1、BS ISO 9276-3:2008 ICS 19.120 NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAW BRITISH STANDARD Representation of results of particle size analysis Part 3: Adjustment of an experimental curve to a reference modelThis British Standard was published under the authority of the Sta
2、ndards Policy and Strategy Committee on 31 July 2008 BSI 2008 ISBN 978 0 580 58413 8 Amendments/corrigenda issued since publication Date Comments BS ISO 9276-3:2008 National foreword This British Standard is the UK implementation of ISO 9276-3:2008. The UK participation in its preparation was entrus
3、ted to Technical Committee LBI/37, Particle characterization including sieving. A list of organizations represented on this committee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its co
4、rrect application. Compliance with a British Standard cannot confer immunity from legal obligations.BS ISO 9276-3:2008Reference number ISO 9276-3:2008(E) ISO 2008INTERNATIONAL STANDARD ISO 9276-3 First edition 2008-07-01 Representation of results of particle size analysis Part 3: Adjustment of an ex
5、perimental curve to a reference model Reprsentation de donnes obtenues par analyse granulomtrique Partie 3: Ajustement dune courbe exprimentale un modle de rfrence BS ISO 9276-3:2008 ISO 9276-3:2008(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobes licensing p
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10、eneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail copyrightiso.org Web www.iso.org Published in Switzerland ii ISO 2008 All rights reservedBS ISO 9276-3:2008 ISO 9276-3:2008(E) ISO 2008 All rights reserved iii Contents Page Foreword iv Introduction v 1 Scope . 1 2 Normative references . 1
11、 3 Symbols and abbreviated terms . 2 4 Adjustment of an experimental curve to a reference model 3 4.1 General. 3 4.2 Quasilinear regression method. 3 4.3 Non-linear regression method. 3 5 Goodness of fit, standard deviation of residuals and exploratory data analysis 6 6 Conclusions 7 Annex A (inform
12、ative) Influence of the model on the regression goodness of fit. 9 Annex B (informative) Influence of the type of distribution quantity on the regression result . 11 Annex C (informative) Examples for non-linear regression. 15 Annex D (informative) 2 -Test of number distributions of known sample siz
13、e 17 Annex E (informative) Weighted quasilinear regression 20 Bibliography . 23 BS ISO 9276-3:2008 ISO 9276-3:2008(E) iv ISO 2008 All rights reservedForeword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work
14、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-government
15、al, 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 mai
16、n 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. Att
17、ention 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 9276-3 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods, Subc
18、ommittee SC 4, Sizing by methods other than sieving. ISO 9276 consists of the following parts, under the general title Representation of results of particle size analysis: Part 1: Graphical representation Part 2: Calculation of average particle sizes/diameters and moments from particle size distribu
19、tions Part 3: Adjustment of an experimental curve to a reference model Part 4: Characterization of a classification process Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution The following part is under preparation: Part 6: Descriptive
20、 and quantitative representation of particle shape and morphology BS ISO 9276-3:2008 ISO 9276-3:2008(E) ISO 2008 All rights reserved v Introduction Cumulative curves of particle size distributions are sigmoids, therefore fitting to a model distribution function or rendering statistical intercomparis
21、on is difficult. These disadvantages can, however, be remedied by transforming these sigmoids into straight lines by means of appropriate coordinate systems, e.g. log-normal, Rosin-Rammler or Gates-Gaudin-Schuhmann (log-log). Target size distributions in particle technology industries can also be de
22、scribed in terms of distribution models. In such systems, a classic linear regression assumes that the squares of the deviations between the experimental points and the theoretical straight line are, on average, equal. This is only valid in the transformed cumulative distribution value system, but n
23、ot in their linear representation, and therefore named a quasilinear regression. In particular, the scale extension makes the values of the squares of the deviations at the extremities of the graph vary by several orders of magnitude. In addition, the sum of the squares of the deviations obtained by
24、 this method is not related to any simple distribution and does not allow any statistical test. Key Q 3 (x) cumulative distribution by volume or mass x particle size Y quantiles of the standard normal distribution 1 quasilinear regression full line quasilinear fit point Q 3(x) data point Figure 1 Ex
25、ample of a functional paper with log-normal plot (cumulative distribution values plotted on a normal ordinate against particle size on a logarithmic abscissa with inverse standard normal distribution transformed) and quasilinear regression full line BS ISO 9276-3:2008 ISO 9276-3:2008(E) vi ISO 2008
26、All rights reservedThe experimental data in Figure 1 are taken from ISO 9276-1:1998 1 , Annex A and represent a sieve- measuring result example between 90 m and 11,2 mm. The mathematical treatment, corresponding to non-linear coordinate systems, mentioned above, agrees with a quasilinear regression.
