ISO/TS 28038:2018

Determination and use of polynomial calibration functions

1.1 This document is concerned with polynomial calibration functions that describe the relationship between a stimulus variable and a response variable. These functions contain parameters estimated from calibration data consisting of a set of pairs of stimulus value and response value. Various cases are considered relating to the nature of any uncertainties associated with the data. 1.2 Estimates of the polynomial function parameters are determined using least‐squares methods, taking account of the specified uncertainty information. It is assumed that the calibration data are fit for purpose and thus the treatment of outliers is not considered. It is also assumed that the calibration data errors are regarded as drawn from normal distributions. An emphasis of this document is on choosing the least‐squares method appropriate for the nature of the data uncertainties in any particular case. Since these methods are well documented in the technical literature and software that implements them is freely available, they are not described in this document. 1.3 Commonly occurring types of covariance matrix associated with the calibration data are considered covering (a) response data uncertainties, (b) response data uncertainties and covariances, (c) stimulus and response data uncertainties, and (d) stimulus data uncertainties and covariances, and response data uncertainties and covariances. The case where the data uncertainties are unknown is also treated. 1.4 Methods for selecting the degree of the polynomial calibration function according to prescribed criteria are given. The covariance matrix associated with the estimates of the parameters in the selected polynomial function is available as a by‐product of the least‐squares methods used. 1.5 For the chosen polynomial function this document describes the use of the parameter estimates and their associated covariance matrix for inverse and direct evaluation. It also describes how the provisions of ISO/IEC Guide 98‐3:2008 (GUM) can be used to provide the associated standard uncertainties. 1.6 Consideration is given to accounting for certain constraints (such as the polynomial passing through the origin) that may need to be imposed and also to the use of transformations of the variables that may render the behaviour of the calibration function more polynomial‐like. Interchanging the roles of the variables is also considered. 1.7 Examples from several areas of measurement science illustrate the use of this document.

Détermination et utilisation des fonctions d'étalonnage polynômial

General Information

• Status: Published
• Publication Date: 16-Dec-2018
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Drafting Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Current Stage: 9093 - International Standard confirmed
• Start Date: 10-May-2023
• Completion Date: 13-Dec-2025

Relations

• Consolidated By: ISO 6709:2022 - Standard representation of geographic point location by coordinates
• Effective Date: 06-Jun-2022

Standards Content (Sample)

TECHNICAL ISO/TS
SPECIFICATION 28038
First edition
2018-12
Determination and use of polynomial
calibration functions
Détermination et utilisation des fonctions d'étalonnage polynômial
Reference number
©
ISO 2018
© ISO 2018
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ii © ISO 2018 – All rights reserved

