modelling of measurement for uncertanty evaluation (1)

8/18/2019 Modelling of Measurement for Uncertanty Evaluation (1)

1/11

INSTITUTE OF PHYSICS PUBLISHING METROLOGIA

Metrologia 43 (2006) S200–S210 doi:10.1088/0026-1394/43/4/S06

Systematic approach to the modelling of

measurements for uncertainty evaluationK D Sommer1 and B R L Siebert2

1 Landesamt f ̈ur Mess- und Eichwesen Thüringen (LMET), Ilmenau, 98693, Germany2 Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany

Received 6 March 2006Published 4 August 2006Online at stacks.iop.org/Met/43/S200

Abstract

Modern evaluation of measurement uncertainty is based on both theknowledge about the measurement process and the (input) quantities whichinfluence the result of measurement. The knowledge about the inputquantities is represented by means of appropriate probability densityfunctions (pdfs), whereas the knowledge about the measurement process isexpressed by a so-called model equation which reflects the interrelationbetween the measurand and the input quantities. The assignment of a pdf fora quantity is based on sound theory but there is no theory on modelling.Neither the Guide to the Expression of Uncertainty in Measurement norother relevant uncertainty documents provide any guidance on systematicmodelling. This paper proposes a systematic and versatile modellingconcept that has evolved from the idea of the classical measuring chain.From this concept, a systematic procedure for modelling of measurements

has been derived which is structured into five fundamental work steps andrequires three types of standard modelling components. The concept utilizesonly a few generic model structures which are strongly related to the methodof measurement employed. These generic model structures can easily betailored to any particular measurement task. The paper provides thetheoretical reasoning for this concept and derives practical procedures for itsapplication.

1. Introduction

In essence, uncertainty is a measure of the ‘trustworthiness’

of the result of a measurement. For the evaluation of theuncertainty associated with thevalueof a measurandone needs

both a model that reflects the interrelation of all quantities

that influence the measurand and knowledge about these

influencing quantities. The measurand is also termed the

output quantity and the influencing quantities are termed input

quantities [1, 2].

In general, knowledge about these (input) quantities is

(unavoidably) incomplete and, therefore, in accordance with

the Bayesian probability concept, expressed by appropriate

probability density functions (pdfs) that, based on the

principle of maximum information theory and Bayes’

theorem, are related to the given information about the

input quantities [2–4]. Therefore, assigning pdfs is basedon a sound theory. The information that forms the

basis for the knowledge about reasonably possible values

can be obtained from repeated measurements or be given

otherwise.

To derive the model which represents the interrelation

between the input quantities and the output quantity, one needsto analyse the measuring process. Basically, due to incomplete

knowledgeabout this interrelation, thismodelwill unavoidably

always only approximate reality. In that sense, modelling can

be seen as a Bayesian learning process. A closed theory for

modellingdoesnotexist, but, asshownin this paper, systematic

approaches are possible.

In thenext section we brieflysummarizethe concept of the

GUM[1] as seen in the light of the forthcoming supplement [2]

and, furthermore, show that modelling of the measurement

process is a key element of modernuncertaintyevaluation. The

third section identifies thecause–effect chain as thebasis of the

modelling concept presented. The next section demonstrates

that any measurement chain can be built up by linking onlythree types of generic modelling components. A stepwise

modelling procedure is presented. The fifth section discusses

0026-1394/06/040200+11$30.00 © 2006 BIPM and IOP Publishing Ltd Printed in the UK S200

http://stacks.iop.org/me/43/S200http://dx.doi.org/10.1088/0026-1394/43/4/S06


2/11

Modelling of measurements for uncertainty evaluation

the relationship between the method of measurement and a

very few generic structures of measurement chains. As shown

in the sixth section, the modelling of complex measuring

systems can be simplified by employing sub-models. The next

section discusses thesources of correlation andthe inclusionof

mutually dependent input quantities in the modelling process.The final section draws some conclusions.

2. GUM concept

Modern uncertainty evaluation is based on both the knowledge

about measurement process and the relevant input quantities.

The knowledge about the measurement process is to be

represented by the so-called model equation which establishes

the mathematical interrelation between the measurand Y and

the input quantities X1, . . . , XN (briefly termed the model).

Basically, the model can be given in any form but must provide

a unique relationship. In most cases [1, 2] it is stated as

Y = f M(X1, . . . , XN ). (2.1)

The model is also valid for the possible values, η of Y and

ξ of X:

η = f M(ξ 1, . . . , ξ N ). (2.2)The model must ensure that for any set of possible input

values one and only one value η is determined. However,

different sets of possible input values can result in the same

value η.

