1 Standards and Calibration Laboratory, SCL 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

2 Standards and Calibration Laboratory, SCL GUM Uncertainty Framework (GUF) “Propagation of Uncertainties”  Measurement Model: Y= f(X 1, X 2,  X N )  Estimate x i of the input quantities X i  Determine u(x i ) associated with each estimate x i and its degrees of freedom  Estimate y = f(x i ) of Y  Calculate the sensitivity coefficient of each x i at X i = x i  Calculate u(y)  Calculate the effective degrees of freedom v eff and coverage factor k with coverage probability p  Calculate the coverage interval: y  k  u(y)

3 Standards and Calibration Laboratory, SCL GUM Uncertainty Framework (GUF) Problems: 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

4 Standards and Calibration Laboratory, SCL Monte Carlo Method (MCM) “Propagation of Distributions”  Measurement Model: Y= f(X 1, X 2,  X N )  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 (x 1,1, x 1,2,  x 1,M )  (x N,1, x N,2,  x N,M )  Calculate M model values y = (f(x 1,1,  x N,1 ),  f(x 1,M,  x N,M ))  Estimate y of Y and associated standard uncertainty u(y)  Calculate the interval [y low,y high ] for Y with corresponding coverage probability p

5 Standards and Calibration Laboratory, SCL Monte Carlo Method (MCM)

6 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

7 Standards and Calibration Laboratory, SCL Adaptive Monte Carlo Procedure

8 Standards and Calibration Laboratory, SCL Validation of GUF Calculate: d low =  y – U p – y low  and d high =  y + U p – y high  If both differences are not larger than , then the GUF is validated.

9 Standards and Calibration Laboratory, SCL Histogram Procedure If the numerical tolerance  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 10 7 Using histogram to approximate the PDF

10 Standards and Calibration Laboratory, SCL Histogram Procedure 1.Build the initial histogram for y with Bin = 100,000 2.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)

11 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

12 Standards and Calibration Laboratory, SCL Determine Coverage Intervals By Inverse linear interpolation [Annex D.5 to D.8 of GS1]

13 Standards and Calibration Laboratory, SCL Shortest Coverage Interval Repeat the method to determine a large number of intervals corresponding to ( , p+  ) and find the minimum value. E.g.  = 0 to 0.05 for 95 % coverage interval. The precision level is related to the incremental step of  in the search. The step uses in this software is , i.e. total 501 steps.

14 Standards and Calibration Laboratory, SCL MCM Software

15 Standards and Calibration Laboratory, SCL GUI of the MCM Code Generator

16 Standards and Calibration Laboratory, SCL Results for example of GS1 PDF for the y values in histogram GUF Gaussian/t-distribution Coverage Intervals MCM and GUF results for y, u(y), y low and y high GUF validation result Number of MCM trials

17 Standards and Calibration Laboratory, SCL Example Calibration of a 10 V Zener Voltage Reference using Josephson Array Voltage Standard Measurement Model: PDF parameters input to the software: Input QuantityPDF Parameter SymbolDescriptionPDF / Constant  vab nQuantum (Step) Number constant63968 fFrequency N( ,  2 ) 75.6 GHz5.13 Hz KJKJ Josephson Constant constant GHz/V VLVL LeakageR(a,b)-5 nV5 nV VOVO OffsetR(a,b) -0.1  V0.1  V VmVm Null VoltageR(a,b)  V3.712  V3.732  V V ran Random Noise t v ( ,  2 ) 0 V30 nV39

18 Standards and Calibration Laboratory, SCL Parameters Input to the MCM Code Generator

19 Standards and Calibration Laboratory, SCL Results Methody (V)u(y) (nV)y low (nV)y high (nV) GUF MCM1 (Fixed number) MCM2 (Adaptive) MCM3 (Histogram) Methodd low (nV) d high (nV) GUF validated?No. of TrialsComputation Time (s) MCM1 (Fixed number) No1,000,000< 2 MCM2 (Adaptive) No6,210,00089 MCM3 (Histogram) No6,270,0008 Computer Configurations: Windows XP; MATLAB R2008b (version 7.7); CPU: Core Due T5600, 1.83 GHz, 2 GB Ram, 80 GB Harddisk

20 Standards and Calibration Laboratory, SCL Thank You