Contents
Foreword . iv
Introduction. v
1  Scope . 1
2  Normative references . 2
3  Terms and definitions . 2
4  Conventions and notation . 4
5  Other Standards using polynomial calibration functions . 6
6  Calibration data and associated uncertainties . 7
7  Polynomials as calibration functions . 10
7.1  General . 10
7.2  Working with polynomials. 10
7.3  Choice of defining interval for the calibration function . 13
7.4  Using the Chebyshev representation of a polynomial . 13
7.5  Assessing suitability of a polynomial function: visual inspection . 16
7.6  Assessing suitability of a polynomial function: monotonicity . 19
7.7  Assessing suitability of a polynomial function: degree . 19
7.8  Validation of the calibration function . 22
7.9  Use of the calibration function . 23
8  Generic approach to determining a polynomial calibration function . 23
9  Statistical models for uncertainty structures . 25
9.1  General . 25
9.2  Response data uncertainties . 25
9.3  Response data uncertainties and covarianc es . 28
9.4  Stimulus and response data uncertainties . 34
9.5  Stimulus and response data uncertainties and covariances . 37
9.6  Unknown data uncertainties . 40
10  Polynomials satisfying specified conditions . 43
11  Transforming and interchanging variables . 44
12  Use of the polynomial calibration function . 45
12.1  General . 45
12.2  Inverse evaluation . 45
12.3  Direct evaluation . 47
Annex A (informative) Checking the monotonicity of a polynomial . 48
Annex B (informative) Standard uncertainty associated with a value obtained by inverse
evaluation. 49
Bibliography . 51
© ISO 2018 – All rights reserved
iii
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
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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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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URL: www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee SC 6, Measurement methods and results.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.
© ISO 2018 – All rights reserved
iv
Introduction
0.1 Calibration is central to measurement science and involves fitting to measured data a function
that describes the relationship of a response (dependent) variable y to a stimulus (independent)
variable x. It also involves the use of that calibration function. This document considers calibration
functions in the form of polynomial models that depend on a set of parameters (coefficients). The
purpose of a calibration procedure is the following.
a) To estimate the parameters of the calibration function given suitable calibration data provided by a
measuring system and evaluate the covariance matrix associated with these parameter estimates.
Any uncertainties provided with the data are taken into consideration.
b) To use an accepted calibration function for inverse evaluation, that is, to determine the stimulus
value corresponding to a further measured response value, and also to obtain the stimulus value
standard uncertainty given the response value standard uncertainty. A calibration function is
sometimes used for direct evaluation, that is, to determine the response value corresponding to a
further stimulus value, and also to obtain the response value standard uncertainty given the
stimulus value standard uncertainty.
This document describes how these calculations can be undertaken using recognized algorithms. It
provides examples from a number of disciplines: absorbed dose determination (NPL), flow meter
characterization (INRIM), natural gas analysis (VSL), resistance thermometry (DFM) and isotope‐based
quantitation (NRC).
0.2 The nature of the calibration data uncertainty information influences the manner in which the
calibration function parameters are estimated and how their associated covariance matrix is provided.
This uncertainty information may include quantified measurement covariance effects relating to
dependencies among the quantities involved.
0.3 Since in any particular instance the degree of the polynomial calibration function is not generally
known, this document recommends the determination of polynomial functions of all degrees up to a
stipulated maximum (limited by the quantity of data available), followed by the selection of one of these
degrees according to suitable criteria. One criterion relates to the requirement that the calibration
function is monotonic (strictly increasing or decreasing) over its domain. A second criterion relates to
striking a balance between the polynomial calibration function providing a satisfactory explanation of
the data and the number of parameters required to describe that polynomial. A further criterion relates
to visual acceptance of the polynomial function.
0.4 The determination and use of a polynomial calibration function thus consist of the following
steps:
1 obtaining calibration data and available uncertainty information including covariance
information when available;
2 determining polynomial functions of all degrees up to a prescribed maximum in a manner that
respects the uncertainty information;
3 selecting an appropriate function from this set of polynomial functions according to the criteria
in Subclause 0.3;
4 providing estimates of the parameters of the chosen polynomial function and obtaining the
covariance matrix associated with those estimates;
© ISO 2018 – All rights reserved
v
5 using the calibration function for inverse evaluation and associated uncertainty evaluation;
6 using the calibration function for direct evaluation and associated uncertainty evaluation.
0.5 This document treats steps 2 to 6 listed in Subclause 0.4 employing the principles of ISO/IEC
Guide 98‐3:2008 (GUM). Therefore, as part of step 1, before using this document, the user should
provide available standard uncertainties and covariances associated with the measured x‐ and y‐values.
Account should be taken of the provisions of the GUM in obtaining these uncertainties on the basis of a
measurement model that is specific to the area of concern.
© ISO 2018 – All rights reserved
vi
TECHNICAL SPECIFICATION ISO/TS 28038:2018(E)

