ASTM D8537-23

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D8537 − 23
Standard Guide for
Analysis of Calibration Data for Nuclear Instruments
This standard is issued under the fixed designation D8537; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 This guide describes data analysis for efficiency calibra- 2.1 ASTM Standards:
tions of nuclear instruments using radioactive sources. It D1129 Terminology Relating to Water
includes the calculation of the calibration parameters, evalua- D7282 Practice for Setup, Calibration, and Quality Control
tion and use of their uncertainties and covariances, and testing of Instruments Used for Radioactivity Measurements
of the calibration data for outliers and overall lack of fit. It also D7902 Terminology for Radiochemical Analyses
provides guidelines for summarizing and reporting the results D8293 Guide for Evaluating and Expressing the Uncertainty
of a calibration. of Radiochemical Measurements
2.2 JCGM Documents:
1.2 The instrument counting efficiency is assumed to be
GUM:JCGM 100:2008 Evaluation of measurement data—
independent of the radiation emission rate.
Guide to the expression of uncertainty in measurement
1.3 Guidance is provided for both single-point calibrations
JCGM 102:2011 Evaluation of measurement data—
and calibration curves.
Supplement 2 to the “Guide to the expression of uncer-
1.4 The guidance presumes the existence of measurement
tainty in measurement”—Extension to any number of
uncertainty models to provide statistical weighting factors for
quantities
the calibration data.
VIM:JCGM 200:2008 International vocabulary of
metrology—Basic and general concepts and associated
1.5 This guide does not cover calibrations involving
terms (VIM)
physically-based computer simulations.
1.6 The system of units for this guide is not specified.
3. Terminology
Dimensional quantities in the guide are presented only as
3.1 Definitions:
illustrations of calculation methods. The examples are not
3.1.1 For definitions of terms used in this practice, refer to
binding on products or test methods treated.
Terminologies D1129 and D7902, Practice D7282,
1.7 This standard does not purport to address all of the
GUM:JCGM 100, JCGM 102, and VIM:JCGM 200.
safety concerns, if any, associated with its use. It is the
3.2 Definitions of Terms Specific to This Standard:
responsibility of the user of this standard to establish appro-
3.2.1 calibration curve, n—functional model that calculates
priate safety, health, and environmental practices and deter-
counting efficiency from the value of a predictor variable and
mine the applicability of regulatory limitations prior to use.
one or more model parameters; also known as an effıciency
1.8 This international standard was developed in accor-
curve.
dance with internationally recognized principles on standard-
3.2.1.1 Discussion—A calibration “curve” might be a linear
ization established in the Decision on Principles for the
or nonlinear function of the predictor variable.
Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
3.2.2 calibration parameter, n—any of the parameters in a
Barriers to Trade (TBT) Committee.
calibration model whose values are determined by a calibration
1 2
This guide is under the jurisdiction of ASTM Committee D19 on Water and is For referenced ASTM standards, visit the ASTM website, www.astm.org, or
the direct responsibility of Subcommittee D19.04 on Methods of Radiochemical contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
Analysis. Standards volume information, refer to the standard’s Document Summary page on
Current edition approved Nov. 15, 2023. Published December 2023. DOI: the ASTM website.
10.1520/D8537-23. Available from www.bipm.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D8537 − 23
and subsequently used together with observed values of the 4.3.2 A calibration curve (or line), typically polynomial,
predictor variable to calculate counting efficiencies. determined by linear least squares (LLS), and
4.3.3 A calibration curve determined by nonlinear least
3.2.3 calibration range, n—interval between the least and
squares (NLLS).
greatest values of the predictor variable for which a calibration
curve is considered valid. 4.4 In each case, it is assumed that some number (n) of
efficiency measurements are made, producing measured effi-
3.2.4 generalized weighting, n—statistical weighting of data
ciencies ɛ , ɛ , ., ɛ . It is assumed that there is a valid
1 2 n
using both variances and covariances.
uncertainty model for the measurements, providing an uncer-
3.2.5 relative residual (%∆ ), n—quotient of a residual, e ,
i i
tainty for each ɛ and an estimated covariance for each pair
i
and the corresponding predicted value, εˆ , typically expressed
i
(ɛ , ɛ ). The parameter-fitting procedure statistically weights the
i j
as a percentage.
measured efficiencies according to their estimated variances.
