EN 61710-2013 en Power law model - Goodness-of-fit tests and estimation methods.pdf

1、BSI Standards PublicationPower law model Goodness-of-fit tests and estimation methodsBS EN 61710:2013National forewordThis British Standard is the UK implementation of EN 61710:2013. It is identical to IEC 61710:2013. It supersedes BS IEC 61710:2000 which is withdrawn.The UK participation in its pre

2、paration was entrusted to TechnicalCommittee DS/1, Dependability.A list of organizations represented on this committee can be obtained onrequest to its secretary.This publication does not purport to include all the necessary provisions ofa contract. Users are responsible for its correct application.

3、 The British Standards Institution 2013.Published by BSI Standards Limited 2013ISBN 978 0 580 75899 7ICS 03.120.30; 21.020; 29.020Compliance with a British Standard cannot confer immunity fromlegal obligations.This British Standard was published under the authority of theStandards Policy and Strateg

4、y Committee on 30 September 2013.Amendments/corrigenda issued since publicationDate Text affectedBRITISH STANDARDBS EN 61710:2013EUROPEAN STANDARD EN 61710 NORME EUROPENNE EUROPISCHE NORM September 2013 CENELEC European Committee for Electrotechnical Standardization Comit Europen de Normalisation El

5、ectrotechnique Europisches Komitee fr Elektrotechnische Normung CEN-CENELEC Management Centre: Avenue Marnix 17, B - 1000 Brussels 2013 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members. Ref. No. EN 61710:2013 E ICS 03.120.01; 03.120.30 English

6、version Power law model - Goodness-of-fit tests and estimation methods (IEC 61710:2013) Modle de loi en puissance - Essais dadquation et mthodes destimation des paramtres (CEI 61710:2013) Potenzgesetz-Modell - Anpassungstests und Schtzverfahren (IEC 61710:2013) This European Standard was approved by

7、 CENELEC on 2013-06-26. CENELEC members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national st

8、andards may be obtained on application to the CEN-CENELEC Management Centre or to any CENELEC member. This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CENELEC member into its own lang

9、uage and notified to the CEN-CENELEC Management Centre has the same status as the official versions. CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Croatia, Cyprus, the Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, Fran

10、ce, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the United Kingdom. BS EN 61710:2013EN 61710:2013 - 2 - Foreword The text of document 56/1500/FD

11、IS, future edition 2 of IEC 61710, prepared by IEC/TC 56 “Dependability“ was submitted to the IEC-CENELEC parallel vote and approved by CENELEC as EN 61710:2013. The following dates are fixed: latest date by which the document has to be implemented at national level by publication of an identical na

12、tional standard or by endorsement (dop) 2014-03-26 latest date by which the national standards conflicting with the document have to be withdrawn (dow) 2016-06-26 Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. CENELEC and/or CEN

13、shall not be held responsible for identifying any or all such patent rights. Endorsement notice The text of the International Standard IEC 61710:2013 was approved by CENELEC as a European Standard without any modification. In the official version, for Bibliography, the following notes have to be add

14、ed for the standards indicated: IEC 61703 NOTE Harmonised as EN 61703. IEC 61164:2004 NOTE Harmonised as EN 61164:2004 (not modified). BS EN 61710:2013- 3 - EN 61710:2013 Annex ZA (normative) Normative references to international publications with their corresponding European publications The follow

15、ing documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. NOTE When an inte

16、rnational publication has been modified by common modifications, indicated by (mod), the relevant EN/HD applies. Publication Year Title EN/HD Year IEC 60050-191 1990 International Electrotechnical Vocabulary (IEV) - Chapter 191: Dependability and quality of service - - BS EN 61710:2013 2 61710 IEC:2

17、013 CONTENTS INTRODUCTION . 7 1 Scope . 8 2 Normative references . 8 3 Terms and definitions . 8 4 Symbols and abbreviations . 8 5 Power law model 9 6 Data requirements 10 6.1 General . 10 6.1.1 Case 1 Time data for every relevant failure for one or more copies from the same population 10 6.1.2 Case

18、 1a) One repairable item . 10 6.1.3 Case 1b) Multiple items of the same kind of repairable item observed for the same length of time . 11 6.1.4 Case 1c) Multiple repairable items of the same kind observed for different lengths of time . 11 6.2 Case 2 Time data for groups of relevant failures for one