The (unavoidably incomplete) knowledge about any input

quantity Xi that contributes to the measurement result is

expressed by means of an appropriate pdf gXi (ξ i). The variable

ξ i represents the possible values of the quantity Xi . In theGUM framework, the expectation of the pdf is taken as the

best estimate of the value of the quantity,

xi = E[Xi] = +∞

−∞gXi (ξ i)ξ i dξ i, (2.3)

and its standard deviation is termed the standard uncertainty

uxi associated with this best estimate xi ,

uxi = {E[(Xi − xi)2]}1/2 = +∞

−∞gXi (ξ i)(ξ i − xi)2 dξ i

1/2.

(2.4)

Depending on the kind of knowledge available, the state-of-knowledge pdf gXi (ξ i) for an input quantity Xi is obtained

either by statistical analysis of repeated observations, i.e.

sampling of the observed quantity, along with the presumption

of an appropriate—mostly Gaussian—data model of this

quantity, or, in the case of non-statistical knowledge, by

utilizing the principle of maximum information entropy (pme)

or Bayes’s theorem [3–5]. In both cases, one obtains

unambiguously a state-of-knowledge pdf for the quantity.

The GUM [1] terms the statistical evaluation method type-

A and the other type-B. The type-A procedure assumes that

the repeatedly measured values are samples of a Gaussian

pdf and takes the arithmetic mean x̄ of the resulting valuesas the unbiased estimate for the expectation and the empiricalstandard deviation s as the unbiased estimate for the standard

deviation of that Gaussian pdf. The experimental standard

deviation of the mean is given by s(x̄) = s/√ n. Uponappropriate transformations, the t -distribution with a degree

of freedom ν = n − 1 serves to express the knowledge gainedby these repeated measurements [4].

Thepdf forthe measurandY , gY (η), is the joint (posterior)

pdf for all relevant input quantities taking their interrelation,as given by the model, into account. It can be calculated using

the Markov formula:

gY (η) = +∞

−∞, . . . ,

+∞−∞

gX1,...,xN (ξ 1, . . . , ξ N )

×δ(η − f M(ξ 1, . . . , ξ N )) dξ 1, . . . , dξ N , (2.5)where f M is the functional relationship between the values of

the contributing input quantities ξ i and the respective value of

the measurand η (see equation (2.2)). The δ- function ensures

that only physically possible values are considered.

Often, one is only interested in stating the value of the

measurand and the uncertainty associated with it:

y = ∞

−∞gY (η)η dη =

∞−∞

, . . . ,

∞−∞

gX1,...,XN (ξ 1, . . . , ξ N )

×f M(ξ 1, . . . . ξ N ) dξ 1, . . . , dξ N (2.6)and

u2y = ∞

−∞gY (η)(η − y)2 dη =

∞−∞

, . . . ,

∞−∞

gX1,...,XN

×(ξ 1, . . . , ξ N )(f M(ξ 1, . . . , ξ N ) − y)2 dξ 1, . . . ,dξ N .(2.7)

However, for thecomputation of theexpanded uncertainty

one needs to compute gY (η) explicitly. Apart from fairly

simple cases that allow analytical solutions one utilizes the

Monte Carlo method as the integration technique [2–6].In many cases in practice, the model is either linear or can

be replaced to a good approximation by a linear model using

a first order Taylor-series expansion about the best estimates

(see section 3.1):

η = f M(ξ) ∼= f M(x) +N i=1

ci(ξ i − xi), (2.8)

where ξ is used for ξ 1, . . . , ξ N and x for x1, . . . , xN . The ciare called sensitivity coefficients and given by

ci =

∂f M(x1, . . . , xN )

∂Xiξ i=xi . (2.9)

Inserting a linear model in equations (2.6)and(2.7) yields

y = E[Y ] = f M(x1, . . . , xN ), (2.10)

and (known as the law of Gaussian uncertainty propagation),

uy =

N i=1

c2i u2xi + 2

N −1i=1

N j =i+1

cicj · uxixj

1/2

, (2.11)

where uxixj = uxi · uxj · rij is the covariance of the quantitiesXi and Xj and rij the respective correlation coefficient.