Determination and use of polynomial calibration functions
1 Scope
1.1 This document is concerned with polynomial calibration functions that describe the relationship
between a stimulus variable and a response variable. These functions contain parameters estimated
from calibration data consisting of a set of pairs of stimulus value and response value. Various cases are
considered relating to the nature of any uncertainties associated with the data.
1.2 Estimates of the polynomial function parameters are determined using least‐squares methods,
taking account of the specified uncertainty information. It is assumed that the calibration data are fit for
purpose and thus the treatment of outliers is not considered. It is also assumed that the calibration data
errors are regarded as drawn from normal distributions. An emphasis of this document is on choosing
the least‐squares method appropriate for the nature of the data uncertainties in any particular case.
Since these methods are well documented in the technical literature and software that implements
them is freely available, they are not described in this document.
1.3 Commonly occurring types of covariance matrix associated with the calibration data are
considered covering (a) response data uncertainties, (b) response data uncertainties and covariances,
(c) stimulus and response data uncertainties, and (d) stimulus data uncertainties and covariances, and
response data uncertainties and covariances. The case where the data uncertainties are unknown is also
treated.
1.4 Methods for selecting the degree of the polynomial calibration function according to prescribed
criteria are given. The covariance matrix associated with the estimates of the parameters in the selected
polynomial function is available as a by‐product of the least‐squares methods used.
1.5 For the chosen polynomial function this document describes the use of the parameter estimates
and their associated covariance matrix for inverse and direct evaluation. It also describes how the
provisions of ISO/IEC Guide 98‐3:2008 (GUM) can be used to provide the associated standard
uncertainties.
1.6 Consideration is given to accounting for certain constraints (such as the polynomial passing
through the origin) that may need to be imposed and also to the use of transformations of the variables
that may render the behaviour of the calibration function more polynomial‐like. Interchanging the roles
of the variables is also considered.
1.7 Examples from several areas of measurement science illustrate the use of this document.
© ISO 2018 – All rights reserved
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements 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/IEC Guide 98‐3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty
in measurement (GUM:1995)
ISO/IEC Guide 99:2007 (corr. 2010), International vocabulary of metrology — Basic and general
concepts and associated terms (VIM)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO/IEC Guide 98‐3:2008 and
ISO/IEC Guide 99:2012 and the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http://www.electropedia.org/
— ISO Online browsing platform: available at https://www.iso.org/obp
3.1
measurement uncertainty
non‐negative parameter characterizing the dispersion of the quantity values being attributed to a
measurand, based on the information used
[SOURCE: ISO/IEC Guide 99:2007 (corr. 2010), 2.26, modified ‐ Notes 1 to 4 have been deleted.]
3.2
standard measurement uncertainty
standard uncertainty
measurement uncertainty (3.1) expressed as a standard deviation
[SOURCE: ISO/IEC Guide 99:2007 (corr. 2010), 2.30.]
3.3
measurement covariance matrix
covariance matrix
symmetric positive‐definite matrix of dimension NN associated with an estimate of a vector quantity
of dimension ,N1 containing on its diagonal the squared standard uncertainties associated with the
components of the estimate of the quantity, and, in its off‐diagonal positions, the covariances associated
with pairs of components of the estimate of the quantity
Note 1 to entry: A measurement covariance matrix V of dimension NN associated with the estimate x of a
x
vector quantity X has the representation
© ISO 2018 – All rights reserved

ux,,x  ux x
  
11 1 N


V    ,
x

ux,,x  ux x
  
NN1 N

where ux ,x u x is the variance (squared standard uncertainty) associated with x and ux ,x is
   
 
ii i i ij
the covariance associated with x and x . ux ,x 0 if elements X and X of X are uncorrelated.
 
i j ij i j
Note 2 to entry: A covariance matrix is also known as a variance‐covariance matrix.
[SOURCE: ISO/IEC Guide 98‐3:2008/Suppl. 1:2008, 3.11 (definition of uncertainty matrix), modified ‐
definition slightly modified, Note 2 deleted, Note 3 becomes Note 2 to entry, slightly modified.]
3.4
correlation matrix
symmetric positive‐definite matrix of dimension NN associated with an estimate of a vector quantity
of dimension ,N1 containing the correlations associated with pairs of components of the estimate
Note 1 to entry: A correlation matrix R of dimension NN associated with the estimate x of a vector quantity
x
X has the representation

rx,,x  rx x
   
11 1 N


R    ,
x

rx,,x  rx x
  
NN1 N

where rx ,x 1 and rx ,x is the correlation associated with x and x . When elements X and X of
 
 
ii ij i j i j
X are uncorrelated, rx ,x 0.

ij
Note 2 to entry: Correlations are also known as correlation coefficients.
Note 3 to entry: R is related to V (see definition 3.3) by
x x
V DRD ,
xxxx
where D is a diagonal matrix of dimension NN with diagonal elements ux ,, ux . Element ij, of
     
