1、raising standards worldwideNO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAWBSI Standards PublicationDD ISO/TS 28037:2010Determination and use ofstraight-line calibrationfunctionsDD ISO/TS 28037:2010 DRAFT FOR DEVELOPMENTNational forewordThis Draft for Development is the UK impl
2、ementation of ISO/TS 28037:2010.This publication is not to be regarded as a British Standard.It is being issued in the Draft for Development series of publications and is of a provisional nature. It should be applied on this pro-visional basis, so that information and experience of its practical app
3、lication can be obtained.Comments arising from the use of this Draft for Development are requested so that UK experience can be reported to the international organization responsible for its conversion to an international standard. A review of this publication will be initiated not later than 3 year
4、s after its publication by the inter-national organization so that a decision can be taken on its status. Notification of the start of the review period will be made in an announcement in the appropriate issue of Update Standards.According to the replies received by the end of the review period, the
5、 responsible BSI Committee will decide whether to support the conversion into an international Standard, to extend the life of the T e c h n i c a l S p e c i f i c a t i o n o r t o w i t h d r a w i t . C o m m e n t s s h o u l d b e s e n t to the Secretary of the responsible BSI Technical Commi
6、ttee at British Standards House, 389 Chiswick High Road, London W4 4AL.The UK participation in its preparation was entrusted to Technical Committee SS/6, Precision of test methods.A list of organizations represented on this committee can be obtained on request to its secretary.This publication does
7、not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. BSI 2010ISBN 978 0 580 66144 0ICS 03.120.30Compliance with a British Standard cannot confer immunity from legal obligations.This Draft for Development was published under the authori
8、ty of the Standards Policy and Strategy Committee on 31 January 2011.Amendments issued since publicationDate T e x t a f f e c t e dDD ISO/TS 28037:2010Reference numberISO/TS 28037:2010(E)ISO 2010TECHNICAL SPECIFICATION ISO/TS28037First edition2010-09-01Determination and use of straight-line calibra
9、tion functions Dtermination et utilisation des fonctions dtalonnage linaire DD ISO/TS 28037:2010ISO/TS 28037:2010(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobes licensing policy, this file may be printed or viewed but shall not be edited unless the typeface
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14、.iso.org Published in Switzerland ii ISO 2010 All rights reservedDD ISO/TS 28037:2010ISO/TS 28037:2010(E)Contents PageForeword: vIntroduction : vi1 Scope : 12 Normative references : 13 Terms and de nitions : 14 Conventions and notation : 45 Principles of straight-line calibration: 55.1 General . . .
15、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 Inputs to determining the calibration function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2.1 Measurement data .
16、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2.2 Associated uncertainties and covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.3 Determining the calibration function . . . . . . .
17、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.4 Numerical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.5 Uncertainties and covariance associated with the calibratio
18、n function parameters . . . . . . . . . . . . . . . . . 75.6 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.7 Use of the calibration function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19、. . . . . . . . . . . . . . . . . . . . . . . . 85.8 Determining the ordinary least squares best- t straight line to data . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Model for uncertainties associated with the yi: 96.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.2 Calibration parameter estimates and associated standard uncertainties and covariance . . . . . . . . . . . . . 106.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21、. . . . . . . . . . . . . . . . . . . . . . . 116.4 Organization of the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Model for uncertainties associated with the xiand the yi: 177.1 General . . . . . . . . . . . . . . . . .
22、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 Calibration parameter estimates and associated standard uncertainties and covariance . . . . . . . . . . . . . 187.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . .
23、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Organization of the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Model for uncertainties associated with the xiand the yiand covariances assoc
24、iated with thepairs (xi;yi) : 248.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Calibration parameter estimates and associated standard uncertainties and covariance . . . . . . . . . . . .
25、 . 249 Model for uncertainties and covariances associated with the yi: 259.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Calibration parameter estimates and associated standard uncertainti
26、es and covariance . . . . . . . . . . . . . 259.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.4 Organization of the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27、 . . . . . . . . . . . . . . . . . . 2710 Model for uncertainties and covariances associated with the xiand the yi: 3110.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3110.2 Calibration paramete
28、r estimates and associated standard uncertainties and covariance . . . . . . . . . . . . . 3110.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411 Use of the calibration function : 3711.1 Prediction . .
29、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2 Forward evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39c ISO 2010
30、| All rights reserved iiiDD ISO/TS 28037:2010ISO/TS 28037:2010(E)AnnexesA (informative) Matrix operations: 41A.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Elementary operations . . . . .
31、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2.1 Matrix-vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2.2 Matrix-matrix multiplication . . . . . . .
32、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2.3 Matrix transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2.4 Identity matrix . . . . . . . . . . . . . . . . . . . . . . .
33、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2.5 Inverse of a square matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.3 Elementary de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.3.1 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.3.2 Invertible matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35、. . . . . . . . . . . . . . . . . . . . . . . 42A.3.3 Lower-triangular and upper-triangular matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.3.4 Orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36、 . . . . . . . . 42A.4 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.4.1 Cholesky factorization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37、. 42A.4.2 Interpretation of the Cholesky factorization of a covariance matrix . . . . . . . . . . . . . . . . . . . . . . 43A.4.3 Solution of a lower-triangular system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.4.4 Solution of an upper-triangular syst
38、em . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.5 Orthogonal factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.5.1 QR factorization . . . . . . . . . . . . . . . . . . .
39、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.5.2 RQ factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B (informative) Application of the Gauss-Newton algorithm to generalized d
40、istance regression : 46C (informative) Orthogonal factorization approach to solving the generalized Gauss-Markov prob-lem : 48C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48C.2 Calibration pa
41、rameter estimates and associated standard uncertainties and covariance . . . . . . . . . . . . . 48C.3 Validation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49D (informative) Provision of uncertainties and covari
42、ances associated with the measured x- andy-values : 52D.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52D.2 Response data 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52D.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52D.2.2 Measurement model for uncertainties and covariances associated with the yi. . . . .
44、. . . . . . . . . . 52D.3 Response data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53D.4 Stimulus data 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45、. . . . . . . . . . . . 53D.5 Stimulus data 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54D.6 Stimulus and response data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46、 . . . . . . . . . . . 55E (informative) Uncertainties known up to a scale factor : 56F (informative) Software implementation of described algorithms : 60G (informative) Glossary of principal symbols : 61Bibliography: 63iv c ISO 2010 | All rights reservedDD ISO/TS 28037:2010ISO/TS 28037:2010(E)ForewordISO (the International Organization for Standardization) is a worldwide federation of national stan
