ISO 9276-3:2008

INTERNATIONAL ISO
STANDARD 9276-3
First edition
2008-07-01
Representation of results of particle size
analysis —
Part 3:
Adjustment of an experimental curve
to a reference model
Représentation de données obtenues par analyse granulométrique —
Partie 3: Ajustement d'une courbe expérimentale à un modèle de
référence
Reference number
©
ISO 2008
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ii © ISO 2008 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
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 (informative) 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) χ -Test of number distributions of known sample size. 17
Annex E (informative) Weighted quasilinear regression. 20
Bibliography . 23

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
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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 9276-3 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods,
Subcommittee 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 distributions
⎯ 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 and quantitative representation of particle shape and morphology
iv © ISO 2008 – All rights reserved

Introduction
Cumulative curves of particle size distributions are sigmoids, therefore fitting to a model distribution function or
rendering statistical intercomparison 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 described 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 not 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 this method is not related to any simple distribution and does not allow any statistical
test.
Key
Q (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
ƒ Q (x) data point
Figure 1 — Example 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
[1]
The experimental data in Figure 1 are taken from ISO 9276-1:1998 , Annex A and represent a sieve-
measuring result example between 90 µm and 11,2 mm.
The mathematical treatment, corresponding to non-linear coordinate systems, mentioned above, agrees with
a quasilinear regression. 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 result 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 adjustment 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 of alignment and the statistical comparison of experimental distributions.
A value of greater than e.g. 0,05 indicates a non-adequate reference model.

vi © ISO 2008 – All rights reserved

INTERNATIONAL STANDARD ISO 9276-3:2008(E)

Representation of results of particle size analysis —
Part 3:
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 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 natural 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 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 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 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 probability distribution
3 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
GGS (Gates-) Gaudin-Schuhmann distribution
LND logarithmic normal probability distribution, defined in ISO 9276-5
n number of size classes
n degrees of freedom, which is the number of data points, n, minus the number of fit model
F
parameters
N number of particles in the measured sample
p set of model 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 distribution, 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 mean square deviation of the quasilinear regression in the transformed scale
ql
s standard deviation of the residuals, square root from residual variance
res
x particle size
x median particle size of distribution with type of quantity, r, intercept parameter of LND
50,r
x intercept parameter of GGS distribution with type of quantity, r
max,r
X(x) transform of x plotted on the x-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 ξ 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 Table 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]
α slope parameter of GGS distribution
ζ integration variable, based on z, in LND
ν exponent of RRSB distribution
ω weighting coefficient
2 © ISO 2008 – All rights reserved