3.2.6 residual (e ), n—difference, ε 2εˆ , between a measured
i i i
Since using estimated variances instead of true variances for
value, ɛ , and the corresponding predicted value, εˆ .
i i this purpose can bias the results, a two-stage procedure is
3.2.7 simple weighting, n—statistical weighting of data employed, in which preliminary values for the calibration
parameters are obtained first and used to refine the variance
using variances but not covariances.
estimates and weights for the final fit.
3.2.8 single-point, adj—relating to a calibration model in
which the instrument counting efficiency is estimated by a 4.5 Optimal weighting requires not only the variances,
single parameter with no predictor variable (a polynomial of u ε , but also the covariances, u ε ,ε . Two options are
~ ! ~ !
c i i j
degree zero). discussed in each case for dealing with covariances. In each
case, Option 1 (“simple weighting”) does not require explicit
3.2.9 standardized residual (ζ ), n—quotient of a residual,
i
calculation of u~ε ,ε !. Option 2 (“generalized weighting”) is
i j
e , and its combined standard uncertainty, u (e ).
i c i
the default option when Option 1 cannot be used.
3.3 Acronyms:
4.6 Guidance is provided for calculating the calibration
3.3.1 GLS—generalized least squares
parameters, their uncertainties, and (when applicable) their
3.3.2 LLS—linear least squares
correlation coefficients; for assessing LOF; and for calculating
3.3.3 LOF—lack of fit
the counting efficiency and associated uncertainty for a subse-
quent sample test source (STS) measurement.
3.3.4 MQO—measurement quality objective
4.7 The emphasis of this guide is not on the details of the
3.3.5 NLLS—nonlinear least squares
least-squares fitting algorithms. For more information, see
3.3.6 OLS—ordinary least squares
Draper and Smith (1), Marquardt (2), or Bevington and
3.3.7 STS—sample test source
Robinson (3).
3.3.8 WCS—working calibration source
5. Significance and Use
3.3.9 WLS—weighted least squares
5.1 The mathematical and statistical techniques described in
this guide support implementation of the calibration require-
4. Summary of Guide
ments of Practice D7282 and the guidance for uncertainty
4.1 Calculation of an instrument counting efficiency re-
analysis given in Guide D8293. The guidance is intended for
quires a mathematical model described in terms of one or more
use either by qualified specialists at a radioanalytical laboratory
calibration parameters. In a single-point efficiency model,
or by developers of software for calibration of nuclear instru-
there is one parameter, which equals the efficiency itself. In all
ments.
other models there is a calibration curve, which calculates the
5.2 Applications for single-point calibrations might include:
efficiency from a predictor variable, such as gamma-ray
5.2.1 Alpha-particle spectrometry,
energy, precipitate mass, or a quench indicator; and also one or
5.2.2 Gas proportional counters used for thin sources with
more calibration parameters. An efficiency calibration mea-
negligible attenuation, and
sures the calibration parameter(s), providing their estimated
5.2.3 Gamma-ray spectrometers used for single nuclides.
values and uncertainties.
5.3 Applications for calibration curves determined by LLS
4.2 A calibration requires at least as many measurements as
might include:
the number of model parameters. Whenever practical, the
5.3.1 Mass attenuation curves for gas proportional counters
number of measurements should exceed the number of
(polynomial), and
parameters, resulting in extra degrees of freedom that can be
5.3.2 Quench calibration curves for liquid scintillation
used to assess lack of fit (LOF). Additional measurements
counters (polynomial).
reduce the total uncertainty of the calibration, and the lack-of-
fit test checks the consistency of the data with the efficiency
5.4 Applications for calibration curves determined by NLLS
and uncertainty models.
might include:
4.3 Three types of calibrations are considered:
4.3.1 A single-point calibration involving multiple
The boldface numbers in parentheses refer to a list of references at the end of
measurements, this standard.