The rationale of the ISO-GUM procedure [1] is that,according to the central limit theorem, the combination of

many pdfs tends to a Gaussian for which the expanded

Metrologia, 43 (2006) S200–S210 S201


3/11

K D Sommer and B R L Siebert

Input Quantities

, y y U , y y u

K

n

o

w

l

e

d

g

e

Output quantity

1X

2X

N X

11, x x u

22, x x u

,N xN x u

1ξ

2ξ

N ξ

PDF expectations,uncertainties

expectation,uncertainty

Gaussianuncertainty propagation

( )f M 2, ,..., N y x 1 x x =

2

2

1

+...N

y Xi i i

∂f u u

∂x =

= Σ

for linearizable

models only

Figure 1. Illustration of the concept of the ISO-GUM procedure [1]. Symbols: Y —measurand; X1, . . . , XN —input quantities;y = E[Y ]—expectation of the pdf for the measurand (taken as best estimate); uy—standard uncertainty associated with y;x1, . . . , xN —expectations of the pdfs for the input quantities; ux1, . . . ,uxN —standard uncertainties associated with x1, . . . , xN ;U y—expanded measurement uncertainty.

uncertainty is known to have thevalue1.96uy (usuallyrounded

to 2uy) fora coverageprobabilityof95%. Figure1 summarizes

the ISO-GUM procedure.

The model equation provides the basis for the propagation

of the pdfs for the input quantities [2] and, in the case of

utilizing Gaussian uncertainty propagation [1] (see figure 1),

for the propagation of their expectation values and associated

uncertainties respectively. Therefore, the modelling of the

measurement process is a key element of modern uncertainty

evaluation, irrespective of the method used for computing the

uncertainty.

3. Basic ideas for a modelling concept

To practitioners, modelling of the measurement process

appears to be the most difficult problem in uncertainty

evaluation. Currently, there is no generally accepted and

practically applicable theory of modelling available.

In general, a model serves to evaluate the original

system or to draw conclusions from its behaviour. In

measurement techniques usually the measurand and other

(system-perturbing) influence quantities may be seen as

causative signals which by the measuring system are

(physically) transformed into effects (indication(s), output

signal(s) etc). Therewith, the system assigns values to the

measurand that are influenced by other system-perturbing

(influence) quantities.

Themodelling concept presented[7,8] isbased on the idea

of the classical measuring chain which constitutes the path of

the measurement signal from cause to effect (see section 3.1).

It mainly refers to the ISO-GUM procedure [1] that only

applies to linear or linearizable models which satisfy equation

(2.8).

The second idea is that the method of measurement [9]

employed is reflected in the structure of the model. This leadsto only a few generic model structures as will be shown in

section 5.

Y

(a)

(b) Graph:

(c) Mathematical relationship: X 1 = h (Y, X 2 , X 3 )

(average)

Bath temperature

Y = t BX 1 = t INDX 2 = δt BX 3 = ∆t Th

X 3

X 2

X 1

Error of indication Reading

Bδt

Th∆t INDt

Bt

Figure 2. (a) Simplified example of a bath-temperaturemeasurement. (b) Graph of the respective cause-and-effectrelationship. (c) Cause-and-effect relationship expressed inmathematical terms. Symbols: t B—(average) bath temperature(measurand); δt B—deviation due to the temperature inhomogeneityof the bath; t Th—indication (scale) error of the thermometer used;t IND—indicated value.

3.1. Cause and effect of a measurement

For the purpose of mathematically expressing the relationship

between the measurand, the indication and the relevant

influence quantities, the cause-and-effect approach has been

proved to be very useful. Figure 2 illustrates this approach

with theexample of a temperature measurement: theindication

depends on both the bath temperature t B, i.e. the measurand,

and the instrumental error t Th of the thermometer used.

Furthermore, the indication is affected by the imperfect

‘coupling’ of the measurand t B with the measuring instrument

(deviation due to the temperature inhomogeneity of the bath).

Then, in accordance with the symbols used in figure 2, the

cause–effect relationship can mathematically be expressed bythe functional dependence X1 = h(Y,X2, X3). The conceptsdescribed here can also deal with the general case, where

S202 Metrologia, 43 (2006) S200–S210


4/11


5/11


X k IN X k OUT

0k G 0k G δ k Z δ

(G 0k + δG 0k )

Figure 5. Generic transmission element. See main text for symbols.

where XkIN—quantity acting at the (physical) input of the

element k; XkOUT—quantity at the output of the element

k; G0k—(nominal) transmission factor at the operating

point; δG0k—(deforming) parameter deviation and δZk—

(superimposing) parameter deviation.

