# BS ISO 13752-1998 (1999)

BRITISH STANDARD BS ISO 13752:1998 Air quality — Assessment of uncertainty of a measurement method under field conditions using a second method as reference ICS 13.040.01BSISO 13752:1998 This British Standard, having been prepared under the directionof the Health and Environment Sector Board, waspublished under the authorityof the Standards Boardand comes into effect on 15 June 1998 © BSI 05-1999 ISBN 0 580 29704 7 National foreword This British Standard reproduces verbatim ISO13752:1998 and implements it as the UK national standard. The UK participation in its preparation was entrusted by Technical Committee EH/2, Air quality, to Subcommittee EH/2/4, General aspects, which has the responsibility to: — aid enquirers to understand the text; — present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep the UK interests informed; — monitor related international and European developments and promulgate them in the UK. A list of organizations represented on this subcommittee can be obtained on request to its secretary. Cross-references The British Standards which implement international or European publications referred to in this document may be found in the BSI Standards Catalogue under the section entitled “International Standards Correspondence Index”, or by using the “Find” facility of the BSI Standards Electronic Catalogue. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i and ii, theISOtitlepage, pagesii toiv, pages1 to12 and a back cover. This standard has been updated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on the inside front cover. Amendments issued since publication Amd. No. Date CommentsBSISO 13752:1998 © BSI 05-1999 i Contents Page National foreword Inside front cover Foreword iii Text of ISO 13752 1ii blankBS ISO13752:1998 ii © BSI 05-1999 Contents Page Foreword iii Introduction 1 1 Scope 1 2 Normative references 1 3 Symbols and abbreviated terms 1 4 Principle 2 5 Requirements 3 6 Parallel measurements 4 7 Graphical analysis of dispersion 4 8 Estimation of coefficients of regression model 4 9 Estimation of measurement uncertainty 7 Annex A (informative) Template of spreadsheet to calculate regressionandvariance function 8 Annex B (informative) Example of spreadsheet to calculate regressionandvariance function 10 Annex C (informative) Bibliography 12 Figure 1 — Flow scheme of maximum likelihood regression 3 Figure B.1 — Example of window Solver Parameters 10 Table A.1 — Formulae of the blank spreadsheet form 8 Table A.2 — Text blank spreadsheet from 9 Table B.1 — Spreadsheet example 11 Descriptors: Air, quality, air pollution, tests, field tests, measuring techniques, estimation, measurement uncertainty, rules of calculation.BS ISO13752:1998 © BSI 05-1999 iii Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75% of the member bodies casting a vote. International Standard ISO13752 was prepared by Technical Committee ISO/TC146, Air quality, Subcommittee SC4, General aspects. Annex A, Annex B and Annex C of this International Standard are for information only.iv blankBSISO 13752:1998 © BSI 05-1999 1 Introduction Performance characteristics for air quality measuring methods are defined in ISO6879. Corresponding test procedures are given in ISO9169 except for accuracy, which is dealt with in this International Standard as measurement uncertainty following the concepts of the Guide to the expression of uncertainty in measurement [5]. The measurement uncertainty under field conditions is also covered in ISO7935 and ISO10849. However, the procedure given in these International Standards is limited to either the determination of a concentration-independent systematic deviation, assuming a concentration-independent dispersion, or a concentration-proportional systematic deviation, assuming a concentration-proportional dispersion. 1 Scope This International Standard specifies a method for assessing the measurement uncertainty of a calibrated measurement method (test method) applied under field conditions using a second method as a reference (reference method). The reference method may not necessarily be a legally prescribed measurement method. The measurement uncertainty is derived from measurements made in parallel on real samples by comparing the measured values of the test method with those of the reference method. The result is only valid within the range of the measurements obtained. The test is designed especially for method validation. 2 Normative references The following standards contain provisions which, through reference in this text, constitute provisions of this International Standard. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. However, parties to agreements based on this International Standard are encouraged to investigate the possibility of applying the most recent editions of the normative documents indicated below. For undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain registers of currently valid International Standards. ISO 6879:1995, Air quality —Performance characteristics and related concepts for air quality measuring methods. ISO 9169:1994, Air quality —Determination of performance characteristics of measurement methods. 3 Symbols and abbreviated terms For the purposes of this International Standard, the following symbols and abbreviated terms apply. a 0 , a 1 , a 2 coefficients of the variance function AQC Air Quality Characteristic (usually concentration) b 0 , b 1 coefficients of the linear regression function or calibration function F F statistic k coverage factor L likelihood l logarithmic likelihood N, N 1 , N 2 number of pairs (x i , y i ) and number of pairs of subpopulation1 and2 respectively P(y i ) probability of y i r i residual at x i s, s i standard deviation as a function of AQC and at value x iof AQC respectively s9 transformed standard deviation as a function of AQC s a0 , s a1 , s a2 standard deviation of a 0 , a 1and a 2respectively s b0 , s b1 standard deviation of b 0and b 1respectively s x , s y standard deviation of all values x iand y irespectively s ycor standard deviation of the measured y-value after correction for the systematic error (bias) s% Y standard deviation (uncertainty) of the systematic error (bias) U expanded uncertainty (coverage factork =2) as a measure of measurement uncertainty X variable of x-method x, x i value of AQC and i-th value of AQC respectively x9 i transformed value x i , mean and weighted mean of all values x iand mean of all values y irespectively Y variable of y-method y i measured value of y-method at x ior output value of y-method at x i x, x ˚ , yBSISO 13752:1998 2 © BSI 05-1999 4 Principle A number N of pairs of measured values [(x 1 , y 2 ), ., (x N , x N )] are obtained from parallel field measurements. The measured values from the reference method (x-method) are considered as true