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
calculated 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 adjusted to measuring data: [x , Q = Q(x )] i = 1,., n
i i i
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:
n
2*
⎡⎤
sQpp=−(;x ) (Qx)⎯⎯→min (1)
()
∑ ii
p
⎣⎦
n
i=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 values 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):
n
sbp=+⎡⎤Xa−Q()x⎯⎯→min (3)
()
ql ∑⎣⎦i
p
n
i=1
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 reference model.
The quasilinear regression is an analytical method, it requires no start approximation. But the non-linear
transformation of the Y-axis results in a non-linear transformation of the Y-deviations, e.g. percentage
deviations have to be considered differently at the tails of a distribution compared to at their centre.
The extension of this method to a weighted quasilinear regression method also does not deliver the optimum
adjustment, see Annex E.
4.3 Non-linear regression method
4.3.1 General
Finding the general optimum model parameters in the linear scale according to Equation (1) is not possible
with analytical equations; a numerical optimization procedure, known as non-linear regression, is required.
A non-linear regression requires a start approximation and a numerical mathematical procedure
(Reference [4]). If, however, this non-linear regression approach is used, an optimum adjustment and a
flexibility may be conveyed to statistical tests of number distributions or to the adjustment of truncated,
multimodal distributions or any other arbitrary models.
The estimation of parameters, for use with various types of standardized distribution used as reference
models (e.g. normal, LND, RRSB or GGS), is based on different strategies, when either a number or a mass
(or volume) distribution is considered (Reference [5]). The star symbol in Equations (4) and (5) indicates the
model estimation while the emboldened symbol p represents the model parameters to be optimized.
Table 1 — Equations used for three statistical models
Model
Quantity
LND (see also ISO 9276-5) RRSB GGS
z α
⎧
⎛⎞
1 ζ
⎛⎞
x
ν
Distri- Qz()=−exp⎜⎟dζ ⎡ ⎤
⎪
∫ ⎛⎞x
⎪⎜⎟ for xxu
⎜⎟
2π max, r
′ ⎢ ⎥
Qx(;d,ν)=−1 exp − Qx =
bution ⎝⎠ ()⎜⎟
⎜⎟ ⎨
−∞ r x
max, r
′
⎢ d ⎥ ⎝⎠
⎝⎠
model
⎣ ⎦ ⎪
1for xx>
with z = (ln x − ln x )/s
⎪ max, r
50,r ⎩
Intercept,
x x
d ′
50,r max
a
Slope, b 1/s
ν α
Y
⎛⎞
1 ζ
QY()=−exp⎜⎟dζ
∫
⎜⎟
2π 2
⎝⎠
−∞
Y(Q) Y = ln [−ln (1−Q)] Y = ln Q
with the standard normal
−1
distribution, Y = Φ (Q)
X(x) ln x ln x ln x
ln x
Linear
50,r
Y = αX − α ln x
YX=− Y = nX − n ln d ′
max
model
ss
All the non-linear (numerical) estimation strategies need a first estimate of the adjustment parameters before
starting the numerical procedure. The best starting estimate may be obtained from the quasilinear regression
with Equation (3).
The numerical procedure may be based for instance on the Levenberg-Marquardt method, which is a popular
alternative to the Gauss-Newton method (References [7], [8]). Some spreadsheet programs include a
non-linear regression tool (add-in) for easy numerical optimization, for instance based on a code from
Reference [9].
4.3.2 Estimation criterion for both a mass (or volume) distribution and a number distribution
The minimum sum of the squares of the deviations (the least squares) between measured Q values, Q(x), and
the model, Q(x, p), is written for the example of a mass-related distribution as
n
2*
⎡⎤
sQ=−(;x p)Q(x)⎯⎯→min (4)
33ii
∑
p
⎣⎦
i=1
Figure 2 shows the quasilinear regression line from Figure 1 as a curve in linear scales and the non-linear
regression from the least squares of the same data, which obviously represents a better adjustment of the
experimental data. The quantification of the goodness of fit is given in Clause 5.

4 © ISO 2008 – All rights reserved

Key
Q cumulative distribution by volume or mass
x particle size
1 quasilinear regression full line
2 non-linear regression — least squares
z quasilinear fit point
6 least squares fit point
■ Q measured
3,i
Figure 2 — Log-normal distribution: the quasilinear regression from Figure 1 in linear scales and the
*
non-linear regression from the least squares of the same data, (Q − Q )
3 3
[1]
Examples of how the experimental sieve-analysis data obtained from ISO 9276-1:1998 , Annex A, can be
approximated and transformed by either RRSB or GGS state models are shown in Annex A.
The influence of the type of quantity of the distribution on the goodness of fit is shown in Annex B. Different
types of quantity place emphasis of adjustment on different size ranges.
Annex C shows the spreadsheet example calculations for the numerical procedure of the non-linear fit in
Figure 2. Furthermore, an example for a bimodal distribution with five model parameters to be optimized is
shown, using the same algorithm.
4.3.3 Estimation criterion for number distributions only and known sample size as particle number, N
Another estimation criterion for non-linear fit, which can be used only for number distributions and known
sample size, N, is the χ -minimum criterion:
**
⎡⎤
n⎡⎤Qx()−−Qx( ) Q(x;pp)−Qx( ; )
{}⎣⎦00ii−−1 0i 0i1
⎣⎦
χ=⎯N ⎯→ min (5)
∑
** p
Qx(;pp) −Qx( ; )
i=1 00ii−1
It can quantify the improvement of accuracy by the measurement of larger particle numbers. This criterion
compares the observed particle number variance in the numerator of Equation (5) with that predicted by
Poisson statistics in the denominator of each size class.
Annex D shows the application for a χ -test of number distributions of known sample size, which quantifies the
importance of large sample sizes for the analysis data interpretation.
5 Goodness of fit, standard deviation of residuals and exploratory data analysis
The basic regression, Equation (1), is used to find the optimum parameters, p = a, b., in such a way that the
mean square deviation between measured Q values and the model Q* is minimized.
Theref
...