The above approach allows one to both graphically

and mathematically describe the cause-and-effect relationship

of a measurement or calibration at any operating point

(see section 4). It is applicable to the description of

linear and linearized non-linear systems and elements alike.Furthermore, under the condition that the working-point

vector x can be kept constant, it includes the case of

mathematical combinationof quantities, suchas multiplication

of a resistance with a current with the view to obtaining the

voltage across the resistance.

4. Practical modelling procedure

In metrological practice, a modelling procedure is needed

that comprehensibly images the measurement process in the

cause–effect direction. The modelling procedure presented,

therefore, is based on graphical depictions of the cause-and-

effect relationship of the measurement. It is a modularprocedure which consists of only three generic modelling

components and five elementary modelling steps.

First approaches to a systematic modular and GUM-

consistent modelling procedure were made by Bachmair [11],

Kessel [12], Kind [13] and Sommer et al [7, 8].

4.1. Standard modelling components

For graphical depiction of the cause-and-effect relationships

of the measurements to be analysed and modelled (see

section 4.2), three generic standard modelling components are

employed [7, 8] as follows.

• Parameter sources (SRC): they provide or reproduce ameasurable quantity, for example the measurand.

• Transmission units (TRANS): they represent any kindof signal processing and influencing, for example

amplification and—as an unwanted effect—mismatching.

The transmission unit more or less complies with the

transmission element depicted in figure 5.

• Indicating units (IND): they indicate their (physical) inputquantities. If the errors and corrections that appear cannot

be singled out and assigned to particular physical causes,

the (summary) error (of indication) of the measuring

instrument [9] XINSTR (see figure 6) is allocated to the

indicating unit.

Figure 6 shows graphical schemes of these standard

modelling components.

4.2. Stepwise modelling procedure

The suggested modelling procedure consists of the following

five elementary steps.

(1) Describing the measurement, identifying the causative

quantities (measurand, influence quantities) and themethod of measurement [9] employed.

(2) Analysing the measurement, decomposing it into its

functional constituents, and, in turn, establishing

graphically the cause-and-effect relationship for the

fictitious ideal (unperturbed) measurement in terms of

the above described standard modelling components (see

section 4.1).

(3) Incorporating all imperfections, effects of incomplete

knowledge about quantities and influences that may

perturb the fictitious ideal measurement.

Establishing graphically and, in turn, mathematically

the cause-and-effect relationship for the real (perturbed)

measurement. Note: the incorporation of imperfections, such as external

influences, in complete knowledge about parameters, and

instabilities, is carried out by utilizing correction factors

and deviations that are related to the fictitious operating

point which is represented by the operating-point vector

x (see sections 3.1 and 3.2).

(4) Identifying mutually dependent input quantities and

including resulting correlations.

Note: there are three possible ways to include correlation.

They are described in section 7.3.

(5) Reversing the mathematical cause-and-effect relationship

with a view to explicitly deriving the model equation,

X1 =

h(Y,X2,X

3, . . . )

⇒ Y

= f

M(X

1, X

2, . . . ).

4.3. Example

The modelling procedure described in section 4.2 is explained

with the (simplified) example of the calibration of a scale.

Step (1) of the modelling procedure

• Description of the measurement/calibration: a non-automatic scale is to be calibrated by means of a standard

weight. This is carried out under prescribed conditions by

direct measurement andcomparison of the indication with

the value of the standard given in a calibration certificate.

• Measurand: instrumental error (deviation) of the scale,

here termed W INSTR.• Causal quantity: mass (given by a weighing value) of the

standard used.

• Measurement method: direct measurement.Step (2) of the modelling procedure

• Analysis of the measuring process: the standard may beconsidered as parameter source. Its ‘imperfect coupling’

with the scale, e.g. caused by air buoyancy, magnetic

susceptibility etc, might be described by a transforming

unit. For a simplified treatment, the scale itself may be

represented by an indicating unit (see figure 7(a)).

• Cause-and-effect relationship of the fictitious idealmeasurement: figure 7(b) graphically illustrates theidealized (unperturbed) measurement/calibration and the

related cause-and-effect relationship.