27、 Here the non-linear transformation of the Y-axis results in a non-linear transformation of the Y-deviations, e.g. another consideration of deviations at the tails of a distribution than at their centre. One possibility to compensate for the non-linear transformation of the Y-differences, in the res
28、ult of the non-linear transformation of the Y values, is the introduction of weighting factors in the quasilinear regression (see Annex E). Moreover, a non-linear regression delivers the best adjustment and allows the most flexibility, such as statistical tests on number distributions, the adjustmen
29、t of truncated or multimodal distributions or any other arbitrary models, but it requires a start approximation and a numerical mathematical procedure. The standard deviation of residuals between experimental points and the model in the non-transformed scale allows the quantification of the degree o
30、f alignment and the statistical comparison of experimental distributions. A value of greater than e.g. 0,05 indicates a non-adequate reference model. BS ISO 9276-3:2008 INTERNATIONAL STANDARD ISO 9276-3:2008(E) ISO 2008 All rights reserved 1 Representation of results of particle size analysis Part 3
31、: Adjustment of an experimental curve to a reference model 1 Scope This part of ISO 9276 specifies methods for the adjustment of an experimental curve to a reference model with respect to a statistical background. Furthermore, the evaluation of the residual deviations, after the adjustment, is also
32、specified. The reference model can also serve as a target size distribution for maintaining product quality. This part of ISO 9276 specifies procedures that are applicable to the following reference models: a) normal distribution (Laplace-Gauss): powders obtained by precipitation, condensation or na
33、tural products (pollens); b) log-normal distribution (Galton MacAlister): powders obtained by grinding or crushing; c) Gates-Gaudin-Schuhmann distribution (bilogarithmic): analysis of the extreme values of the fine particle distributions; d) Rosin-Rammler distribution: analysis of the extreme values
34、 of the coarse particle distributions; e) any other model or combination of models, if a non-linear fit method is used (see bimodal example in Annex C). This part of ISO 9276 can substantially support product quality assurance or process optimization related to particle size distribution analysis. 2
35、 Normative references The 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 9276-2, Representation
36、of results of particle size analysis Part 2: Calculation of average particle sizes/diameters and of moments from particle size distributions ISO 9276-5, Representation of results of particle size analysis Part 5: Methods of calculation relating to particle size analyses using logarithmic normal prob
37、ability distribution BS ISO 9276-3:2008 ISO 9276-3:2008(E) 2 ISO 2008 All rights reserved3 Symbols and abbreviated terms a straight line intercept (equation of a straight line) b slope (gradient) of the straight regression line (equation of a straight line) d intercept parameter of RRSB distribution
38、 GGS (Gates-) Gaudin-Schuhmann distribution LND logarithmic normal probability distribution, defined in ISO 9276-5 n number of size classes n Fdegrees of freedom, which is the number of data points, n, minus the number of fit model parameters N number of particles in the measured sample p set of mod
39、el parameters, vector q density of particle size distribution Q(x) observed cumulative distribution, total of the particles finer than x, between 0 and 1 Q*(x; p) model estimation, theoretical cumulative distribution depending on the reference model with parameters, p r type of quantity of a size di
40、stribution, r = 0: number, r = 3: volume or mass RRSB Rosin-Rammler (Sperling and Bennet) distribution (derived from Weibull-distribution) s standard deviation of LND, logarithm of geometric standard deviation ISO 9276-5 s qlmean square deviation of the quasilinear regression in the transformed scal
41、e s resstandard deviation of the residuals, square root from residual variance x particle size x 50,rmedian particle size of distribution with type of quantity, r, intercept parameter of LND x max,rintercept parameter of GGS distribution with type of quantity, r X(x) transform of x plotted on the x-
42、axis X = x for a normal distribution and X = ln x or lg x for a log- normal, Rosin-Rammler or bilogarithmic (log-log) distribution, X is equivalent to in ISO 9276-1 and ISO 9276-5 Y(Q) transform of Q plotted on the y-axis (Y = inverse of standard normal distribution for a normal distribution, see Ta
43、ble 1 for other model types) Y* = a + bX general expression of the equation for the straight regression line of a model cumulative particle size distribution z dimensionless normalization variable in LND ISO 9276-5 slope parameter of GGS distribution integration variable, based on z, in LND exponent
44、 of RRSB distribution weighting coefficient BS ISO 9276-3:2008 ISO 9276-3:2008(E) ISO 2008 All rights reserved 3 4 Adjustment of an experimental curve to a reference model 4.1 General The estimation of parameters to be used in the regression equations appearing in this part of ISO 9276 are calculate
45、d from either particle size distribution values, Q, fractions of these particle size values, dQ, or density values, q. These particle size distribution parameters may also be used as parameters for other regression equations. Generally a certain distribution model Q*(x; p) = Q*(x; a,b) should be adj
46、usted to measuring data: x i , Q i= Q(x i ) i = 1,., n The intention and capability of the regression equation is to find the optimum parameters p = a, b. such that the mean square deviation between measured Q values, Q(x), and the model, Q*(x; p), will be minimized: () 2 2* 1 1 (;) () m i n n ii i
47、sQ xQ x n = = p pp(1) 4.2 Quasilinear regression method The non-linear (or rather non-linear) optimization problem in Equation (1) can be transformed by Y to a linear Equation (2) for the various statistical models used in this part of ISO 9276. The values of X are the transformed particle size valu
48、es obtained from any particle size distribution. Y* = Y*(Q*) = a + bX (2) The solution and optimization using a linear regression with Equation (2) in the transformed state, delivers an approximation for Equation (1), which can be replaced with the following quasilinear regression Equation (3): () 2
49、 2 ql 1 1 () m i n n i i sb X a Q x n = =+ p p (3) The solution of Equation (3) minimizes the absolute deviations in the transformed format (see Figure 1). This quasilinear regression can also be used for all standardized particle size distributions using the various transformation equations listed in Table 1 (Reference 3). The ordinates, designated Y, are the transforms of the Q (x) cumulative distribution values obtained by the formula of the relevant refe