1 N
x
V is given by
x
ux,,x r x x u x ux .

  
ij ij i j
[SOURCE: ISO/IEC Guide 98‐3:2008/Suppl. 2:2011, 3.21, modified ‐ definition slightly modified, Notes 4
and 5 deleted.]
© ISO 2018 – All rights reserved
3.5
measurement model
mathematical relation among all quantities known to be involved in a measurement
[SOURCE: ISO/IEC Guide 99:2007 (corr. 2010), 2.48, modified ‐ Notes 1 and 2 deleted.]
3.6
calibration
operation that, under specified conditions, in a first step, establishes a relation between the quantity
values with measurement uncertainties provided by measurement standards and corresponding
indications with associated measurement uncertainties and, in a second step, uses this information to
establish a relation for obtaining a measurement result from an indication
Note 1 to entry: A calibration may be expressed by a statement, calibration function, calibration diagram,
calibration curve, or calibration table. In some cases, it may consist of an additive or multiplicative correction of
the indication with associated measurement uncertainty (3.1).
Note 2 to entry: Calibration should not be confused with adjustment of a measuring system, often mistakenly
called ‘self‐calibration’, nor with verification of calibration.
Note 3 to entry: Often the first step alone in the above definition is perceived as being calibration.
[SOURCE: ISO/IEC Guide 99:2007 (corr. 2010), 2.39.]
3.7
stimulus interval
interval in the stimulus variable over which a calibration function is defined
3.8
stimulus
quantity that effects a response (3.9) in a measuring system
3.9
response
quantity resulting from stimulating a measuring system
3.10
inverse evaluation
use of a calibration function to provide the stimulus value corresponding to a response value
3.11
direct evaluation
use of a calibration function to provide the response value corresponding to a stimulus value
4 Conventions and notation
For the purposes of this document the following conventions and notations are adopted.
4.1 The quantity whose values are provided by measurement standards is termed the independent
variable x (also called ‘stimulus’) and the quantity described by measuring system indication values is
termed the dependent variable y (also called ‘response’).
© ISO 2018 – All rights reserved
4.2 x and y denote the measured values of the Cartesian co‐ordinates of the ith point xy,,
 
i i ii
im1, , , in a calibration data set of m points. Vector and matrix notation is frequently used. The
T
values of x and y are often expressed as vectors, with ‘ ’ denoting ‘transpose’:
i i
T T
 
xxx,, , yyy,, .
1 m 1 m
 
A matrix or vector of zeros is denoted by .
4.3 True values (that would be achieved with perfect measurement) of the co‐ordinates of the ith
point are denoted by and . Measured values of points expressed in Cartesian co‐ordinates and
 
i i
corresponding true values are related by:
xd , ye ,
ii i ii i
where d and e denote the errors in x and y , respectively. Errors are unknowable, but can often be
i i i i
estimated.
4.4 The standard uncertainties associated with x and y are denoted by ux and uy ,
   
i i i i
respectively. The covariance associated with x and x is denoted by ux ,x . Similarly, the

i j ij
covariance associated with and y is denoted by uy ,y .
y
 
i j ij
NOTE This document does not consider cross‐variances ux ,y since no practical calibration application
 
ij
has been identified in which cross‐variances are prescribed.
4.5 The uncertainty information for the specification of a polynomial calibration problem is
represented by matrices V and V each of dimension mm holding the variances (squared standard
x y
2 2
uncertainties) ux ux ,x and uy uy,,y and the covariances ux ,x and
      
 
iii iii ij
uy ,y . Formula (1) denotes the covariance matrix associated with x and Formula (2) denotes the

ij
covariance matrix associated with y:

ux,,x  ux x
  
11 1 m


V    , (1)
x

ux,,x  ux x
  
mm1 m


uy,,y  u y y
   
11 1 m


V    . (2)
y

uy,,y  uy y
  
mm1 m

© ISO 2018 – All rights reserved
For a particular calibration problem, either of V and V may be equal to 0.
x y
NOTE This document is concerned with problems in which the ux or the uy are generally different
   
i i
(heteroscedastic case).
4.6 If the covariances ux ,x (i  j) are all zero, V is a diagonal matrix:
 
ij x
2
ux


22 

V   diagux , , ux (3)
  
x 1 m
 
  
 
ux

m

and similarly for the uy ,y .

ij
4.7 The elements below the main diagonal of a symmetric matrix are generally not displayed. Thus,
for instance, the representation of the matrix

12,,07 0,8 12,,07 0,8

07,,25 05, is 25,,05 .