D8537 − 23
and:
5.4.1 Gamma-ray spectrometry across a range of gamma-
ray energies,
u R ,R
~ !
B i B j
u ε ,ε
~ !
i j
5.4.2 Mass attenuation curves for gas proportional counters,
~A ·I ·DF !~A ·I ·DF !
i i i j j j
(3)
and
u~A ,A ! u~I ,I !
i j i j
1ε ε 1
S D
5.4.3 Quench calibration curves for liquid scintillation i j
A ·A I ·I
i j i j
counters.
where:
5.5 Although this guide focuses on efficiency calibrations
i, j = index numbers of two different calibration sources,
for nuclear instruments, the same general principles and
t = source count time,
S i
paradigms should apply to other types of calibrations and to
t = background count time, and
B i
other instruments, as long as there are valid uncertainty models
φ = relative standard uncertainty due to variability of the
CS
for the calibration data.
calibration sources and possibly to model error.
6.4.1 This guide assumes a constant value for the factor φ ,
CS
6. Overview
which accounts for variations in counting geometry and
6.1 This section provides an overview of the recommended
imperfections of the mathematical model, although in principle
statistical procedures. Section 7 provides additional details for
its value might vary with that of the predictor variable.
a single-point calibration. Section 8 describes procedures for
6.5 In Eq 3, u R ,R 5u R if the same measured back-
~ !
~ B i B j! B
fitting a calibration curve using linear least squares, and
ground R is used for both ɛ and ɛ . Otherwise, u R ,R may
~ !
B i j B i B j
Section 9 describes procedures for fitting a calibration curve
be 0.
using nonlinear least squares. Section 10 provides instructions
6.6 If u R ,R is always zero or negligible, and if each of
for assessing the fit in any of the three scenarios. Section 11 ~ !
B i B j
the relative covariances, u A ,A ⁄ A ·A and u I ,I ⁄ I ·I , is
~ ! ~ ! ~ ! ~ !
describes the content and organization of a calibration report. i j i j i j i j
the same for all i ≠ j, one may use Option 1, which requires
Topic Section
only simple weights, not a full weighting matrix. For example,
Single-point calibration 7
Calibration curve—Linear least squares 8
suppose I = I for each i, and each source activity A is
i i
Calibration curve—Nonlinear least squares 9
calculated as:
Assessing the fit 10
The calibration report 11
A 5 AC·V ·Y (4)
i i i
Estimating additional variability Appendix X1
where:
6.2 A single-point calibration is used when the counting
AC = activity concentration (or massic activity) of the cali-
efficiency is modeled as a constant, with no predictor variable.
bration reference material,
In all other situations, a calibration curve is used, in which case
V = volume (or mass) of the reference material used in the
the choice of least-squares fitting techniques (linear or nonlin- i
source, and
ear) is determined by whether the calibration model is linear or
Y = chemical yield (for a working calibration source,
i
nonlinear in the calibration parameters.
WCS).
6.3 For illustration, it is assumed that each measured effi-
If all the yields, Y , are determined using the same yield
i
ciency ɛ is calculated with an equation such as:
i
tracer or carrier solution, with activity or mass concentration,
R 2 R
S i B i
c , and if u V ,V '0 and u R ,R '0, then:
~ !
~ !
T i j B i B j
ε 5 , i 5 1,2,…,n, (1)
i
A ·I ·DF
i i i
2 2
u~A ,A ! u ~AC! u ~c !
i j T
5 1 (5)
where: 2 2
A ·A AC c
i j T
i = index number of the calibration source,
u~V ,V !
i j
R = gross count rate,
'0 (6)
S i
V ·V
i j
R = background count rate,
B i
u I ,I u I
~ ! ~ !
