1.7 Evaluation of measurement results in calibrating
1.7.4 Determination of the linear calibration functions of measuring
In the most general case the problem of determining the calibration functions of mea- suring instruments can be formulated in the following manner. It is known that between the measurandXat the input of a measuring instrument and output quantityYof this instrument there is a functional dependenceY DF .X /. In the process of calibration the values ofXandY are measured or given with some uncertainties. It is required to determine a parametric form of the dependenceF .X /, to evaluate its parameters, and to calculate the uncertainty of the curve fitted to the experimental data.
The form ofF .X / is by no means always known in advance, for example, from certain chemical regularities. Usually, in practice the matter concerns some kind of approximation ofF .X / with functions of the kind givenf .x/, which satisfies the requirements for the target uncertainty associated with fitting.
The accuracy of determining the calibration functions of a measuring instrument is caused by the uncertainty of the experimental data.xi,yi/, by the algorithm for evaluating the parameters of thef .X /dependence, and by the correctness of the ap- proximation of the theoretical calibration dependence of the function selected. These factors make a joint impact on the uncertainty of the calibration function estimate, which can be reduced at the expense of the optimal design of the measurement exper- iment when evaluating the calibration functions of measuring instruments.
Section 1.7 Evaluation of measurement results at calibration 45 The importance of measurement design increases when developing the measure- ment procedures intended for the routine calibration of measuring instruments, where it is necessary to provide the target uncertainty of the calibration function. The mea- surement design consists of selecting a number of valuesxi, i D1,: : :,N, at which measurements of the dependent quantityyiare made, and the location of these values xi,i D 1,: : :,N within the measurement range and the number of repeated mea- surementsyij,j D 1,: : :,ni at the specified valuexi are found, and, and finally, the requirements for the uncertainties associated with the valuesxi, i D 1,: : :,N themselves (values of the independent variableX) are determined.
In view of this the problem of determining the calibration curves of the measuring instruments is divided into the following stages [100, 106, 144, 430, 463]:
(1) selection of a model of the calibration functionf .X /;
(2) design of a measurement experiment;
(3) evaluation of calibration curve parameters and associated uncertainties.
The error of determining the calibration function can be represented in the form of two summands:
_
f .x/F .x/Df .x/_ f .x/Cf .x/F .x/,
where F .x/ is the theoretical (true) calibration dependence, f .x/ is the accepted model of the calibration curve, and
_
f .x/ is the estimate of the model on the basis of experimental data.
The summand
_
f .x/f .x/is the transformed error that is completely determined with the uncertainties of the initial experimental data and the data processing algo- rithm. The second summandf .x/F .x/is the methodical error that has a systematic character and characterizes the inadequacy of the model selected.
Provided that the extreme cases are being considered, when one of the summands is converted into zero, then the equality of the second summand to zero leads to a sta- tistical problem of evaluating the parameters of a regression, the classical statement of which corresponds to the absence of errors of an independent variablexi. The equality of the first summand to zero corresponds to the case of the absence of the experimental data errors, which leads to the classical problem of the function interpolation on the basis of its valuesyi, iD1,: : :,N given at specified pointsxi, i D1,: : :,N.
There is always a certain “threshold” inconsistency of the accepted model with real data. Therefore, the second summand always differs from zero. It is important to take into account this inconsistency when evaluating the uncertainties of determining the functional dependence or when ensuring that this inconsistency does not significantly influence the calculation of the measurement uncertainty.
Usually, in practice the second way is selected. A certain form of the calibration function is assumed, reasoning from the experience of a researcher, the simplicity re- quirements, and its suitability for control at repeated applications. Then the parameters
of the accepted model are evaluated. After that, the adequacy of the model for the real data is checked using the criteria of agreement – for example, the criterion2. If the model and data are consistent, then the data on the uncertainty of the model is ne- glected.
As previously mentioned, the selection of the model defies formalized description.
It is only possible to formulate the general principles, the main ones of which are
the rational relationship between the model complexity and the uncertainty of the model evaluation. It should be taken into account that with a limited volume of ex- perimental data the increase in the number of parameters being evaluated results in an increase in the combined uncertainty of the calibration function (with a probable decrease in the systematic error due to the model inadequacy);
ensuring the uncertainty required for determining the functional dependence rea- soning for the purposes of its further application.
Based on these principles, in practice the linear calibration functions are rather of- ten used. Provided that the accuracy required in the range being considered is still achieved, then the range is divided into subranges, and a piecewise-linear approxima- tion of the calibration curve is used.