values. The difference between the values of a measurement pair is attributed to measurement deviation of the test method (y- method). It is assumed that there is a linear relationship between the X and Y variable estimated by: The regression coefficients b 0and b 1can be calculated arithmetically if one of the following assumptions on dispersion of the y-values holds: — standard deviation of the test method is independent of x (i.e. standard deviation is constant) and estimated by: — standard deviation of the test method is proportional to x (i.e. coefficient of variation is constant) and estimated by: NOTE 1The first assumption can be considered as fluctuations of the background or intercept value b 0without fluctuations of the slope b 1and the second as fluctuations of the slope without fluctuations of the background or intercept value. NOTE 2The value of the coefficients of the regression function (estimation of bias) is not seriously affected by deviations from the assumption on the standard deviation. However, the estimated random part of measurement uncertainty heavily depends on the assumption. The general variance function used in this International Standard accounts not only for the variability of intercept and slope but also for statistical noise, the standard deviation of which is proportional to the square root of the value itself (approximately proportional to the square root of x): NOTE 3Coefficients have been taken as squares because the coefficient rather than its square reflects the physical meaning. NOTE 4The calculation procedure of the general variance function according to ISO9169 cannot be used because repetitive measurements are not available. The coefficients of this model [b 0and b 1of equation(1) and a 0 , a 1and a 2of equation (4)] cannot be calculated arithmetically. They are estimated iteratively on the criterion of maximum likelihood as an indicator of best fit (see Figure 1). After selecting a set of start values for the coefficients and using the assumption on normality, the probability,P(y i ), of every data point, (x i , y i ), belonging to the line can be calculated: The likelihood L is the mathematical product of the individual probabilities of y-values: The likelihood L is the indicator of fit. The coefficients are changed and the likelihood computed until a maximum value for L is obtained. The corresponding coefficients are the most likely coefficients for the regression model. For the determination by maximum likelihood a computerized optimization procedure is necessary. The uncertainty of a measured value for any AQC value is derived from the regression function and the variance function respectively. y9 i transformed value y i estimated value of Y at value x of AQC estimated value of Y at value x iof AQC y cor measured value of the y-method after correction for the systematic error (bias) %y systematic error (bias) at value x of AQC … random number from normal distribution with central value0 and standard deviation1 6 i weighting factor at x i .(1) .(2) .(3) y ˆ y i ˆ .(4) .(5) .(6)BSISO 13752:1998 © BSI 05-1999 3 5 Requirements 5.1 General The method described in this International Standard requires that: — a linear relationship exists between the variables compared; if the relationship is different but mathematically known the procedure may be adapted. — the measurement errors of the test method are normally distributed. — the measurement uncertainty of the reference method is insignificant compared to that of the test method; if not, it is falsely attributed to the test method and this will result in overestimating its measurement uncertainty. — the impact of differences in composition between the air sampled by the two methods is negligible compared to the expected uncertainty of the test method; if not, this error component is falsely attributed to the test method and will result in overestimating measurement uncertainty. The uncertainty of the coefficients calculated by this International Standard will be reduced by increasing the number of measurement pairs. Therefore, it is recommended that at least30 measurement pairs be obtained if the general variance model described in this International Standard is applied. Figure 1 — Flow scheme of maximum likelihood regressionBSISO 13752:1998 4 © BSI 05-1999 5.2 Test method (y-method) Specify all steps of the measurement method which will be subject to the assessment and execute the measurements as specified. 5.3 Reference method (x-method) In view of the provisions and environmental conditions, e.g. interfering, substances, temperature etc., to be expected at the test site, investigate whether the assumption is justified that the x-method yields insignificant uncertainty compared to the y-method. This investigation may be based on the properties of the measurement principle, literature data or results of laboratory tests or field tests. Describe the x-method in detail and execute the measurements accordingly. 5.4 Test conditions Make sure that the test conditions resemble the conditions under which the test method is going to be used (test period, range of the air quality characteristic, range of physical and chemical influence variables, and operational conditions). Describe the provisions and the environmental conditions at the test site. The measurement equipment of both methods should be installed so that: — the difference in composition between parallel samples is insignificant and, — the equipment for one method does not influence that for the other. 5.5 Data processing In the case of the general variance model, data processing requires computer facilities for finding the maximum fit (likelihood) by adjusting the values of a 0 , a 1 , a 2 , b 0and b 1 . 6 Parallel measurements Execute parallel measurements representative of the conditions under which the test method is going to be used. Record the pairs of measured values. 7 Graphical analysis of dispersion The dispersion of measured values of a method is either constant or increases with the AQC value. An impression of the variance as a function of the AQC value can easily be obtained graphically by plotting, for all data pairs (x i , y i ), the absolute residuals| r i| against x iwhere r i= y i– and is the predicted value for a conventional linear least-square fit: — if the values of the residuals are independent of x igo to 8.2; — if the values of the residuals are proportional to x igo to 8.3; — if the values of the residuals are neither independent of nor proportional to x igo to 8.4. In those cases where the first or second relationships only applies to part of the range, the range must be curtailed accordingly. The coefficients of the regression function and variance function in 8.2 and 8.3 can be calculated arithmetically (simple variance model). Those of 8.4 require iterative computati