0,,8 05 17, sym. 17,


4.8 A polynomial calibration function relating y and x is denoted by px , where n is the degree
 
n
of the polynomial. It is denoted by px, a when it is necessary to indicate that it depends on n  1

n
T
parameters 
aaa,, .
0 n
ˆ
q
4.9 An estimate of a quantity is denoted by q. Model values corresponding to the data point
ˆ
ˆˆ ˆ ˆ
x , y , namely, satisfying yp x , a are denoted by and .
x y
  
ii in i i i
4.10 The function that is minimized to estimate the polynomial function parameters a is termed the
objective function.
4.11 While data values in examples are provided to a given number of decimal digits, results of
calculations are sometimes provided to a greater number, for comparison purposes, for example.
5 Other Standards using polynomial calibration functions
Other Standards concerned with polynomial calibration are as follows.
[23]
a) ISO 6143:2006 is concerned with comparison methods for determining and checking the
composition of calibration gas mixtures. It contains clauses on the determination (and use) of
‘analysis functions’ given calibration data. The analysis functions considered are polynomials of
degrees 1, 2 and 3 representing the stimulus as a function of response. Uncertainties are permitted
© ISO 2018 – All rights reserved
in the stimulus data values and the response data values. Covariances are permitted in the stimulus
data, but not in the response data.
[24]
b) ISO 7066‐2:1988 covers basic methods for determining and using polynomial calibration
functions in the context of the measurement of fluid flow: assessment of uncertainty in the
calibration and use of flow measurement devices. It handles, in the language of this document,
standard uncertainties associated with the data y‐values, and inverse evaluation.
[20]
c) ISO 11095:1996 specifically addresses reference materials, outlining general principles needed
to calibrate a measuring system and to maintain that system in a state of statistical control. It
provides a basic method for estimating a straight‐line calibration function when stimulus values are
known exactly.
[21]
d) ISO 11843‐2:2000 concerned with capability of detection, uses straight‐line calibration functions
when the standard uncertainties in the response values are constant or depend linearly on stimulus.
[22]
ISO 11843‐5:2008 extends the provisions of ISO 11843:2000 to the non‐linear case.
[25]
e) ISO/TS 28037:2010 covers the same uncertainty structures as in the current document, and is
concerned with straight‐line calibration. The current document can be regarded as an extension of
ISO/TS 28037 to polynomial functions of general degree.
6 Calibration data and associated uncertainties
6.1 Calibration consists of two stages (definition 3.6). The first stage establishes a relation between
(stimulus) values provided by measurement standards and corresponding instrument response values.
The second stage uses this relation to obtain stimulus values from further instrument response values
(inverse evaluation). The relation also allows a response value to be obtained given a further stimulus
value (direct evaluation). In this document the relation takes the form of a polynomial calibration
function, which is described by a set of parameters, estimates of which are deduced from the calibration
data and the associated uncertainties.
NOTE This document is not concerned with determining a mathematical form from which a stimulus value
can be determined explicitly given a response value. Such a form is known in some fields of application as an
analysis function.
6.2 The calibration of a measuring system should take into account prescribed calibration data
uncertainties and any prescribed covariances.
6.3 An acceptable calibration function will satisfy a statistical test for compatibility with the
calibration data and the accompanying uncertainties. In many circumstances it will also have to be
monotonic (strictly increasing or decreasing).
6.4 Standard uncertainties and covariances accompany the parameter estimates, and the information
concerning the calibration function is used to provide a stimulus value (or response value) and the
associated standard uncertainty corresponding to a given response value (or stimulus value,
respectively).
6.5 Any particular set of calibration data xy, , im1,,  , will have an uncertainty structure
 
ii
specific to that data. At one extreme, nothing is known about the uncertainties and covariances and, to
proceed, assumptions are necessary. At the other extreme, all standard uncertainties ux and uy
   
i i
© ISO 2018 – All rights reserved
and all covariances ux ,x and uy ,y are prescribed. In practice, the provided information often
  
