Evaluation of Measurement Uncertainty using Adaptive Monte Carlo Methods

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1 Evaluation of Measurement Uncertainty using Adaptive Monte Carlo Methods Gerd Wübbeler, Peter M. Harris, Maurice G. Cox, Clemens Elster ) Physikalisch-Technische Bundesanstalt (PTB) ) National Physical Laboratory (NPL) Emerging Topics in Mathematics for Metrology From Measurement Uncertainty to Metrology of Complex Systems Physikalisch-Technische Bundesanstalt (PTB) - June 00, Berlin, Germany

2 Content Evaluation of measurement uncertainty according to GUM S GUM S adaptive Monte Carlo scheme Alternative approach: Stein s Two-stage scheme

3 GUM Supplement (GUM S) PDF based method Numerical evaluation by a Monte Carlo Method (MCM) Probability density function (PDF) Measured data probability PDF density Standard uncertainty Unsicherheit Further information Schätzwert (Messergebnis) Estimate Messgröße Measurand 3

4 Propagation of distributions PDFs for input quantities Measurement model PDF for measurand Change-of-variables 4

5 GUM S Monte Carlo Method (MCM) Model Y = f ( X, K, X N ) PDF of input quantities g X, K, X N ( ξ K, ξ ), N ( ) ξ, K,ξ N random draw from g X X ( ξ, ξn ), K, N, K η = f ( ξ, K, ξ N ) evaluation of measurement model g Y ( η) = g ( ξ) δ f X [ η ( ξ) ] dξ η random sample from g Y ( η) many repetitions PDF ( ) η g Y 5

6 Illustration trials Y = X X X 3 6

7 Convergence X i ~ N(0,) Law of large numbers 7

8 Repetition of the MCM calculation X i ~ N(0,) MCM results exhibit random variations 8

9 GUM S Adaptive Monte Carlo scheme (7.9) Goal Estimation of the expectation with accuracy with a coverage probability of about 95 %. y δ Sequential batch-processing mode (e.g trials per batch) y i mean of the trials within batch i y i for sufficiently large batch size Gaussian distributed (central limit theorem) ( y, K, y h ) y( h) s y ( h) = h h i= = h y i h i= ( y i y( h)) 9

10 GUM S Adaptive Monte Carlo scheme (7.9) Start: Batch and y, y h = Stopping-rule s y ( h) h δ yes y ˆ = y( h) no h = h + (new batch) y h 0

11 Assessment of the adaptive schemes Model Estimates Uncertainties Y = X + X x u = x = ( x ) = u( x ) = Gaussian distributions (uncorrelated) Rice distribution probability density y =.8 9 u(y) = Y

12 Assessment of the adaptive schemes Goal Determination of the expectation value of the Rice distribution with an accuracy of δ = (95 %) Assessment Adaptive scheme is repeatedly executed times Result per run Estimate of the expectation value Number of required batches

13 Assessment of the GUM S adaptive scheme Distribution of 0 5 estimates y [ y δ, y + δ ] Only 80 % of the results found in the specified accuracy interval 95 % Interval (Batch size 0 4, δ=0.005) ŷ 3

14 Assessment of the GUM S adaptive scheme Distribution of the number of batches Large number of early terminations at h = batches (about 3 %) (Batch size 0 4, δ=0.005) 4

15 Reasons for the behavior of GUM S adaptive scheme Sequential estimation scheme Random sample size (random number of batches) Multiple (dependent) testing for termination of sampling Confidence level of GUM S stopping rule adequate for fixed sample size Multiple testing and random sample size not taken into account Resulting confidence level does not necessarily meet the confidence level applied in individual test 5

16 Alternative Adaptive Monte-Carlo scheme Goal Carry out the MCM until a prescribed accuracy is achieved at a specified confidence level (with lowest possible numerical effort) 6

17 Stein s Two-stage scheme Given y, y,k i.i.d. from N ( µ, σ ), µ, σ unknown Goal h and y( h) = h h i= y i so that [ y ( h ) δ, y ( h ) + δ ] is a confidence interval for µ at confidence level -α Step Make h > random draws y, K, y h Variance s y ( h ) = h h i= ( y i y( h )) 7

18 Two-stage scheme Step h Number of additionally required drawings h = ( t ) s y( h ) h, α / max h δ +, 0 h > 0 make h further random draws h = 0 no additional random draws are made y ( h ) + h Pr( y( h + h ) δ µ y( h + h ) + δ ) α Proof C Stein, 945, Ann. Math. Statist

19 Application of the two-stage scheme within GUM S Sequential batch-processing mode y i u i ( y ) Mean of the trials in batch i Variance of the trials in batch i unbiased sufficiently large batch size (CLT) y i, ( y ) u i approximately Gaussian distributed Two-stage scheme applicable for Estimate y Squared uncertainty u ( y ) 9

20 Assessment of the adaptive schemes Model Estimates Uncertainties Y = X + X x u = x = ( x ) = u( x ) = Gaussian distributions (uncorrelated) Rice distribution probability density y =.8 9 u(y) = Y 0

21 Assessment of the two-stage scheme Distribution of 0 5 estimates y [ y δ, y + δ ] Interval covers 95 % of the estimates ŷ (h =0, Batch size 0 3, δ=0.005, α=0.05)

22 Assessment of the two-stage scheme Distribution of the number of batches No early terminations (h =0, Batch size 0 3, δ=0.005, α=0.05)

23 Assessment of the two-stage scheme Determination of the estimate of the measurand: dependence on batch size Batch size 0 00 Success rate 0.95(0.00) 0.95(0.00) 0.95(0.00) ANT * 45 80() 46 76(8) 46 43(8) (0.00) (8) (0.00) 57 7(96) ( h = 0, δ = 0.005, α = 0.05 ) Result for a requested confidence level of 99.9 % ( α = 0.00, Batch size 000) Success rate (0.000 ) ANT (97) * ) ANT: Average Number of Trials 3

24 Summary: Adaptive schemes for GUM S GUM S adaptive scheme does not (intend to) meet a 95 % confidence level Alternative approach: Two-stage scheme Attains specified confidence level for a Gaussian distribution (Proof by C. Stein, important for metrological applications) Applicable for the estimate and the squared uncertainty Allows prediction of computation time (number of trials) Wübbeler, Harris, Cox, Elster Metrologia 47 (00)

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