Report

Evaluation of Measurement Uncertainties Using the Monte Carlo Method Speaker: Chung Yin, Poon Standards and Calibration Laboratory (SCL) The Government of the Hong Kong Special Administrative Region 1 Standards and Calibration Laboratory, SCL GUM Uncertainty Framework (GUF) “Propagation of Uncertainties” Measurement Model: Y= f(X1, X2, XN) Estimate xi of the input quantities Xi Determine u(xi) associated with each estimate xi and its degrees of freedom Estimate y = f(xi) of Y Calculate the sensitivity coefficient of each xi at Xi = xi Calculate u(y) Calculate the effective degrees of freedom veff and coverage factor k with coverage probability p Calculate the coverage interval: yku(y) 2 Standards and Calibration Laboratory, SCL GUM Uncertainty Framework (GUF) x1, u(x1) x2, u(x2) Problems: Y = f (X) y, u(y) xN, u(xN) The contributory uncertainties are not of approximately the same magnitude Difficult to provide the partial derivatives of the model The PDF for output quantity is not a Gaussian distribution or a scaled and shifted t-distribution 3 Standards and Calibration Laboratory, SCL Monte Carlo Method (MCM) “Propagation of Distributions” Measurement Model: Y= f(X1, X2, XN) Assign probability density function (PDF) to each X Select M for the number of Monte Carlo trials Generate M vectors by sampling from the PDF of each X (x1,1, x1,2, x1,M) (xN,1, xN,2, xN,M) Calculate M model values y = (f(x1,1, xN,1), f(x1,M, xN,M)) Estimate y of Y and associated standard uncertainty u(y) Calculate the interval [ylow,yhigh] for Y with corresponding coverage probability p 4 Standards and Calibration Laboratory, SCL Monte Carlo Method (MCM) gX1(x1) gX2(x2) Y = f (X) gY(h) gXN(xN) 5 Standards and Calibration Laboratory, SCL Operation Modes For MCM • There are three modes of operations – Fixed-Number-of-Trials Mode – Adaptive Mode – Approximated Adaptive (or Histogram) Mode 6 Standards and Calibration Laboratory, SCL Adaptive Monte Carlo Procedure Calculate sy, su(y), syhigh, sylow Set number of significant digit ndig (usually = 2) Use h M model values calculate u(y) and then d Set M = max (100/(1-p), 104) h=1 Y Perform MCM trial y(h)=(y1(h),y2(h),...yM(h)) Any 2sy, 2su(y), 2sylow or 2syhigh > d/5? N Calculate u(y(h)), ylow(h), yhigh(h) h>1? N Y Use h M model values calculate y, u(y), ylow, yhigh h=h+1 7 Standards and Calibration Laboratory, SCL Validation of GUF • Calculate: dlow = y – Up – ylow and dhigh = y + Up – yhigh • If both differences are not larger than d, then the GUF is validated. 8 Standards and Calibration Laboratory, SCL Histogram Procedure • If the numerical tolerance d is small, the value of M required would be larger. This may causes efficiency problems for some computers • Experiences show that a very precise measurement will require a M of up to 107 • Using histogram to approximate the PDF 9 Standards and Calibration Laboratory, SCL Histogram Procedure 1. 2. Build the initial histogram for y with Bin = 100,000 Continue generate the model Update y and u(y) for each iteration Check stabilization. (Same as the adaptive procedure, i.e. check the four s values) Update the histogram Store the outliers (i.e. those values beyond the boundaries of the histogram) 10 Standards and Calibration Laboratory, SCL Histogram Procedure 4. When stabilized: Build complete histogram to include the outliers Transform the histogram to a distribution function Use this discrete approximation to calculate the coverage interval 11 Standards and Calibration Laboratory, SCL Determine Coverage Intervals • By Inverse linear interpolation [Annex D.5 to D.8 of GS1] GY(h) p ylow = y(r ) + (y(r+1) - y(r ) ) pr+1 a pr+1 - pr Similary pr y(r+1) ylow y(r) y GY(h) p ps+1 p+a ps y(s+1) yhigh y(s) 12 a - pr p + a - ps yhigh = y(s) + (y(s+1) - y(s) ) ps+1 - ps y Standards and Calibration Laboratory, SCL Shortest Coverage Interval • Repeat the method to determine a large number of intervals corresponding to (a, p+a) and find the minimum value. E.g. a = 0 to 0.05 for 95 % coverage interval. • The precision level is related to the incremental step of a in the search. • The step uses in this software is 0.0001, i.e. total 501 steps. 13 Standards and Calibration Laboratory, SCL MCM Software MCM Engine (MATLAB Codes) MCM Code Generator (VB Codes) MATLAB Script File - Perform MCM Procedure -GUF Validation -Graphical Output of Results 14 Standards and Calibration Laboratory, SCL GUI of the MCM Code Generator 15 Standards and Calibration Laboratory, SCL Results for example 9.4.3.2 of GS1 • PDF for the y values in histogram • GUF Gaussian/t-distribution • Coverage Intervals • MCM and GUF results for y, u(y), ylow and yhigh • GUF validation result • Number of MCM trials 16 Standards and Calibration Laboratory, SCL Example Calibration of a 10 V Zener Voltage Reference using Josephson Array Voltage Standard n f V L V O V m V ran • Measurement Model: KJ • PDF parameters input to the software: y Input Quantity 17 PDF Parameter Symbol Description PDF / Constant n Quantum (Step) Number constant 63968 f Frequency N(, 2) 75.6 GHz KJ Josephson Constant constant 483597.9 GHz/V VL Leakage VO v a b R(a,b) -5 nV 5 nV Offset R(a,b) -0.1 V 0.1 V Vm Null Voltage R(a,b) 3.722 V 3.712 V 3.732 V Vran Random Noise tv(, 2) 0V 5.13 Hz 30 nV 39 Standards and Calibration Laboratory, SCL Parameters Input to the MCM Code Generator 18 Standards and Calibration Laboratory, SCL Results 19 Method y (V) u(y) (nV) ylow (nV) yhigh (nV) GUF 10 67 -131 +131 MCM1 (Fixed number) 10 66 -120 +120 MCM2 (Adaptive) 10 66 -120 +120 MCM3 (Histogram) 10 66 -120 +120 Method dlow (nV) dhigh (nV) GUF validated? No. of Trials Computation Time (s) MCM1 (Fixed number) -11 +11 No 1,000,000 <2 MCM2 (Adaptive) -11 +11 No 6,210,000 89 MCM3 (Histogram) -11 +11 No 6,270,000 8 Computer Configurations: Windows XP; MATLAB R2008b (version 7.7); CPU: Core Due T5600, 1.83 GHz, 2 GB Ram, 80 GB Harddisk Standards and Calibration Laboratory, SCL Thank You 20 Standards and Calibration Laboratory, SCL