A = activity of the calibration source at its reference date, i j
i
5 (7)
I ·I I
I = radiation emission probability, and
i j
i
DF = decay factor.
i
Since all the relative covariance terms are equal, Option 1
NOTE 1—Eq 1 is only an example. Other equations are possible.
may be used. In this case, φ will denote the sum of the shared
ε
6.4 Given the efficiency equation, one uses uncertainty
relative variance/covariance components. For example,
propagation to write equations for the combined variances
2 2 2
u AC u c u I
2 ~ ! ~ ! ~ !
T
u ε and covariances u ε ,ε , as described in Guide D8293. 2
~ ! ~ !
c i i j φ 5 1 1 (8)
ε 2 2 2
AC c I
T
For example, when using Eq 1,
Note that u~ε ,ε !5ε ε φ for i ≠ j.
2 2 i j i j ε
R ⁄ t 1R ⁄ t
u A u I
S i S i B i B i ~ ! ~ !
2 i i
2 2
u ε
~ !
5 1ε 1 1φ
c i S D
2 i 2 2 CS
(2) 6.7 It might also be possible to decompose the variances
~A ·I ·DF ! A I
i i i i i
u (V ) into components due to random and fixed effects and to
i
D8537 − 23
include the latter component in φ . covariances of the calibration parameters are calculated.
ε
6.10 Option 2—Generalized Weighting with Covariance
6.8 Option 2 should be used if the conditions required for
Matrix:
Option 1 are not met. Option 2 statistically weights the data
6.10.1 Uncertainty propagation is used to estimate the total
using the full covariance matrix of the measured efficiencies.
combined variance, u ~ε !, and covariance u~ε ,ε !, as described
c i i j
6.9 Option 1—Simple Weighting without Covariances:
in 6.4. See Eq 2 and Eq 3 for examples.
6.9.1 Option 1 uses partial variance estimates, denoted by
2 ˜
u ε , to calculate weighting factors. The relative variance 6.10.2 The preliminary measurement covariance matrix U
~ !
Ɛ
cP i
2 2
is constructed and inverted to obtain the preliminary weighting
terms that contribute to φ above are omitted from u ~ε !. So,
ε cP i
˜
matrix, W.
2 2 2 2
u ε 5 u ε 2 ε φ (9)
~ ! ~ !
cP i i i ε
˜ ˜
W 5 U (16)
For example, if Eq 8 is used, then: Ɛ
where:
R ⁄ t 1R ⁄ t
u ~A !
S i S i B i B i
2 cP i 2
2 2
u ε u ε u ε ,ε … u ε ,ε
~ ! 5 1ε 1φ ~ ! ~ ! ~ !
cP i S D c 1 1 2 1 n
2 2
i CS
(10)
A ·I·DF A
~ !
i i i 2
u ε ,ε u ε … u ε ,ε
~ ! ~ ! ~ !
2 1 c 2 2 n
˜
U 5 (17)
Ɛ
where u A denotes a partial variance of A calculated
~ ! ¡ ¡ ¢ ¡
cP i i
1 2
2 2 2
without the relative variance terms u (AC) / AC and u (c )/c
u ε ,ε u ε ,ε … u ε
T T ~ ! ~ ! ~ !
n 1 n 2 c n
from Eq 5.
6.10.3 Preliminary estimates of the calibration parameters,
6.9.2 The preliminary weights w˜ ,w˜ ,…,w˜ are given by: ˜
1 2 n
β , are found and used to calculate preliminary values for the
j
predicted efficiencies, ε˜ . These values are then used to refine
1 i
w˜ 5 (11)
i the estimates of variance, u ~ε !, and covariance, u ~ε ,ε !. For
p i p i j
u ~ε !
cP i
example,
These weights are used for the preliminary fit as described
ε˜ ·A ·I ·DF ⁄ t 1R 1 ⁄ t 11 ⁄ t
~ !
i i i i S i B i S i B i
below for each calibration scenario. For calculations with 2
u ~ε !
p i
A ·I ·DF
~ !
matrices, the weights are used to construct the preliminary i i i
(18)
2 2
˜ u A u I
~ ! ~ !
weighting matrix, W. i i
2 2
1ε˜ 1 1φ
S D
i 2 2 CS
A I
i i
w˜ 0 … 0
1 and:
0 w˜ … 0
˜
u R ,R
~ !
