evaluation of measurement uncertainties using the monte carlo method
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1Standards and Calibration Laboratory, SCL
Evaluation of Measurement Uncertainties Using the
Monte Carlo Method
Speaker: Chung Yin, PoonStandards and Calibration Laboratory (SCL)The Government of the Hong Kong Special
Administrative Region
2Standards 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 YCalculate 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 pCalculate the coverage interval: yku(y)
3Standards and Calibration Laboratory, SCL
GUM Uncertainty Framework (GUF)
Problems:The contributory uncertainties are not of
approximately the same magnitudeDifficult to provide the partial derivatives of the
modelThe PDF for output quantity is not a Gaussian
distribution or a scaled and shifted t-distribution
Y = f (X) y, u(y)
x1, u(x1)
x2, u(x2)
xN, u(xN)
4Standards 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 XSelect M for the number of Monte Carlo trialsGenerate 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
5Standards and Calibration Laboratory, SCL
Monte Carlo Method (MCM)
Y = f (X)gY(h)
gX1(x1)
gX2(x2)
gXN(xN)
6Standards 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
7Standards and Calibration Laboratory, SCL
Adaptive Monte Carlo ProcedureSet number of significant digit
ndig (usually = 2)
Set M = max (100/(1-p), 104)
h = 1
Perform MCM trial
y(h)=(y1(h),y2
(h),...yM(h))
Calculate u(y(h)), ylow(h), yhigh
(h)
h > 1 ?
h = h + 1
Calculate sy, su(y), syhigh, sylow
Use h M model values calculate u(y) and then d
Any 2sy, 2su(y), 2sylow or 2syhigh > d/5?
Use h M model values calculate y,
u(y), ylow, yhighN
Y
Y
N
8Standards and Calibration Laboratory, SCL
Validation of GUF
• Calculate: dlow = y – Up – ylow anddhigh = y + Up – yhigh
• If both differences are not larger than d, then the GUF is validated.
9Standards 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
10Standards and Calibration Laboratory, SCL
Histogram Procedure
1. Build the initial histogram for y withBin = 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)
11Standards 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
12Standards and Calibration Laboratory, SCL
Determine Coverage Intervals• By Inverse linear interpolation [Annex D.5 to D.8 of GS1]
y(r)
ylow
y(r+1)
pr
pr+1
a
y
p GY(h)
y(s)
yhigh
y(s+1)
y
p GY(h)
ps+1
ps
p+a
13Standards 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.
14Standards and Calibration Laboratory, SCL
MCM Software
MCM Code Generator
(VB Codes)
MATLAB Script File
MCM Engine(MATLAB Codes)
- Perform MCM Procedure-GUF Validation
-Graphical Output of Results
15Standards and Calibration Laboratory, SCL
GUI of the MCM Code Generator
16Standards 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
17Standards and Calibration Laboratory, SCL
ExampleCalibration of a 10 V Zener Voltage Reference using
Josephson Array Voltage Standard
• Measurement Model:• PDF parameters input to the software:
ranmOLJ
VVVVKfny
Input Quantity PDF ParameterSymbol Description PDF /
Constant v a b
n Quantum (Step) Number
constant 63968
f Frequency N(, 2) 75.6 GHz 5.13 Hz
KJ Josephson Constant
constant 483597.9 GHz/V
VL Leakage R(a,b) -5 nV 5 nV
VO 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) 0 V 30 nV 39
18Standards and Calibration Laboratory, SCL
Parameters Input to the MCM Code Generator
19Standards and Calibration Laboratory, SCL
Results
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
20Standards and Calibration Laboratory, SCL
Thank You
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