ij ij
lies between these extremes.
NOTE In this document any uncertainty or covariance that is not prescribed is taken as zero.
6.6 The following five cases can be distinguished, the first four in approximately increasing order of
complexity of uncertainty structure. The fifth is different in character in the sense that the uncertainty
information is unknown.
a) Response data uncertainties. Standard uncertainties u(y), i = 1,.,m, prescribed.
i
b) Response data uncertainties and covariances. As 6.6 a) with covariances u(y , y), i = 1,.,m, j = 1,.,m
i j
(i ≠ j), also prescribed.
c) Stimulus and response data uncertainties. As 6.6 a) with standard uncertainties u(x), i = 1,.,m, also
i
prescribed.
d) Stimulus and response data uncertainties and covariances. As 6.6 c) with covariances u(x , x) and
i j
u(y , y), i = 1,.,m, j = 1,.,m (i ≠ j), also prescribed.
i j
e) Unknown data uncertainties.
In cases 6.6 a) to 6.6 d), the prescribed uncertainties and covariances are summarized as covariance
matrices V and V as appropriate according to Subclause 4.5.
y
x
NOTE Cases 6.6 a) to 6.6 c) can be treated as special cases of 6.6 d), but computationally less efficiently.
6.7 The key distinction between calibration data with prescribed uncertainties and calibration data

with unknown uncertainties made in this document is the following.
a) For calibration data with prescribed uncertainties and covariances [cases 6.6 a) to 6.6 d)] a metric,
such as the chi‐squared statistic (Subclause 7.7.1), that uses the uncertainties and covariances may
be employed to decide whether a candidate calibration function, in this document a polynomial of a
particular degree, is statistically valid. This approach assumes the plausibility of the specified
uncertainty information.
b) For calibration data with unknown uncertainties [case 6.6 e)] a chi‐squared statistic can still be
calculated for candidate polynomial models. The assumptions are made that the data errors in the
response variable are homogeneous and the data errors in the stimulus variable are negligible. The
value of the chi‐squared statistic can be used to estimate the response variable standard
uncertainty and the provisions of 6.7 a) then applied.
6.8 A polynomial is selected from a set of candidate polynomials of various degrees according to a
suitable criterion such as AIC (Subclause 7.7.3). For some data sets with prescribed uncertainties there
might be no suitable polynomial (or any other smooth) representation consistent with this information.
For the data in Figure 1 a), the only uncertainties are associated with the y‐values, the vertical bars
represent ± 1 standard uncertainty, and the covariances are zero. The smallness of the standard
uncertainties prevent a monotonic function that is consistent with the data from being obtained. For the
data in Figure 1 b), identical to those in Figure 1 a) except that the standard uncertainties are some
three times as large, a monotonic polynomial of low degree is suitable. An acceptable calibration
[30]
function should be both monotonic (Subclause 7.6) and statistically adequate .
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a) Statistical inadequacy b) Statistical adequacy
Key
X stimulus (a.u.)
Y response (a.u.)
NOTE Error bars denote ±1 standard uncertainty. ‘a.u.’ denotes arbitrary units.
Figure 1 —Statistical inadequacy and adequacy
NOTE Figure 1 a) appears to relate to mis‐specification of the standard uncertainties associated with the
calibration data; their possible rectification is beyond the scope of this document.
6.9 Estimates of the calibration function parameters depend on the calibration data and, apart from
case 6.6 e), the prescribed data uncertainties and covariances. The law of propagation of uncertainty
(LPU) in the ISO/IEC Guide 98‐3:2008 (GUM) can be applied to propagate calibration data uncertainties
and covariances through the computation of the calibration function parameters to obtain parameter
uncertainties and covariances. When there is no uncertainty associated with the stimulus values
(Subclauses 9.2, 9.3 and 9.6), the propagation is exact, since the parameters of a polynomial calibration
function depend linearly on the data response values and LPU applies with no approximation error in
such cases (see Subclause 7.2.1). For other cases (Subclauses 9.4 and 9.5), the propagation is
approximate, based on a linearization about the parameter estimates. The approximation incurred by
the linearization will often be fit for purpose for practical calibration problems.
NOTE If linearization is unfit for purpose, such as when the stimulus value uncertainties are large, the
propagation of distributions may be used to obtain parameter estimates, uncertainties and covariances. This
approach (ISO/IEC Guide 98‐3:2008/Suppl. 2:2011), which uses a Monte Carlo method, is beyond the scope of this
document.
6.10 Uncertainty information concerning the calibration function parameters takes the form of a
covariance matrix for (estimates of) those parameters. That information can equally be represented as
the standard uncertainties associated with those parameters together with their correlation matrix
(definition 3.4), which may be a more useful form. Either form can be used in the evaluation of the
standard uncertainty associated with inverse or direct evaluation.
6.11 When the calibration function is used for inverse evaluation (Subclause 12.2), the application of
LPU is approximate, even for polynomials of degree one, because when used inversely the polynomial is
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non‐linear in its parameters. Again, the approximation incurred by the linearization will often be fit for
purpose.
NOTE When it is acceptable to express a polynomial calibration function as x in terms of y, the polynomial
calibration function determined is used directly and there is no linearization error in that stage of the calculation.
7 Polynomials as calibration functions
7.1 General
7.1.1 Given calibration data, this clause considers the determination of a relationship yp x
 