The initial data for evaluating the calibration function are the pairs of values with the associated measurement uncertainties:ạ.yi,u .yi//,.xi,u .xi//,i D1,: : :,Nº. As a rule, the uncertainties of measuring instrument indications u .yi/are calculated ac- cording to the type A evaluation on the basis of repeated indications of the measur- ing instrument or given variance of indication repeatability of the measuring instru- ment. The uncertainty of the values of the independent variableu .xi/is calculated on the basis of the available information about the uncertainty of giving the pointsxi – for example, about the procedure of preparing calibration solutions based on certified materials.
Then after evaluating the parameters, the linear curve can be easily represented in the formY D a0Cb.X NX /[430]. For such representation the least-squares method makes it possible to obtain unbiased and uncorrelated estimates.a0,b/, in case of the equal measurement uncertainties and uncorrelated dataạ.yi,u.y//,.xi,u.x//, iD1,: : :,Nºthey have
a0D PN
iD1yi
N , bD
PN
iD1yi.xi Nx/
PN
iD1.xi Nx/2 .
The standard uncertainty of the calibration curve at the pointxis calculated by the formula
u2.x/D
u2.y/Cb2u2.x/
1
N C .x Nx/2 PN
iD1.xi Nx/2
!
. (1.22)
Section 1.7 Evaluation of measurement results at calibration 47 The expanded uncertainty is calculated by the formulaUP.x/ D k u.x/; the coverage factor is assumed to be equal tok D 2 for the confidence levelP D 0.95 andkD3 for the confidence levelP D0.99.
For the arbitrary covariance matrix of initial dataạ.xi,yi/, i D1,: : :,Nºthe gen- eralized LSM is used. The corresponding estimates of parameters.a0,b/are obtained using the numerical methods. In [311] an approach to evaluating parameters of the linear calibration curve on the basis of the calculation of a joint probability density of the parameters.a0,b/, taking into account the available a priori information, is con- sidered.
The design of a measurement experiment for evaluating the calibration curves is realized taking into account the accepted algorithm for processing thd emeasurement results, since a particular dependence of the uncertainty associated with.a0,b/on the parameters of the experiment plan is determined by the applied algorithm for data processing.
The distinction of defining a problem of designing a measurement experiment in metrology from the similar procedure in the theory of an optimal experiment is re- vealed in the following.
(1) Usually the matter concerns the availability of providing the target uncertainty of the calibration curve, whereas in the theory of an optimal experiment there is a minimum or maximum of the accepted criterion of optimality.
(2) The uncertainty of the calibration curve is defined not only by random factors the influence of which can be decreased at the expense of increasing the number of experimental data, but also by uncertainties associated with valuesxi, which are usually caused by the systematic factors. It is not always possible to neglect the uncertainties of independent quantitiesxi. Consequently, in the general case the re- quirements for the uncertainty of the calibration curve are finally transformed into the requirements for the uncertainty associated with valuesxi. Therefore, the de- sign of a measurement experiment includes not only the determination of a number of valuesxi and their location within the range, but also the required uncertainty of these values.
(3) Usually when determining the calibration curve the coefficients of a model of the given form, which can be different from a “true” one, are evaluated. Therefore, there is always a systematic error caused by the model inadequacy that has to be taken into account in the criterion of optimality.
It is well known that the optimal design spectrum of the measurement experiment for determining the linear regression is concentrated at the ends of the range. However, this theoretical result cannot be applied in practice, in view of the presence of system- atic errors and deviations of the calibration curve from the linear regression.
An experiment plan".M /is identified as the following set of quantities:
²x1,x2,: : :,xN n1,n2,: : :,nN
³
, X
ni DM.
A set of pointsạx1,: : :,xNºis called the design spectrum.
There are a great number of different criteria regarding the optimality of experiment plans, which may be divided into two groups: criteria connected with the covariance matrix of estimates of the model parameters, and criteria related to the uncertainty associated with the curve fitted. The basic criteria in the groups listed are as follow.
(A) Criteria related to the covariance matrixD."/of the estimates of the calibration curve parameters:
min
" jD."/jisD-optimality;
min
" Sp ŒD."/ , (Sp ŒD."/ DPm
iD1Di i) isA-optimality;
min
" max
i i i."/(where i i are the proper numbers of the covariance matrix) is E-optimality.
(B) criteria related to the uncertainty associated with the curve fitted:
min
" max
x Ed2.x,"/isG-optimality;
min
"
REd2.x,"/dxisQ-optimality,
whereEd2.x,"/is the mathematical expectation of the square of an error of the func- tional dependence at the pointx:d.x,"/D j Of .x/ftrue.x/j.
For metrology the criteria of optimality in the form of a functional are the more natural. Therefore, later on we shall consider theG-optimal plans.
The problem of finding an optimal plan of the measurement experiment while fitting the linear calibration curves will be considered below, taking into account deviations of the real calibration curve from the linear one. Let theG-optimal plan for evaluating the parameters of the linear regressionyDaCbxbe determined for the case when the model error is evaluated by a deviation from the parabolic regressiony D c0C c1xCc2x2. In addition it is assumed that the uncertainty of the independent variable is negligible, the experimental data are uncorrelated, and their uncertainties are constant within the range.
