# ISO 11843-7-2012

BS ISO 11843-7:2012© ISO 2012Capability of detection —Part 7: Methodology based on stochastic properties of instrumental noiseCapacité de détection —Partie 7: Méthodologie basée sur les propriétés stochastiques du bruit instrumentalINTERNATIONAL STANDARDISO11843-7First edition2012-06-01Reference numberISO 11843-7:2012(E)Licensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)ii © ISO 2012 – All rights reservedCOPYRIGHT PROTECTED DOCUMENT© ISO 2012All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO’s member body in the country of the requester.ISO copyright officeCase postale 56 • CH-1211 Geneva 20Tel. + 41 22 749 01 11Fax + 41 22 749 09 47E-mail copyright@iso.orgWeb www.iso.orgPublished in SwitzerlandLicensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)© ISO 2012 – All rights reserved iiiContents PageForeword ivIntroduction v1 Scope 12 Normative references . 13 Terms and definitions . 14 Quantitative analysis and background noise 24.1 Error sources of analysis 24.2 Random processes in background 35 Theories for precision 35.1 Theory based on auto-covariance function . 35.2 Theory based on power spectrum 56 Practical use of FUMI theory 96.1 Estimation of noise parameters 96.2 Procedures for estimation of SD .10Annex A (informative) Symbols and abbreviated terms used in this part of ISO 11843 13Annex B (informative) Derivation of Formula (7) 14Annex C (informative) Derivation of Formulae (14) to (16) 15Bibliography .17Licensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)ForewordISO (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.International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.The main task of technical committees is to prepare International Standards. 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.Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights.ISO 11843-7 was prepared by Technical Committee ISO/TC 69, Application of statistical methods, Subcommittee SC 6, Measurement methods and results.ISO 11843 consists of the following parts, under the general title Capability of detection:— Part 1: Terms and definitions— Part 2: Methodology in the linear calibration case— Part 3: Methodology for determination of the critical value for the response variable when no calibration data are used— Part 4: Methodology for comparing the minimum detectable value with a given value— Part 5: Methodology in the linear and non-linear calibration cases— Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson distributed measurements by normal approximations— Part 7: Methodology based on stochastic properties of instrumental noiseiv © ISO 2012 – All rights reservedLicensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)IntroductionThe series of ISO 11843 is based on the probability distributions of the net state variable (measurand) for both the linear and nonlinear calibration situations. The focus is implicitly, though sometimes explicitly, on the uncertainty associated with an estimate of the measured response predominantly coming from the baseline noise in instrumental analysis. In many, if not most, analytical instruments, the baseline noise is considered the prime cause of uncertainty when the sample amount is as low as the minimum detectable value. Within its domain of applicability, the method given in this part of ISO 11843 can dispense with the repetition of real samples, thus helping to improve global environments by saving time and energy that would be required by repetition.The basic concept of ISO 11843-7 is the mathematical description of the probability distribution of the response variable in terms of mathematically well-defined random processes. This description straightforwardly leads to the minimum detectable value. As for the relation of the response and measurand, linear and nonlinear calibration functions can be applied. In this manner, compatibility with ISO 11843-2 and ISO 11843-5 is ensured.The definition and applicability of the minimum detectable value are described in ISO 11843-1 and ISO 11843-2; the definition and applicability of the precision profile are described in ISO 11843-5. The precision profile expresses how the precision changes depending on the net state variable. ISO 11843-7 specifies the practical use of the fundamental concepts in ISO 11843 in case of the background noise predominance in instrumental analysis.The minimum detectable value, xd, is generally expressed in the unit of the net state variable. If the calibration function is linear, the SD or CV of the response variable estimated in this part of ISO 11843 can linearly be transformed to the SD or CV of the net state variable, which in turn can be used for the estimation of the minimum detectable value, xd.If the calibration function is nonlinear, the precision profile of the response variable in this part of ISO 11843 needs to be transformed to the precision profile of the net state variable as shown in ISO 11843-5. In this situation, the contents of ISO 11843-5 can be used for this purpose without the slightest modification.© ISO 2012 – All rights reserved vLicensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012Licensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012Capability of detection —Part 7: Methodology based on stochastic properties of instrumental noise1 ScopeBackground noise exists ubiquitously in analytical instruments, whether or not a sample is applied to the instrument. This part of ISO 11843 is concerned with mathematical methodologies for estimating the minimum detectable value in case that the most predominant source of measurement uncertainty is background noise. The minimum detectable value can directly and mathematically be derived from the stochastic characteristics of the background noise.It specifies basic methods to— extract the stochastic properties of the background noise,— use the stochastic properties to estimate the standard deviation (SD) or coefficient of variation (CV) of the response variable, and— calculate the minimum detectable value based on the SD or CV obtained above.The methods described in this part of ISO 11843 are useful for checking the detection of a certain substance by various types of measurement equipment in which the background noise of the instrumental output predominates