IEO [so GUIDE 98-3/Suppl.1 Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) Supplement 1: Propagation of distributions using a Monte Carlo method First edition 2008 @ISO/IEC2008 Copyright Intemational Organization for Standardization without icense from IHS Not for Resale ISO/IECGUIDE98-3/Suppl.1:2008(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat accepts no liability in this area. Adobe is a trademark of Adobe Systems Incorporated. Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation COPYRIGHTPROTECTEDDOCUMENT @ ISO/IEC2008 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, ISO's member body in the country of the requester. ISO copyright office Case postale 56 : CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail [email protected] Web www.iso.org Published in Switzerland @ ISO/IEC 2008 All rights reserved Not fr Resale d without license from IHS ISO/IECGUIDE98-3/Suppl.1:2008(E) Contents Page Foreword... Introduction. .vi 1 Scope. 2 Normative references.. 3 Terms and definitions. 4 Conventions and notation .... 5 Basic principles... .8 5.1 Main stages of uncertainty evaluation .... .8 5.2 5.3 Obtaining summary information... ..9 5.4 Implementations of the propagation of distributions.... 10 5.5 Reportingtheresults.. 11 5.6 GuMm uncertainty framework .... 5.7 Conditions forvalid application of the GUM uncertaintyframeworkforlinear models 5.8 Conditions for valid application of the GUM uncertainty framework for non-linear models .....14 5.9 Monte Carlo approach to the propagation and summarizing stages ... 5.10 Conditions for the valid application of the described Monte Carlo method........6. 5.11 Comparison of the GUM uncertainty framework and the described Monte Carlo method ... 6 Probability density functions for the input quantities... 18 6.1 General... 18 6.2 Bayes' theorem.... 6.3 Principle of maximum entropy... ..19 6.4 Probabilitydensityfunctionassignmentforsomecommoncircumstances .20 6.4.1 General....... .20 6.4.2 Rectangulardistributions............. 6.4.3 Rectangular distributionswith inexactlyprescribed limits.. ..20 6.4.4 Trapezoidal distributions.... 22 6.4.5 6.4.6 Arc sine (U-shaped) distributions... 6.4.7 Gaussian distributions. 25 6.4.8 Multivariate Gaussian distributions... 25 6.4.9 .distributions..... 6.4.10 Exponential distributions ..28 6.4.11 Gamma distributions... 28 6.5 Probability distributions from previous uncertainty calculations .29 7 ImplementationofaMonteCarlomethod.. .29 7.1 General ....... 7.2 Number of Monte Carlo trials .. 29 7.3 Sampling from probability distributions.. 29 7.4 Evaluation of the model...... 7.5 Discrete representation of the distribution function for the output quantity 30 7.6 Estimate of the output quantity and the associated standard uncertainty. 31 7.7 Coverage interval for the output quantity.... 7.8 Computation time......... 7.9 Adaptive Monte Carlo procedure. 32 7.9.1 General.... 32 7.9.2 Numerical tolerance associated with a numerical value.. 7.9.3 Objective of adaptive procedure..... ili hout icense from IHS Not for Resale ISO/IECGUIDE98-3/Suppl.1:2008(E) 7.9.4 Adaptiveprocedure .33 8 Validation of results 8.1 Validationof theGUM uncertaintyframework usinga MonteCarlomethod, .35 8.2 Obtaining results fromaMonte Carlomethodforvalidationpurposes .35 9 Examples ..36 9.1 Illustrations of aspects of this Supplement.... 9.2 ..37 Additive model... 9.2.1 Formulation.. .37 9.2.2 Nor