaVd, AverageVd, sVd, DeltaVh, AverageVh, sVh As Single Cells(3, 7).Value = "": Cells(5, 7).Value = "": Cells(6, 7).Value = "": Cells(9, 5).Value = "": Cells(10, 5).Value = "" Cells(8, 7).Value = "": Cells(9, 7).Value = "" : N = Cells(3, 3).Value: Pi = 3.141592654 For i = 1 To N: Cells(3, 7).Value = i ‘Uncertainty of V r1 = Rnd(): r2 = Rnd() d = Sqr(-2 * Log(r1)) * Cos(2 * Pi * r2) * Sheets("Data").Cells(14, 6).Value + Sheets("Data").Cells(12, 6).Value h = Sqr(-2 * Log(r1)) * Sin(2 * Pi * r2) * Sheets("Data").Cells(14, 10).Value + Sheets("Data").Cells(12, 10).Value V = Pi * d * d * h / 4: Volumes(i) = V DeltaV = V - AverageV: AverageV = AverageV + DeltaV / i: sV = sV + DeltaV * (V - AverageV) 'Contribution of the diameter r1 = Rnd(): r2 = Rnd() d = Sqr(-2 * Log(r1)) * Cos(2 * Pi * r2) * Sheets("Data").Cells(14, 6).Value + Sheets("Data").Cells(12, 6).Value h = Sheets("Data").Cells(12, 10).Value V = Pi * d * d * h / 4 DeltaVd = V - AverageVd: AverageVd = AverageVd + DeltaVd / i: sVd = sVd + DeltaVd * (V - AverageVd) 'Contribution of the height r1 = Rnd(): r2 = Rnd() d = Sheets("Data").Cells(12, 6).Value h = Sqr(-2 * Log(r1)) * Sin(2 * Pi * r2) * Sheets("Data").Cells(14, 10).Value + Sheets("Data").Cells(12, 10).Value V = Pi * d * d * h / 4 DeltaVh = V - AverageVh: AverageVh = AverageVh + DeltaVh / i: sVh = sVh + DeltaVh * (V - AverageVh) Next i ‘Display uncertainty and contributions Cells(5, 7).Value = AverageV: Cells(6, 7).Value = Sqr(sV / (N - 1)) Cells(9, 5).Value = 100 * sVd / (N - 1) / Cells(6, 7).Value ^ 2: Cells(10, 5).Value = 100 * sVh / (N - 1) / Cells(6, 7).Value ^ 2
- 19. http://economie.fgov.be Program (2) 'Construction of PDF and CDF Bin = 12 * Sqr(sV / (N - 1)) / 100 For i = 1 To 100 Sheets("Histograms").Cells(i, 1).Value = (Cells(1, 7).Value - 6 * Sqr(sV / (N - 1))) + i * Bin Sheets("Histograms").Cells(i, 2).Value = "" ‘Clear CDF cells Next i For i = 1 To N: Cells(3, 7).Value = i: j = 1 While Volumes(i) > Sheets("Histograms").Cells(j, 1).Value: j = j + 1: Wend Sheets("Histograms").Cells(j, 2).Value = Sheets("Histograms").Cells(j, 2).Value + 1 Next i ‘Search the uncertainty limits (CDF < 2.5 % and CDF > 97.5 %) j = 1: While Sheets("Histograms").Cells(j, 3).Value < 2.5: j = j + 1: Wend Vmin = Sheets("Histograms").Cells(j, 1) - Sheets("Histograms").Cells(j - 1, 1) Vmin = Vmin * (2.5 - Sheets("Histograms").Cells(j - 1, 3)) Vmin = Vmin / (Sheets("Histograms").Cells(j, 3) - Sheets("Histograms").Cells(j - 1, 3)) Vmin = Vmin + Sheets("Histograms").Cells(j - 1, 1) Cells(8, 7).Value = Vmin j = 1: While Sheets("Histograms").Cells(j, 3).Value < 97.5: j = j + 1: Wend Vmax = Sheets("Histograms").Cells(j, 1) - Sheets("Histograms").Cells(j - 1, 1) Vmax = Vmax * (97.5 - Sheets("Histograms").Cells(j - 1, 3)) Vmax = Vmax / (Sheets("Histograms").Cells(j, 3) - Sheets("Histograms").Cells(j - 1, 3)) Vmax = Vmax + Sheets("Histograms").Cells(j - 1, 1) Cells(9, 7).Value = Vmax 'Normalize contributions Dim Total As Single Total = Cells(9, 5).Value + Cells(10, 5).Value Cells(9, 5).Value = Cells(9, 5).Value / Total * 100: Cells(10, 5).Value = Cells(10, 5).Value / Total * 100
- 20. http://economie.fgov.be Evaluation of measurement uncertainty (3) 0 10000 20000 30000 40000 50000 60000 0,00 20,00 40,00 60,00 80,00 100,00 120,00 PDF CDF
- 21. http://economie.fgov.be Evaluation of measurement uncertainty (4) Bins PDF CDF
- 22. http://economie.fgov.be Evaluation of measurement uncertainty (5) 118,030 cm3 V 116,653 cm3 (95 %)
- 23. http://economie.fgov.be Evaluation of measurement uncertainty (6)
- 24. http://economie.fgov.be Evaluation of measurement uncertainty (7) Comparison GUM ↔ GUM+1: GUM u = 0,350 cm3 → U(k=2) = 0,700 cm3 GUM+1 U(95%) = 0,694 cm3 The difference is due to the departure from a normal distribution of the PDF
