ECA EIA 61710-2017 Power Law Model - Goodness-of- Fit Tests and Estimation Methods.pdf

1、 EIA STANDARD Power Law Model - Goodness-of-Fit Tests and Estimation Methods EIA 61710 (IEC 61710:2013 Ed.2.0, IDT) May 2017 EIA 61710 ANSI/EIA 61710-2017 Approved: May 11, 2017 NOTICE EIA Engineering Standards and Publications are designed to serve the public interest through eliminating misunderst

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5、 or Publication. This EIA Standard is identical (IDT) with the International Standard IEC Publication 61710:2013: Power Law Model - Goodness-of-Fit Tests and Estimation Methods. This document is the EIA Standard EIA 61710 Edition 2.0: Power Law Model - Goodness-of-Fit Tests and Estimation Methods. T

6、he text, figures and tables of IEC 61710:2013 are used in this Standard with the consent of the IEC and the American National Standards Institute (ANSI). The IEC copyrighted material has been reproduced with permission from ANSI. The IEC Foreword and Introduction are not part of the requirements of

7、this standard but are included for information purposes only. This Standard does not purport to address all safety problems associated with its use or all applicable regulatory requirements. It is the responsibility of the user of this Standard to establish appropriate safety and health practices an

8、d to determine the applicability of regulatory limitations before its use. (From Standards Proposal No. 5372.05, formulated under the cognizance of the EIA Dependability Standards Committee). Published by Electronic Components Industry Association 2017 Standards any IEC National Committee interested

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16、other IEC Publications. 8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is indispensable for the correct application of this publication. 9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be

17、the subject of patent rights. IEC shall not be held responsible for identifying any or all such patent rights. International Standard IEC 61710 has been prepared by IEC technical committee 56: Dependability. This second edition cancels and replaces the first edition, published in 2000, and constitut

18、es a technical revision. The main changes with respect to the previous edition are listed below: the inclusion of an additional Annex C on Bayesian estimation for the power law model. The text of this standard is based on the following documents: FDIS Report on voting 56/1500/FDIS 56/1508/RVD Full i

19、nformation on the voting for the approval of this standard can be found in the report on voting indicated in the above table. EIA 61710 Page 2 This publication has been drafted in accordance with the ISO/IEC Directives, Part 2. The committee has decided that the contents of this publication will rem

20、ain unchanged until the stability date indicated on the IEC web site under “http:/webstore.iec.ch“ in the data related to the specific publication. At this date, the publication will be reconfirmed, withdrawn, replaced by a revised edition, or amended. IMPORTANT The colour inside logo on the cover p

21、age of this publication indicates that it contains colours which are considered to be useful for the correct understanding of its contents. Users should therefore print this document using a colour printer. EIA 61710 Page 3 INTRODUCTION This International Standard describes the power law model and g

22、ives 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 pow

23、er 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 o

24、f the 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 presen

25、ts algorithms from which computer programs should be easy to construct. EIA 61710 Page 4 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 intervals for the fa

26、ilure 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 the same cond

27、itions (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 edition of the refe

28、renced 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 Symbols and abbrevi

29、ations 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 goodness-of-fit test

30、 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 failure EIA 61710 Pag

31、e 5 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 number of items L, U multipliers used in calculation

32、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 tN accumulated number of failures up to time t itN

33、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 test jT total accumulated relevant time for jth item RURLTT

34、, lower, upper prediction limits for the length of time to the Rth future failure TN1estimated median time to (N1)th failure it accumulated relevant time to the ith failure jit ith failure time for jth item tNtotal accumulated relevant time for failure terminated test jNt total accumulated relevant

35、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 intensity 5 Power law model The statistical procedures for the power law model use

36、 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 failures up to te

37、st time t is given by: EIA 61710 Page 6 ttNE with 0,0,0 t where is the scale parameter; is the shape parameter ( 10 corresponds to a decreasing failure intensity; 1 corresponds to a constant failure intensity; 1 corresponds to an increasing failure intensity). The failure intensity at time t is give

38、n by: 1)( ttNEdtdtz with 0t Thus the parameters and both affect the failure intensity in a given time. Methods are given in 7.2 for maximum likelihood estimation of the parameters of and . Subclause 7.3 gives goodness-of-fit tests for the model and 7.4 and 7.5 give confidence interval procedures. Su

39、bclause 7.6 gives prediction interval procedures and 7.7 gives tests for the equality of the shape parameters. The model is simple to evaluate. However when 1 , theoretically 0z (i.e. )(tz tends to infinity as t tends to zero) and 0)( z (i.e. )(tz tends to zero as t tends to infinity); but this theo

40、retical limitation does not generally affect its practical use. 6 Data requirements 6.1 General 6.1.1 Case 1 Time data for every relevant failure for one or more copies from the same population The normal evaluation methods assume the observed times to be exact times of failure of a single repairabl

41、e item or a set of copies of the same repairable item. The figures below illustrate how the failure times are calculated for three general cases. 6.1.2 Case 1a) One repairable item For one repairable item observed from time 0 to time T, the relevant failure time, it , is the elapsed operating time (

42、that is, excluding repair and other down times) until the occurrence of the i-th failure as shown in Figure 1. Key A operating time, B down time Figure 1 One repairable item BA0 123 T IEC 996/13 EIA 61710 Page 7 Time terminated data are observed to *T , which is not a failure time, and failure termi

43、nated data are observed to Nt , which is the time of the Nth failure. Time terminated and failure terminated data use slightly different formulae. 6.1.3 Case 1b) Multiple items of the same kind of repairable item observed for the same length of time It is assumed there are k items, which all represe

44、nt the same population. That is, they are nominally identical items operating under the same conditions (e.g. environment and load). When all items are observed to time *T , which is not a failure time (i.e. time terminated data), then the failure time data are combined by superimposing failure time

45、s Niti.,2,1, for all k items on the same time line as shown in Figure 2. Key A item 1 B item 2 C item k D superimposed process Figure 2 Multiple items of the same kind of repairable item observed for same length of time 6.1.4 Case 1c) Multiple repairable items of the same kind observed for different

46、 lengths of time When all items do not operate for the same period of time, then the time at which observation of the jth item is terminated kjTj,.,2,1 , where kTTT .21, is noted. The failure data are combined by superimposing all the failure times for all k items on the same time line as shown in F

47、igure 3. The times to failure are Niti,.,2,1, , where N = the total number of failures observed accumulated over the k items. D T* 0CBAt 1 t2 t3 tN-1 tNIEC 997/13 EIA 61710 Page 8 Key A item 1 B item 2 C item 3 D item k t time Figure 3 Multiple repairable items of the same kind observed for differen

48、t lengths of time If each item is a software system then the repair action should be done to the other systems which did not fail at that time. 6.2 Case 2 Time data for groups of relevant failures for one or more repairable items from the same population This alternative method is used when there is

49、 at least one copy of an item and the data consist of known time intervals, each containing a known number of failures. The observation period is over the interval ),0( T and is partitioned into d intervals at times )(.)2()1(0 dttt . The ith interval is the time period between )1( it and )(it , where 1,2,., , 0 0 idtand td T . It is important to note that the interval lengths and the numb