n
describing the dependent variable y as a polynomial function of degree n of the independent variable x.
7.1.2 If the degree n is not known in advance, as is commonly the case, an appropriate polynomial
degree is to be determined. Subclause 7.7 describes determination of the degree such that the resulting
function satisfies suitable criteria.
7.1.3 The information used to determine the polynomial calibration function is the calibration data and
any calibration data uncertainties and covariances. In this document, the calibration data are denoted
by xy,, im1 , , ,that is, m pairs of measured values of x and y. The highest degree of
n

ii max
polynomial to be considered is also to be specified, where n is less than the number of distinct
max
values of x .
i
NOTE Annex D of ISO/TS 28037:2010 indicates how the uncertainties and covariances associated with the
measured response and stimulus variables can be provided in some cases, giving an interpretation of that
information.
7.2 Working with polynomials
7.2.1 For a degree higher than one a polynomial is non‐linear in terms of its variable x, but it is linear in
its parameters (coefficients). A polynomial of degree n (order n + 1) has n + 1 coefficients. It can be
expressed in monomial form, with coefficients hh, , , as:
0 n
n
2 nr
pxh hxhxhx hx . (4)


nn01 2 r
r0
2 n
7.2.2 The functions 1, , xx , , x are known as the monomial basis functions for polynomials of degree
n. A polynomial of degree 1 is a straight line, degree 2 a quadratic function, degree 3 a cubic function,
etc. An immediate appeal of polynomials is that their evaluation requires only some n additions and n
multiplications (Subclause 7.4).
7.2.3 Polynomials are often suitable for representing a smooth curve or data generated from a smooth
curve over a given interval. They are extremely flexible: mathematically a polynomial of an appropriate
degree can approximate any smooth (continuous) curve to a given numerical precision. Polynomials of
modest degree are less appropriate for representing curves with abrupt changes in value or gradient, or
describing a saturation effect.
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7.2.4 Whilst the description of polynomial functions in the monomial form [Formula (4)] makes clear
their nature, the use of this form can lead to difficulties with numerical computation and interpretation
r
of the contribution of individual terms. A first difficulty is that for x 1, the terms x become very
r
large in magnitude as r increases. Similarly, for x 1, the terms x become very small in magnitude
t
as r increases. This imbalance is alleviated by working with polynomials in a normalized variable
lying in the interval [−1, 1] that depends linearly on x, thus ensuring that the (transformed) polynomial
 
is also of degree n in t. If x lies in the interval xx,,
 min max
2xxx
min max
t , (5)
xx
max min
with all its powers lying in the interval [−1, 1]. The polynomial can then be expressed as
n
2 nr
Ptq qtqtqt qt (6)


nn01 2 r
r0
for some coefficients qq,, . A second difficulty arises from the fact that, especially for large r, the
0 n
r2 r
monomial basis function t looks similar to t in the interval [−1, 1]. Figure 2 a) depicts the
2 4 6 8
monomials t (uppermost curve), t , t and t (lowermost curve). The similarity of these basis
functions leads to ill-conditioning in determining the monomial parameters, which will mean a loss of
numerical precision. This ill‐conditioning worsens rapidly as the degree increases, with the
consequence that the loss of numerical precision can become catastrophic for higher polynomial
degrees. A third difficulty relates to the interpretation of the coefficients in the monomial form in the
original variable, namely, Formula (4). However, in Formula (6), the monomial form in the transformed
r
t, the term involving t contributes an amount lying between and , with at least one
variable  q q
r r
of these values attained at the endpoints of the interval [−1, 1].