over the other sources of measurement uncertainty. Feasible choices are visible and ultraviolet absorption spectrometry, atomic absorption spectrometry, atomic fluorescence spectrometry, luminescence spectrometry, liquid chromatography and gas chromatography.2 Normative referencesThe following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.ISO 11843-1:1997, Capability of detection — Part 1: Terms and definitionsISO 11843-2:2000, Capability of detection — Part 2: Methodology in the linear calibration caseISO 11843-5:2008, Capability of detection — Part 5: Methodology in the linear and non-linear calibration casesISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probabilityISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statisticsISO 3534-3, Statistics — Vocabulary and symbols — Part 3: Design of experimentsISO 5725-1, Accuracy (trueness and precision) of measurement methods and results — Part 1: General principles and definitions3 Terms and definitionsFor the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-2, ISO 3534-3, ISO 5725-1, ISO 11843-1, ISO 11843-2, ISO 11843-5 and the following apply. A list of symbols and abbreviated terms used in this document is provided in Annex A.INTERNATIONAL STANDARD ISO 11843-7:2012(E)© ISO 2012 – All rights reserved 1Licensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)3.1 precision profilemathematical description of the standard deviation (SD) of the response variable [σY(X)] or net state variable [σX(X)] as a function of the net state variable.[ISO 11843-5:2008, 3.4]NOTE 1 The coefficient of variation (CV) of the response variable or net state variable as a function of the net state variable is also referred to as a precision profile.NOTE 2 Precision means the SD or CV of the observed response variable or SD or CV of the net state variable when estimated by the calibration function (ISO 11843-5).3.2 minimum detectable value of the net state variablexdvalue of the net state variable in the actual state that will lead, with probability 1 – β, to the conclusion that the system is not in the basic stateNOTE 1 Under the assumption that the SD, σX(X), of the net state variable is constant [(σX(X) = σX], the minimum detectable value, xd, is defined asxd= (kc+ kd)σX(1)where kcdenotes a coefficient to specify the probability of an error of the first kind;kdis a coefficient to specify the probability of an error of the second kind.If the SD, σY, of the response variable is assumed to be constant [σY(X) = σY], then the minimum detectable value can be calculated by the following equation:xd= (kc+ kd)(σY/|dY/dX|) (2)where |dY/dX| denotes the absolute value of the slope of the linear calibration function and is constant.NOTE 2 If the net state variable is normally distributed, the coefficients kc= kd= 1,65 specify the probabilities of an error of the first and second kinds (= 5 %) and Formula (1) can simply be written as xd= 3,30σX.NOTE 3 If kc= kd= 1,65, Formula (1) takes the form that σX/ xd= 1/3,30 = 30 %. Therefore, xdcan be found in the precision profile. xdis located at X, the CV of which is 30 %.NOTE 4 Different types of precision profiles are defined, but they can be transformed to each other. For example, the SD, σY(X), of the response variable can be transformed to the SD, σX(X), of the net state variable by means of the absolute value of the derivative, |dY/dX|, of the calibration function [Y = f(X)]: σX(X) = σY(X)/|dY/dX| (ISO 11843-5). This treatment is an approximation, the extent of which depends on local curvature, involving d2Y/dX2.NOTE 5 Adapted from ISO 11843-5:2008.4 Quantitative analysis and background noise4.1 Error sources of analysisThe quantitative analysis to obtain a measurand from a sample is generally considered to consist of preparation, instrumental analysis, data handling and calibration. These steps of analysis are mechanically independent of each other and so are probabilistically independent as well.This part of ISO 11843 applies only to instrumental analysis. However, the errors from the other steps affect the error of the final value of the measurand, as well. That is, the combined uncertainty associated with an estimate 2 © ISO 2012 – All rights reservedLicensed copy: University of Auckland Library, University of Auckland Library, Version correct as of 06/07/2012 21:20, (c) The British Standards Institution 2012BS ISO 11843-7:2012ISO 11843-7:2012(E)of the measurand depends on the propagation of all uncertainties relating to the relevant steps. The following conditions are necessary for the use of ISO 11843-7.At concentrations near the minimum detectable value in chromatography, the error from the sample injection into a chromatograph is even less important (e.g. CV = 0,3 % in a recent apparatus) than the background noise (CV = 30 % by definition). If the importance of a factor other than noise is comparable to that of the noise, the methodologies of this part of ISO 11843 are not applicable.Data handling is usually a process to extract a signal component from noisy instrumental output such as peak height or area, which is a relative height of a summit of a peak-shaped signal or integration of intensities over a signal region, respectively. The statistical influences of this process are the major concern of this part of ISO 11843. The use of a digital or analogue filter can also be taken into account, if the noise after the filtration is analysed for this purpose.4.2 Random processes in backgroundTypical examples of the response variable are area and height measured in chromatography. In this part of ISO 11843, intensity difference [Formula (6)] and area [Formulae (10) and (11)] are taken as the difference and summation of intensities Yiof instrumental output. The response variables are usually independent of each other even if they are obtained from consecutive measurement by the same instrument. On the other hand, the consecutive intensities Yiare formulated as a time-dependent random process, and in many cases, can be considered 1/f noise.[1]The power spectrum, P(f), of 1/f noise has a slope inversely proportional to frequency, f:Pff()∝1(3)when f is near zero.In mathematical theory, the simplest model of random processes is the white noise. Let widenote the random variable of the white noise at point i. B