19、 or more repairable items from the same population . 12 6.3 Case 3 Time data for every relevant failure for more than one repairable item from different populations 12 7 Statistical estimation and test procedures . 13 7.1 Overview . 13 7.2 Point estimation 13 7.2.1 Case 1a) and 1b) Time data for eve

20、ry relevant failure . 13 7.2.2 Case 1c) Time data for every relevant failure 14 7.2.3 Case 2 Time data for groups of relevant failures . 15 7.3 Goodness-of-fit tests . 16 7.3.1 Case 1 Time data for every relevant failure. 16 7.3.2 Case 2 Time data for groups of relevant failures . 17 7.4 Confidence

21、intervals for the shape parameter 18 7.4.1 Case 1 Time data for every relevant failure. 18 7.4.2 Case 2 Time data for groups of relevant failures . 19 7.5 Confidence intervals for the failure intensity 20 7.5.1 Case 1 Time data for every relevant failure. 20 7.5.2 Case 2 Time data for groups of rele

22、vant failures . 20 7.6 Prediction intervals for the length of time to future failures of a single item 21 7.6.1 Prediction interval for length of time to next failure for case 1 Time data for every relevant failure 21 7.6.2 Prediction interval for length of time to Rth future failure for case 1 Time

23、 data for every relevant failure . 22 7.7 Test for the equality of the shape parameters k ., ,2123 7.7.1 Case 3 Time data for every relevant failure for two items from different populations 23 7.7.2 Case 3 Time data for every relevant failure for three or more items from different populations 24 Ann

24、ex A (informative) The power law model Background information 30 Annex B (informative) Numerical examples 31 BS EN 61710:201361710 IEC:2013 3 Annex C (informative) Bayesian estimation for the power law model 41 Bibliography 56 Figure 1 One repairable item . 10 Figure 2 Multiple items of the same kin

25、d of repairable item observed for same length of time 11 Figure 3 Multiple repairable items of the same kind observed for different lengths of time 12 Figure B.1 Accumulated number of failures against accumulated time for software system 32 Figure B.2 Expected against observed accumulated times to f

26、ailure for software system 32 Figure B.3 Accumulated number of failures against accumulated time for five copies of a system . 35 Figure B.4 Accumulated number of failures against accumulated time for an OEM product from vendors A and B . 37 Figure B.5 Accumulated number of failures against time for

27、 generators . 38 Figure B.6 Expected against observed accumulated number of failures for generators 39 Figure C.1 Plot of fitted Gamma prior (6,7956, 0,0448) 47 for the shape parameter of the power law model . 47 Figure C.2 Plot of fitted Gamma prior (17,756 6, 1447,408) for the expected number of f

28、ailures parameter of the power law model 47 Figure C.3 Subjective distribution of number of failures 51 Figure C.4 Plot of the posterior probability distribution for the number of future failures, M . 54 Figure C.5 Plot of the posterior cumulative distribution for the number of future failures, M .

29、55 Table 1 Critical values for Cramer-von-Mises goodness-of-fit test at 10 % level of significance. 25 Table 2 Fractiles of the Chi-square distribution 26 Table 3 Multipliers for two-sided 90 % confidence intervals for intensity function for time terminated data . 27 Table 4 Multipliers for two-side

30、d 90 % confidence intervals for intensity function for failure terminated data 28 Table 5 0,95 fractiles of the F distribution 29 Table B.1 All relevant failures and accumulated times for software system 31 Table B.2 Calculation of expected accumulated times to failure for Figure B.2 . 33 Table B.3

31、Accumulated times for all relevant failures for five copies of a system (labelled A, B, C, D, E) . 34 Table B.4 Combined accumulated times for multiple items of the same kind of a system 34 Table B.5 Accumulated operating hours to failure for OEM product from vendors A and B 36 Table B.6 Grouped fai

32、lure data for generators . 38 Table B.7 Calculation of expected numbers of failures for Figure B.6 . 40 Table C.1 Strengths and weakness of classical and Bayesian estimation . 42 BS EN 61710:2013 4 61710 IEC:2013 Table C.2 Grid for eliciting subjective distribution for shape parameter . 46 Table C.3