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2 4 6 8
a) Monomial functions t, t, t and t b) Chebyshev polynomials
on the interval [1,1] T(t), T(t), T(t), and T(t)
2 3 4 5
Key
X independent variable t
Y dependent variable
Figure 2 — Monomial functions and Chebyshev polynomials
7.2.5 There are other forms for the basis functions that have even better properties than Formula (6).
The Chebyshev polynomials Tt , used in this document, are one such set of basis functions. They are
 
r
defined by recurrence on the interval [−1,1] (see reference [6], page 1):
Tt1,, Tt t Tt2tTtTt, r2. (7)
         
01 rr1 r2
Chebyshev polynomials can also be defined using the trigonometrical relationship
Trcoscos , cosrt. (8)

r
Figure 2 b) depicts Tt (least oscillatory), Tt , Tt and Tt (most oscillatory). The
       
2 3 4 5
intertwining of the Tt can be shown generally to lead to much better numerical conditioning than
 
r
r
the use of the t . The Chebyshev representation of a polynomial of degree n is
n
pxP t aT taT t aT t . (9)
    
nn 00 nn rr

r0
[16]
NOTE Another class of basis functions due to Forsythe is described in reference [7] where an algorithm is
given for converting from the Forsythe form to the Chebyshev representation. The Forsythe form is based on
generating a set of basis functions that are orthogonal with respect to the data x‐values. Although the Forsythe
form has excellent numerical properties, the Forsythe basis functions depend on the data x‐values, rendering their
use in conjunction with polynomials from other sources inconvenient. Moreover, the Forsythe form has not been
generalized to data possessing x‐value uncertainties or any covariances.
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7.3 Choice of defining interval for the calibration function
7.3.1 Consider the use of a determined calibration function p(x) for inverse evaluation (Subclause
n
12.2), that is, to provide the value x of the stimulus variable x corresponding to a value y for the
0 0
response variable y, particularly in the case where x would lie near an extremity of the interval
[x , x ] over which p(x) is defined. Assume p(x) is strictly increasing over [x , x ]; a similar
min max n n min max
argument applies in the decreasing case. The y‐values at the interval endpoints are y = p(x ) and
min n min
y = p(x ). For any value y in the interval [y , y ], x is given uniquely by solving the equation
max n max 0 min max 0
p(x) = y. However, since y is subject to uncertainty, it may lie outside the interval [y , y ], with the
n 0 0 0 min max
consequence that x would lie outside the defining interval [x , x ].
0 min max
NOTE There is no problem in the above respect for direct evaluation.
7.3.2 There are two ways of treating such a situation. The first is to allow only values of y within the
interval [y , y ], which would limit the applicability of the calibration function. The second way is to
min max
extend the interval over which the calibration function is defined. One possibility is to extend the
interval [min x , max x] as little as possible, say to [min x  Δx, max x + Δx], where
i i i i i i i i
Δx = 0,1(max x − min x). Some experimentation may be required to determine an appropriate interval
i i i i
in any particular case. There may be application‐specific reasons to select an appropriate interval. The
most extreme case arises when the gradient of the calibration curve is small in magnitude, since a small
change in response induces a large change in stimulus [illustrated in the optical density‐absorbed dose
calibration function shown in Figure 5 b) for response values of approximately 0,45]. For the examples
given in this document, suitable intervals [x , x ] were chosen.
min max
7.3.3 The interval [min x, max x] should be extended as little as possible to reduce extrapolation
i i i i
beyond the span of the data, which is generally considered unsafe.
NOTE For the optical density‐absorbed dose example (Subclauses 7.5.3 and 9.2), the use of
Δx = 0,1(max x − min x) is inadequate for inverse interpolation for some y close to y as defined in Subclause
i i i i 0 max
7.3.1 with u(y) = 0,003, but the replacement of 0,1 by 0,15 proves satisfactory.
7.4 Using the Chebyshev representation of a polynomial
7.4.1 By using Chebyshev polynomials in a normalized variable [Formula (5)] it is possible to use
[2]
polynomial functions of mode
...