Evaluation of Measurement Uncertainties Using the Monte Carlo Method

1. 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. 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: yku(y)

3. 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. 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

5. Monte Carlo Method (MCM)

6. Operation Modes For MCM • There are three modes of operations • Fixed-Number-of-Trials Mode • Adaptive Mode • Approximated Adaptive (or Histogram) Mode

7. Adaptive Monte Carlo Procedure

8. Validation of GUF • Calculate: dlow = y – Up – ylow  and dhigh = y + Up – yhigh  • If both differences are not larger than , then the GUF is validated.

9. 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 107 • Using histogram to approximate the PDF

10. Histogram Procedure • 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)

11. Histogram Procedure • 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. Determine Coverage Intervals • By Inverse linear interpolation [Annex D.5 to D.8 of GS1]

13. 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 0.0001, i.e. total 501 steps.

14. MCM Software

15. GUI of the MCM Code Generator

16. 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

17. ExampleCalibration of a 10 V Zener Voltage Reference using Josephson Array Voltage Standard • Measurement Model: • PDF parameters input to the software:

18. Parameters Input to the MCM Code Generator

19. Results Computer Configurations: Windows XP; MATLAB R2008b (version 7.7); CPU: Core Due T5600, 1.83 GHz, 2 GB Ram, 80 GB Harddisk

20. Thank You