W 5 diag~w˜ ,w˜ ,…,w˜ ! 5 (12) B i B j
1 2 n
u ~ε ,ε !
¡ ¡ ¢ ¡ 5
p i j
1 2
A ·I ·DF A ·I ·DF
~ !~ !
i i i j j j
0 0 … w˜ (19)
n
u A ,A u I ,I
~ ! ~ !
i j i j
NOTE 2—Throughout this guide, a tilde (~) over a symbol denotes a
1ε˜ ε˜ 1
S D
i j
A ·A I ·I
i j i j
preliminary estimate or value, which will be recalculated and replaced
later.
p
6.10.4 The refined covariance matrix U is constructed and
Ɛ
˜
6.9.3 Preliminary estimates of the calibration parameters, β , inverted to obtain the refined weighting matrix, W.
j
are found and used to calculate preliminary values for the
p 21
W 5 U (20)
Ɛ
predicted efficiencies, ε˜ . These values are then used to refine
i
the variance estimates, u ε . For example, where:
~ !
p i
u ~ε ! u ~ε ,ε ! … u ~ε ,ε !
p 1 p 1 2 p 1 n
ε˜ ·A ·I·DF ⁄ t 1R 1 ⁄ t 11 ⁄ t
~ !
i i i S i B i S i B i
u ~ε ! 2
p i
2 u ~ε ,ε ! u ~ε ! … u ~ε ,ε !
p 2 1 p 2 p 2 n
A ·I·DF
~ ! p
i i
U 5 (21)
Ɛ
(13)
2 ¡ ¡ ¢ ¡
u A
~ ! 1 2
cP i
2 2
1ε˜ 1φ
S D
i 2 CS u ε ,ε u ε ,ε … u ε
~ ! ~ ! ~ !
p n 1 p n 2 p n
A
i
The weighting matrix W is used for the final fit as described
6.9.4 The refined weights w , w , …, w are then given by:
1 2 n
below for each scenario.
6.10.5 After the final fit is obtained, the predicted
w 5 (14)
i 2
u ε 2
~ !
p i
efficiencies, εˆ , and their variances, u εˆ , are calculated. The fit
~ !
i p i
is assessed and an outlier test might be performed. If the results
These weights are used for the final fit as described below for
are acceptable, the total combined standard uncertainties and
each scenario. For calculations with matrices, the weights are
covariances of the calibration parameters are calculated.
used to construct the final weighting matrix, W.
6.11 Both Options:
w 0 … 0
6.11.1 For a multi-parameter calibration curve, the correla-
0 w … 0
W 5 diag w ,w ,…,w 5 (15) tion coefficient for each unordered pair of calibration param-
~ !
1 2 n
¡ ¡ ¢ ¡
1 2
eters is calculated.
0 0 … w
n
6.11.2 The efficiency, ɛ , for a subsequent sample test
STS
6.9.5 After the final fit is obtained, the predicted efficiencies, source measurement and its combined standard uncertainty,
εˆ , and their (partial) variances, u εˆ , are calculated. The fit is u (ɛ ), are calculated. The uncertainty includes a component
~ !
i p i c STS
assessed, and an outlier test might be performed. If the results φ ·ε accounting for the variability of sample test sources,
STS STS
are acceptable, the total combined standard uncertainties and and in the case of a calibration curve, also for model error.
D8537 − 23
7. Single-Point Calibration—Average Value 7.3.3 Calculate the preliminary efficiency estimate as a
generalized weighted average, ε˜.