- 25. http://economie.fgov.be Supplement 1 to the GUM 1. Comparison GUM – GUM+1 2. Monte-Carlo methods 3. Evaluation of measurement uncertainty using a MC method 4. Calibration of a thermometer (GUM H3)
- 26. http://economie.fgov.be Use of method of least squares to obtain linear calibration curve for a thermometer Mathematical model: Uncertainty of temperature measurements is negligible → uncertainty determined by uncertainty of parameters y1 and y2 Calibration of a thermometer GUM H3 (1) b(t) = y1 + y2.(t – t0) (t0 = 20 °C)
- 27. http://economie.fgov.be Calibration of a thermometer GUM H3 (2) k tk (°C) tk-20 (°C) bk (°C) 1 21,521 1,521 -0,171 2 22,012 2,012 -0,169 3 22,512 2,512 -0,166 4 23,003 3,003 -0,159 5 23,507 3,507 -0,164 6 23,999 3,999 -0,165 7 24,513 4,513 -0,156 8 25,002 5,002 -0,157 9 25,503 5,503 -0,159 10 26,010 6,010 -0,161 11 26,511 6,511 -0,160 r2 = 0,5427 -0,173 -0,171 -0,169 -0,167 -0,165 -0,163 -0,161 -0,159 -0,157 -0,155 1 2 3 4 5 6 7 b(tk) (°C) bk - b(tk) (°C) -0,1679 -0,0031 -0,1668 -0,0022 -0,1657 -0,0003 -0,1646 0,0056 -0,1635 -0,0005 -0,1625 -0,0025 -0,1614 0,0054 -0,1603 0,0033 -0,1592 0,0002 -0,1581 -0,0029 -0,1570 -0,0030 y1 = -0.1712 °C s(y1) = 0.0029 °C y2 = 0.00218 s(y2) = 0.00067 b(t) = y1 + y2.(t – t0) (t0 = 20 °C) Input for the Monte Carlo simulations Input for the least squares
- 28. http://economie.fgov.be Calibration of a thermometer GUM H3 (3)
- 29. http://economie.fgov.be Calibration of a thermometer GUM H3 (4) 'Declarations Dim i, N As Long Dim y1, y2, sy, sy1, sy2, t, r, r1, r2, b, Deltab, Meanb, sb, bMin, bMax As Single Dim bValues(10000000#) As Single Dim j As Integer 'Cleaning and initializing Cells(13, 11).Value = "": Cells(14, 11).Value = "" For j = 8 To 18: Cells(j, 4).Value = Cells(j, 2).Value: Next j y1 = Cells(20, 18).Value y2 = Cells(21, 18).Value sy = Cells(23, 18).Value sy1 = Cells(26, 18).Value sy2 = Cells(27, 18).Value 'Input N = Cells(22, 7).Value: t = Cells(20, 7).Value 'Iteration For i = 1 To N: Cells(22, 7).Value = i r1 = Rnd() * 2 - 1 r2 = Rnd() * 2 - 1 For j = 8 To 18 Cells(j, 4).Value = y1 + r1 * sy1 + (y2 + r2 * sy2) * (Cells(j, 1).Value - 20) Next j b = Cells(7, 11).Value + Cells(8, 11).Value * (t - 20) bValues(i) = b Deltab = b - Meanb: Meanb = Meanb + Deltab / i: sb = sb + Deltab * (b - Meanb) Next i
- 30. http://economie.fgov.be Calibration of a thermometer GUM H3 (5) 'Output Cells(13, 11).Value = Meanb: Cells(14, 11).Value = Sqr(sb / (N - 1)) 'Construction of PDF and CDF Bin = 12 * Sqr(sb / (N - 1)) / 100 For i = 1 To 100 Sheets("Histograms").Cells(i, 1).Value = (Cells(13, 11).Value - 6 * Sqr(sb / (N - 1))) + i * Bin Sheets("Histograms").Cells(i, 2).Value = "" Next i For i = 1 To N: Cells(3, 7).Value = i: j = 1 While bValues(i) > Sheets("Histograms").Cells(j, 1).Value: j = j + 1: Wend Sheets("Histograms").Cells(j, 2).Value = Sheets("Histograms").Cells(j, 2).Value + 1 Next i 'Search the uncertainty limits (CDF < 2.5 % and CDF > 97.5 %) j = 1: While Sheets("Histograms").Cells(j, 3).Value < 2.5: j = j + 1: Wend bMin = Sheets("Histograms").Cells(j, 1) - Sheets("Histograms").Cells(j - 1, 1) bMin = bMin * (2.5 - Sheets("Histograms").Cells(j - 1, 3)) bMin = bMin / (Sheets("Histograms").Cells(j, 3) - Sheets("Histograms").Cells(j - 1, 3)) bMin = bMin + Sheets("Histograms").Cells(j - 1, 1) Cells(13, 14).Value = bMin j = 1: While Sheets("Histograms").Cells(j, 3).Value < 97.5: j = j + 1: Wend bMax = Sheets("Histograms").Cells(j, 1) - Sheets("Histograms").Cells(j - 1, 1) bMax = bMax * (97.5 - Sheets("Histograms").Cells(j - 1, 3)) bMax = bMax / (Sheets("Histograms").Cells(j, 3) - Sheets("Histograms").Cells(j - 1, 3)) bMax = bMax + Sheets("Histograms").Cells(j - 1, 1) Cells(14, 14).Value = bMax
- 31. http://economie.fgov.be Calibration of a thermometer GUM H3 (6) GUM b = - 0,149 4 °C s(b) = 0,004 1 °C GUM + 1 (VBA) b = - 0,149 4 °C s(b) = 0,004 2 °C u(k=2) = 0,007 6 °C GUM + 1 (C++) b = - 0,149 4 °C s(b) = 0,004 2 °C u(k=2) = 0,007 6 °C