33、 Grid for eliciting subjective distribution for expected number of failures parameter 46 Table C.4 Comparison of fitted Gamma and subjective distribution for shape parameter 48 Table C.5 Comparison of fitted Gamma and subjective distribution for expected number of failures by time 20 000 hT = parame

34、ter 48 Table C.6 Times to failure data collected on system test 49 Table C.7 Summary of estimates of power law model parameters 50 Table C.8 Time to failure data for operational system 53 BS EN 61710:201361710 IEC:2013 7 INTRODUCTION This International Standard describes the power law model and give

35、s step-by-step directions for its use. There are various models for describing the reliability of repairable items, the power law model being one of the most widely used. This standard provides procedures to estimate the parameters of the power law model and to test the goodness-of-fit of the power

36、law model to data, to provide confidence intervals for the failure intensity and prediction intervals for the length of time to future failures. An input is required consisting of a data set of times at which relevant failures occurred, or were observed, for a repairable item or a set of copies of t

37、he same item, and the time at which observation of the item was terminated, if different from the time of final failure. All output results correspond to the item type under consideration. Some of the procedures can require computer programs, but these are not unduly complex. This standard presents

38、algorithms from which computer programs should be easy to construct. BS EN 61710:2013 8 61710 IEC:2013 POWER LAW MODEL GOODNESS-OF-FIT TESTS AND ESTIMATION METHODS 1 Scope This International Standard specifies procedures to estimate the parameters of the power law model, to provide confidence interv

39、als for the failure intensity, to provide prediction intervals for the times to future failures, and to test the goodness-of-fit of the power law model to data from repairable items. It is assumed that the time to failure data have been collected from an item, or some identical items operating under

40、 the same conditions (e.g. environment and load). 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest editi

41、on of the referenced document (including any amendments) applies. IEC 60050-191:1990, International Electrotechnical Vocabulary (IEV) Chapter 191: Dependability and quality of service 3 Terms and definitions For the purposes of this document, the terms and definitions of IEC 60050-191 apply. 4 Symbo

42、ls and abbreviations The following symbols and abbreviations apply: shape parameter of the power law model estimated shape parameter of the power law model UBLB , lower, upper confidence limits for 2C Cramer-von-Mises goodness-of-fit test statistic ()MC21 critical value for the Cramer-von-Mises good

43、ness-of-fit test statistic at level of significance 2 Chi-square goodness-of-fit test statistic ()2 th fractile of the 2 distribution with degrees of freedom d number of intervals for groups of failures ()tNE expected accumulated number of failures up to time t jtE expected accumulated time to jth f

44、ailure BS EN 61710:201361710 IEC:2013 9 ( ) itNEestimated expected accumulated number of failures up to ( )it jtEestimated expected accumulated time to jth failure ( )21,F th fractile for the F distribution with ( )21, degrees of freedom i general purpose indicator j general purpose indicator k numb

45、er of items L, U multipliers used in calculation of confidence intervals for failure intensity scale parameter of the power law model estimated scale parameter of the power law model M parameter for Cramer-von-Mises statistical test N number of relevant failures jN number of failures for jth item (

46、)tN accumulated number of failures up to time t ( ) itN accumulated number of failures up to time ( )it R difference between the order number of future (predicted) failure and order number of last (observed) failure T accumulated relevant time *T total accumulated relevant time for time terminated t

47、est jT total accumulated relevant time for jth item RURLTT , lower, upper prediction limits for the length of time to the Rth future failure TN+1estimated median time to (N+1)th failure it accumulated relevant time to the ith failure jit ith failure time for jth item tNtotal accumulated relevant tim

48、e for failure terminated test jNt total accumulated relevant time to Nth failure of jth item ( ) ( )itit ,1 endpoints of ith interval of time for grouped failures ( )tz failure intensity at time t ( )tzestimated failure intensity at time t UBLBzz , lower, upper confidence limits for failure intensit

49、y 5 Power law model The statistical procedures for the power law model use the relevant failure and time data from the test or field studies. The basic equations for the power law model are given in this clause. Background information on the model is given in Annex A and examples of its application are given in Annex B. The expected accumulated number of fa