7.1 Assume a single-point efficiency is measured n times
n n
(n > 1). Let the measured efficiencies be denoted by
˜
ε W
(i51 i(j51 ij
ε , ε , …, ε . The calibration parameter will be calculated as a
ε˜ 5 (27)
1 2 n
n n
˜
W
(i51(j51 ij
weighted or simple average of these values. See Section 6 to
choose Option 1 or 2.
th
˜ ˜
where W denotes the ij entry of W. Alternatively, use the
ij
arithmetic mean ε˜5ε‾. See Eq 23.
7.2 Option 1—Weighted Average without Covariances:
7.2.1 Use uncertainty propagation to estimate a partial 2
7.3.4 Calculate refined estimates of the variances, u ~ε ! and
p i
variance u ε for each measured efficiency, as described in
~ !
cP i
covariances, u (ε , ε ), as described in 6.10.3 (with each ε˜ 5ε˜).
i
p i j
6.9.1. See Eq 10 for example.
See Eq 18 and Eq 19 for examples.
7.2.2 Calculate the preliminary estimate of the efficiency, ε˜, p
7.3.5 Construct the refined covariance matrix, U , and
ε
as a weighted average.
weighting matrix, W, as described in 6.10.4. See Eq 20 and Eq
n 21.
ε
i
( 2
u ε
i51 ~ !
cP i 7.3.6 Calculate the final efficiency estimate as a generalized
ε˜ 5 (22)
n
weighted average, εˆ:
(
u ε
i51 ~ !
cP i
n n
ε W
(i51 i(j51 ij
εˆ 5 (28)
Alternatively, and especially if all the partial variances are n n
W
(i51(j51 ij
roughly equal, use the arithmetic mean, ε‾, for the preliminary
th
estimate.
where W denotes the ij entry of W.
ij
n 7.3.7 Calculate the estimated variance of the weighted
ε˜ 5 ε‾ 5 ε (23)
average εˆ.
( i
n
i51
n n 21
The predicted efficiency ε˜ has the same value, ε˜, for each i. 2
i
u εˆ 5 W (29)
~ ! S D
p ( ( ij
i51j51
7.2.3 Calculate a refined estimate of each variance, u ε , as
~ !
p i
described in 6.9.3. See Eq 13 for example. 7.3.8 Assess the fit as described in Section 10 (with m = 1
and with each εˆ 5εˆ).
i
7.2.4 Calculate the final efficiency estimate as a weighted
average, εˆ. 7.3.9 Calculate the combined standard uncertainty of εˆ.
n
ε
i u εˆ 5 =u εˆ (30)
~ ! ~ !
c p
( 2
u ~ε !
i51
p i
εˆ 5 (24) 7.4 Both Options—Single-Point Calibration:
n
7.4.1 The laboratory may establish an upper limit for the
( 2
u ~ε !
i51
p i
acceptable relative combined standard uncertainty, u εˆ ⁄ εˆ,
~ !
c
The predicted efficiency εˆ has the same value, εˆ, for each i.
i based on the uncertainty requirements of the measurement
method or other measurement quality objectives (MQOs).
7.2.5 Calculate its variance.
n 21
7.4.2 Record the estimated efficiency, εˆ, and its combined
u ~εˆ! 5 (25)
S D standard uncertainty, u εˆ .
p ( 2 ~ !
c
u ~ε !
i51
p i
7.4.3 The efficiency for a subsequent measurement of a
7.2.6 Assess the fit as described in Section 10 (with m = 1
sample test source, ɛ , is estimated by:
STS
and with each εˆ 5εˆ).
i
ε 5 εˆ (31)
STS
7.2.7 Calculate the combined standard uncertainty of εˆ as:
and its combined standard uncertainty is:
2 2 2
u εˆ 5 =u εˆ 1εˆ φ (26)
~ ! ~ !
c p ε
2 2 2
u ε 5 =u εˆ 1εˆ φ (32)
~ ! ~ !
