ISO 18315:2018

INTERNATIONAL ISO
STANDARD 18315
First edition
2018-11
Nuclear energy — Guidance to
the evaluation of measurement
uncertainties of impurity in uranium
solution by linear regression analysis
Énergie nucléaire — Lignes directrices pour l'évaluation des
incertitudes de mesure des impuretés en solution d'uranium par
analyse de régression linéaire
Reference number
©
ISO 2018
© ISO 2018
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ii © ISO 2018 – All rights reserved

Contents Page
Foreword .iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Principle . 3
5 Uncertainty evaluation . 4
5.1 Regression line fitting . 4
5.2 Adequacy check of fitted regression line . 5
5.3 Combined uncertainty . 5
5.4 Effective degrees of freedom . 6
5.5 Expanded uncertainty . 7
6 Reflection of reference solution uncertainties in evaluation . 7
7 Bias correction . 7
8 Uncertainty evaluation report . 7
Annex A (informative) Practical example of uncertainty evaluation . 9
Annex B (informative) Flowchart of uncertainty evaluation process .13
Annex C (informative) Non-uniform variances and weighting method .15
Bibliography .18
Foreword
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iv © ISO 2018 – All rights reserved

INTERNATIONAL STANDARD ISO 18315:2018(E)
Nuclear energy — Guidance to the evaluation of
measurement uncertainties of impurity in uranium
solution by linear regression analysis
1 Scope
This document provides a method for evaluation of the measurement uncertainty arising when an
impurity content of uranium solution is determined by a regression line that has been fitted by the
“method of least squares”. It is intended to be used by chemical analyzers.
Simple linear regression, hereinafter called “basic regression”, is defined as a model with a single
independent variable that is applied to fit a regression line through n different data points (x , y ) (i = 1,…,
i i
n) in such a way that makes the sum of squared errors, i.e. the squared vertical distances between the
data points and the fitted line, as small as possible. For the linear calibration, “classical regression”
or “inverse regression” is usually used; however, they are not convenient. Instead, “reversed inverse
[2]
regression” is used in this document .
Reversed inverse regression treats y (the reference solutions) as the response and x (the observed
measurements) as the inputs; these values are used to fit a regression line of y on x by the method of
least squares. This regression is distinguished from basic regression in that the x ’s (i = 1,…, n) vary
i
according to normal distributions but the y ’s (i = 1,…, n) are fixed; in basic regression, the y ’s vary but
i i
the x ’s are fixed.
i
The regression line fitting, calculation of combined uncertainty, calculation of effective degrees of
freedom, calculation of expanded uncertainty, reflection of reference solutions’ uncertainties in the
evaluation result, and bias correction are explained in order of mention. Annex A presents a practical
example of uncertainty evaluation. Annex B provides a flowchart showing the steps for uncertainty
evaluation. In addition, Annex C explains the use of weighting factors for handling non-uniform
variances in reversed inverse regression.
NOTE 1 In the case of classical regression, the fitted regression line is inverted prior to actual sample
[3]
measurement . In the case of inverse regression, the roles of x and y are not consistent with the convention that
the variable x represents the inputs, whereas the variable y represents the response. For these reasons, the two
regressions are excluded from this document.
NOTE 2 The term “reversed inverse regression” was suggested taking into account the history of regression
analysis theory. Instead, it can be desirable to use some other term, e.g. “pseudo-basic regression”.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
3.1
calibration
fitting of a regression line of y on x through n data points using the method of least squares
Note 1 to entry: The n data points are typically obtained by measuring n different reference solutions. After the
fitting, the fitted regression line is used as a measurement formula for determining the physical or chemical
quantity of a sample.
3.2
calibration quality control factor
factor that is used to check the adequacy of the fitted regression line from the aspect of calibration quality
3.3
calibration uncertainty
uncertainty due to such possible variations of the slope and intercept supposing that the regression line
fitting is repeated according to the same procedure using a “new set of n different reference solutions”
each time
Note 1 to entry: In this case, the fitted regression line will be different each time, i.e. the slope and intercept of
the fitted regression line will vary.
3.4
combined uncertainty
uncertainty obtained by combining the calibration uncertainty (3.3) and the random uncertainty of
sample measurement (3.9) according to the error propagation rule
3.5
effective degrees of freedom
degrees of freedom calculated by Welch-Satterthwaite approximate formula
3.6
expanded uncertainty
multiplication of the combined uncertainty u by a coverage factor k given depending on the effective
y
degrees of freedom of the combined uncertainty (3.4)
Note 1 to entry: The probability that the true value of the physical or chemical quantity will be within ± the “final
expanded uncertainty” from the determined and bias-corrected value is “exactly” or “approximately” 95 %; in
most cases, “approximately” is more suitable expression.
3.7
predicted y value
y value of a point on the fitted regression line
Note 1 to entry: The predicted y value given by the regression line formula y = a + bx indicates the physical or
chemical quantity that will be determined in response to the light or current signal intensity “x” measured by
instrument. The square root of the estimate for the variance of the predicted y value is treated as the calibration
uncertainty in this document.
3.8
prediction interval
possible vertical distance between the additional data point and the previously fitted regression line
after a regression line has been fitted using a set of n data points
Note 1 to entry: Another data point is additionally produced by measuring “a new reference solution”.
2 © ISO 2018 – All rights reserved

3.9
random uncertainty of sample measurement
possible uncertainty that may arise during such a sample measurement supposing that the
intensity of the light with a specific wavelength emitted from a sample is measured to determine the
sample’s impurity content following the regression line fitting
Note 1 to entry: This uncertainty is typically estimated based on multiple measurements of the sample and
should not be inferred from the mean squared error.
3.10
uncertainty
concept corresponding to the square root of “variance (or estimate for that variance)” that is handled
mainly in statistics
3.11
weighting factors
numbers by which the variances are multiplied in weighted least squares (WLS) regression
Note 1 to entry: OLS (ordinary least squares) regression handles the uniform variances. However, in WLS
regression, the variances are assumed to be non-uniform and the weighting factors are used to handle the non-
uniform variances.
2 2
Note 2 to entry: Let σ ,…, σ be the variances of the variables x ,…, x respectively. Then, in OLS regression,
x1 xn 1 n
2 2 2 2 2
σ = ∙∙∙ = σ (= σ ), whereas, in WLS regression, typically σ ≠∙∙∙≠ σ . However, even in the case of WLS
x1 xn x x1 xn
2 2 2
regression, the equality wσ = ∙∙∙ = wσ (= σ ) can be established by utilizing the weighting factors w ’s
1 x1 n xn x i
(i = 1,…, n).
4 Principle
In practice, a single regression line fitting will be run for calibration. Nevertheless, supposing that
the regression line fitting is repeated using a “new set of n different reference solutions” each time,
the slope and intercept of the fitted regression line will be different each time; as a result, it can be
observed that the predicted y value varies according to an approximately normal distribution. Such a
possible variation of the predicted y value is the cause of the calibration uncertainty that arises during
the regression line fitting. In this regard, there is a helpful statistical theory. With the aid of that theory,
although calibration experts do not repeat the
...