ISO/TS 28038:2018

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/TS 28038:2018(E)
©
ISO 2018

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ISO/TS 28038:2018(E)

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ISO/TS 28038:2018(E)
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
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ISO/TS 28038:2018(E)
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 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
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO’s adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following
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.
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ISO/TS 28038:2018(E)
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;
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ISO/TS 28038:2018(E)
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.
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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.
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ISO/TS 28038:2018(E)
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
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ISO/TS 28038:2018(E)

ux,,x  ux x
  
11 1 N


V    ,
x

ux,,x  ux x
  
NN1 N

2
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.]
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ISO/TS 28038:2018(E)
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’).
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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
 
0
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

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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

1

22 

V   diagux , , ux (3)
  
x 1 m
 
  
2
 
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
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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
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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
...