2 c STS c STS
where φ is the sum of the previously omitted relative
ε
where:
variance terms described in 6.6 and 6.9.1.
φ = relative standard uncertainty due to variations among
STS
7.2.8 Continue at 7.4.
individual sample test sources.
7.3 Option 2—Generalized Weighted Average with Covari-
The relative uncertainty component φ may equal φ ,
STS CS
ance Matrix:
especially when the calibration is performed with working
7.3.1 Use uncertainty propagation to estimate the total
calibration sources (WCSs) prepared in the laboratory.
combined variances, u ε , and covariances, u(ε , ε ), as de-
~ !
c i i j
scribed in 6.4. See Eq 2 and Eq 3 for examples.
8. Calibration Curve—Linear Least Squares
7.3.2 Construct the preliminary measurement covariance
8.1 Assume the calibration model has the functional form:
˜ ˜
matrix, U , and calculate the preliminary weighting matrix, W,
Ɛ
as described in 6.10.2. See Eq 16 and Eq 17. ε 5 β X 1β X 1· · ·1β X (33)
1 1 2 2 m m
D8537 − 23
where m denotes the number of calibration parameters (m ≥ ˜ ˜
M g˜ 2 M g˜
22 1 12 2
˜
β 5 (41)
1), each X is a specified function of the predictor variable, and 1
j ˜ ˜ ˜ ˜
M M 2 M M
11 22 12 21
the coefficients β are calibration parameters. It is assumed that
j
the uncertainty of each X is negligible. Typically, the efficiency
and:
j
function is a polynomial in some predictor variable X, and X is
j
˜ ˜
M g˜ 2 M g˜
11 2 21 1
a power of X. For example, ˜
β 5 (42)
˜ ˜ ˜ ˜
M M 2 M M
2 m21 11 22 12 21
ε 5 β 1β X1β X 1· · ·1β X (34)
1 2 3 m
˜
where each M is given by Eq 39 and each ˜g is given by Eq
jk j
8.2 Make n measurements of the efficiency (n > m) across
40.
the required range of the predictor variable, obtaining mea-
sured results ε , ε , …, ε . Let X , X , …, X denote the values
1 2 n i1 i2 im 8.4.6 Calculate preliminary values for the predicted
th
of X , X , …, X for the i measurement. Let X denote the
1 2 m efficiencies, ε˜ .
i
n × m design matrix, given by:
˜ ˜ ˜
ε˜ 5 β X 1β X 1· · ·1β X ,
i 1 i1 2 i2 m im
X X … X
(43)
11 12 1m
i 5 1,2,…,n.
X X … X
21 22 2m
X 5 (35)
¡ ¡ ¢ ¡
8.4.7 Use these preliminary values to refine the variance
1 2
X X … X
n1 n2 nm estimates for the measured efficiencies ε , as described in 6.9.3.
i
See Eq 13 for example.
Let Ɛ denote the n × 1 column vector of measured efficien-
8.4.8 Calculate the refined weights w , w , …, w , as de-
cies:
1 2 n
scribed in 6.9.4. See Eq 14.
T
~ε ε … ε !
Ɛ 5 (36)
1 2 n
8.4.9 Use WLS with the weights w , w , …, w to solve the
1 2 n
ˆ ˆ
8.3 See Section 6 to choose Option 1 or 2. The fitting
equation Xβ>Ɛ for β. The solution satisfies the matrix equa-
technique for Option 1 is weighted least squares (WLS). The
tion:
fitting technique for Option 2 is generalized least squares
T ˆ T
X WX β 5 X WƐ (44)
(GLS). ~ !
8.4 Option 1—Weighted Least Squares (WLS) without Co-
where:
variances:
W 5 diag w ,w ,…,w (45)
~ !