The parameters are evaluated using the LSM:
a_0DX yi=N,
_
bDX
yi.xi Nx/=X
.xi Nx/2, N
xDX
xi=N, a_ Da_0bx._
Section 1.7 Evaluation of measurement results at calibration 49 The optimal plan for the experiment is defined from the condition of minimizing a squared maximum value of the error of the calibration curve within the interval of the independent variable values:
d Dmax
x2XEạa_Cbx_ c0c1xc2x2º2.
The problem is solved in the following manner. The optimal plan is defined for some N, and then the value ofN is selected from the condition of providing the required uncertainty of the calibration curve. If only the symmetrical plans are considered, then the expression for the criteriondafter simple calculations will be reduced to the form
d Dmax
x2X
c22
²
.x Nx/2
P.xi Nx/2 N
³2
Cu2.y/
² .x Nx/2 P.xi Nx/2C 1
N
³ . In finding the maximum it is convenient to pass on to a new variablez D.x NTx/2 whereT is half the measurement range. In new designationsd is written in the form
d D max
0<z<1
c22T4
² z2
Pzi2 N
³2
Cu2.y/
² z2 Pzi2C 1
N
³ .
If c2 D 0 (i.e., the linear curve is true), then the maximum is reached atz D 1 (and is equal tou2.y/ạN1 C P1ziº). The minimumd of the criterion, which is equal to2uN2.y/, is reached atzi=1 (xi=x?TN ). This is a well known result. Ifu.y/D0 (i.e., there is not any measurement error, and the basic error is caused by the inadequacy of the model), then the maximum is achieved atz D0, and the minimum of the criterion is achieved atzi D1=2.xi D Nx˙ p22T /and is equal toc224T4. In the general case the design spectrum is concentrated at the points
xi D Nx˙p
zT, (1.23)
where
zD 8ˆ ˆˆ ˆ<
ˆˆ ˆˆ : 1 4
´ 1C
s
1C8u2.y/
c22T4N μ
, 8u2.y/
c22T4N 1
1, 8u2.y/
c22T4N 1.
The z values are within the range.0.5, 1/ depending on the relationship of the measurement uncertainty associated with experimental data and deviation of calibra- tion curve from the linear one. Thedcriterion value for the optimal plan is equal to c22T4z2Cu2N.y/.
To provide the target uncertaintyu2target of the calibration curve it is necessary to meet the condition
c22T4z2Cu2.y/
N u2target. (1.24)
The second summand in the left side of inequality (1.24) can be decreased at the expense of increasing the number of repeated measurements at the points of design spectrum (1.23). The first summand reflects the uncertainty component caused by the model nonlinearity and can be decreased only at the expense of reducing the rangeT or dividing it into subranges.
Thus, if the calibration curve for a particular measuring instrument is studied rather carefully and the correctness of approximating it with the linear function is grounded, then, performing the calibrations repeatedly, it is useful to conduct measurements at the calibration points corresponding to design spectrum (1.23).
In practice a uniform design spectrum is often applied. Such a design spectrum is in a certain sense a compromise and is used in two ways: on the one hand it is used for evaluating the parameters of the linear curve, on the other hand for checking the compliance of the model and experimental data. These two tasks are different; the second one relates to the design of discriminating experiments and it is not considered in this chapter.
When applying the uniform design spectrum, the uncertainty of the calibration curve can be evaluated on the basis of deviations of the experimental data from the curve being fitted. In contrast to the calculation uncertainty, according to (1.22) this way of uncertainty evaluation may be seen as experimental:
u2.x/DuA2.y/ 1
N C .x Nx/2 PN
iD1.xi Nx/2
!
, uA.y/D vu ut 1
N2 XN iD1
.yiy_i/2, (1.25) wherey_iis the estimate obtained on the basis of the curve beging fitted.
Using the estimates of form (1.25) it is necessary to take into account that the dis- persion of the experimental data around the calibration curve is due to a number of reasons (but it is not possible to say that these reasons are fully taken into account in equation (1.25)): the repeatability of measuring instrument indications, the deviation from the linearity, and the uncertainty of values of independent quantity.
The design of the measurements for evaluating calibration curves is the methodi- cal base for developing a procedure of calibration which will be able to provide the target uncertainty of the calibration function evaluation. The target uncertainty of the calibration function is dictated by its further application. In particular this can be the use of a given measuring instrument for carrying out measurements with a required accuracy, or the attribute of a given measuring instrument to a definite accuracy class of instruments by results of its calibration, etc.
Section 1.8 Summary 51