1 2 n
8.4.1 Use uncertainty propagation to estimate a partial
In the special case when m = 2, the solution can be obtained
variance u ~ε ! for each measured efficiency, as described in
cP i
in the manner described in 8.4.5. In general, matrix algebra can
6.9.1. See Eq 10 for example.
be used.
8.4.2 Calculate the preliminary weights w˜ ,w˜ ,…,w˜ as de-
1 2 n
ˆ
8.4.10 Calculate the partial covariance matrix for β.
scribed in 6.9.2. See Eq 11.
T 21
V 5 ~X WX! (46)
8.4.3 Use WLS with the weights w˜ ,w˜ ,…,w˜ to obtain a
1 2 n
˜ NOTE 4—When using curve-fitting software that implements only
preliminary estimate of the parameter vector, β. The solution
ordinary (unweighted) linear least squares, one can achieve the results of
satisfies the matrix equation:
WLS by dividing each measured efficiency, ε , and each matrix entry, X ,
i ij
by the uncertainty u (ε ) before fitting the curve. Afterwards, given the
p i
T˜ ˜ T˜
~X WX!β 5 X WƐ (37)
ˆ
solution, β, and covariance matrix, V, continue at 8.4.12 with the original
˜
unmodified design matrix X.
where W is the (diagonal) weighting matrix, given by:
T
8.4.11 In general, matrix algebra is used to invert X WX in
˜
W 5 diag w˜ ,w˜ ,…,w˜ (38)
~ !
1 2 n
Eq 46. However, in the special case when m = 2, if M denotes
T
T˜ the 2 × 2 matrix X WX, then:
8.4.4 The m × m (symmetric) matrix X WX and the m × 1
T
˜
column vector X Wε can be calculated as follows:
M M
22 11
V 5 , V 5 , (47)
11 22
D D
n n
X X
ij ik
T˜
~X WX! 5 w˜ X X 5 (39)
( i ij ik ( 2
jk
u ε
i51 i51 ~ !
cP i
and
n n
X ε
ij i 2M
T
˜
~X WƐ! 5 w˜ X ε 5 (40)
j ( i ij i ( V 5 V 5 (48)
u ε 12 21
~ !
i51 i51
cP i D
NOTE 3—When using curve-fitting software that implements only
where D = M M  − M M .
ordinary (unweighted) linear least squares, one can achieve the results of
11 22 12 21
WLS by dividing each measured efficiency, ε , and each coefficient, X , by
i ij 8.4.12 Calculate the predicted efficiencies, εˆ .
i
the uncertainty u (ε ) before fitting the curve.
cP i
ˆ ˆ ˆ
T εˆ 5 β X 1β X 1· · ·1β X ,
˜
i 1 i1 2 i2 m im
8.4.5 In general, matrix algebra is used to invert X WX or to
(49)
i 5 1,2,…,n.
˜
solve for β by other means. However, in the special case when
T
˜ ˜ 2
m = 2, if M denotes the 2 × 2 matrix X WX and ˜g denotes the 8.4.13 Calculate the variance, u εˆ , of each predicted
~ !
p i
T˜
2 × 1 column vector X WƐ, then: efficiency.
D8537 − 23
2 T
u εˆ 5 X VX 8.5.7 Use these preliminary values to refine the variance and
~ !
p i i i
m m21 m
covariance estimates for the measured efficiencies, ε , as
(50)
i
5 X V 12 X X V
( ij jj ( ( ij ik jk
described in 6.10.3.
j51 j51 k5j11
p
8.5.8 Construct the refined covariance matrix, U , and
Ɛ
where:
weighting matrix, W, as described in 6.10.4. See Eq 20 and Eq
th
X = i row of the design matrix X,
i
21.
th
V = j diagonal entry of V, and
jj
th
8.5.9 Use GLS with the weighting matrix W to calculate the
V = jk off-diagonal entry of V.
jk
ˆ
final least-squares solution, β. The solution satisfies the matrix
8.4.14 Assess the fit as described in Section 10.
equation:
8.4.15 Calculate the total combined standard un
...