S204 Metrologia, 43 (2006) S200–S210


6/11


( )P ∆

X IN

δX INDX

INX

OUT

X T1

X Tm

... X Tδ ( )P

(a)

(b)

X SRC

SRC

X SRC∆ ( )P

TRANS IND

(c)h (X IN , X T1, ...,X Tm )

X IND

X INSTR

X SRC0

δ X M

+

+ + +

for materialmeasures

δ

Figure 6. Graphical depiction of the standard modelling components: (a) parameter source (including material measures), (b)transformation unit and (c) indicating unit. Symbols: XSRC—quantity provided or reproduced by SRC; XSRC0—nominal value of XSRC;XSRC(P )

= XSRC0

−XSRC(P )—instrumental error of the material measure depending on parameters P ; XIN,XT1, . . . , XTm—input

quantities; XOUT—output quantity; δXT(P),δXM(P )—(additive) disturbances/deviations depending on measurement conditions andexternal influences P ; XINSTR—instrumental error of IND; δXIND—deviation due to the limited resolution and XIND—indicated quantity.

Standardweight

Scale

IndicationW S*

W IND*

INDTRANSSRC

1W * S

Measurand

W IND*

∆W *INSTR

Figure 7. A simplified example of a fictitious ideal calibration of a scale by means of a standard weight (left) and the respectivecause-and-effect relationship. Symbols: W ∗S —weighing value provided by the standard; W

∗IND—indicated weighing value and

W ∗INSTR—error (of indication) of the scale (measurand) (right).


• Graphical depiction of the cause-and-effect relationship for the real measurement: by means of deviations

and correction factors, the following influences and

imperfections are introduced into the graphical cause-and-

effect relationship of the described calibration: W S =W S0 − W S—error of the nominal value of the standardused; kB = (1 − ρaρ−1S )/(1 − ρ1,2ρ−18000)—air buoyancy(correction) factor where ρa—air density, ρS—density of

thestandard,ρ1,2 = 1.2kgm−3 andρ8000 = 8000kg m−3;δW CPL(P )—deviation due to the influences of the

(temperature-dependent) air convection and the magneticfield strengthH mag, i.e. P CAL = P CAL(t a,H mag), thereforeδW CPL(P ) characterizes the ‘imperfect coupling’ of the

quantity W S with the instrument that depends on several

parameters, denoted by P ; δW M(t a)—deviation of the

scale due to the influence of the ambient temperature t a;

W INSTR is the deviation of the scale; it is the measurand

and δW IND is the finite resolution of the reading W IND.

Figure 8(a) illustrates the real calibration of the scale and

the relevant influences. Figure 8(b) shows the cause-and-

effect relationship modelled for the real measurement.

• Mathematical representation of the cause-and-effect relationship for the real measurement: the cause-

and-effect equation (measurement equation) can easilybe derived from its graphical scheme: in the cause–

effect direction, one has just to map the (mathematical)

operations indicated by the graphical scheme. Expressed

in mathematical terms, the cause-and-effect relationship

for the real measurement of the given example reads

W IND = (W S0 −W S)kB + δW CPL(P ) + δW M(t a)+W INSTR + δW IND (4.1)


• Identifying mutually dependent input quantities and including resulting correlations: kB, δW CPL(P ) and

δW M(t a) depend on temperature. For the sake of

keeping this example simple, this is neglected here.However, some consideration of correlation in modelling

is described in section 7.2.


• Model equation/model for the evaluation of themeasurement uncertainty: from the cause-and-effect

relationship for the real measurement (see equation (4.1)),

the following model equation is obtained:

W INSTR = W IND − δW IND − δW M(t a) − δW CPL(P )−(W S0 −W S)kB. (4.2)

The sensitivity coefficients can be read off equation (4.2),easily. Upon completion of the modelling procedure, the

next important step of the uncertainty evaluation consists in

Metrologia, 43 (2006) S200–S210 S205


7/11


Figure 8. (a) Simplified example of a calibration of a scale. (b) Respective cause-and-effect relationship modelled for the real measurement.Symbols W S—weighing value provided by the standard; W S0—nominal value of the standard and W S = W S0 −W S—instrumental error of the standard, e.g. given in a calibration certificate; for other symbols see text.

Table 1. Example: quantitative evaluation of input quantities in accordance with the ISO-GUM [1] method type-B for the calibration of ascale. For example and symbols see section 4.3.

Expectation StandardQuantity Knowledge pdf value uncertainty

W IND Digital indication, single reading: 9998.8g Constant value 9998.8g 0gδW IND Resolution: 0.1 g Rectangular 0g 0.029 gW S0 Nominal value of the standard Constant value 10 000 g 0 g

W S Error of the nominal value of the standard Gaussian 0.001 g 0.020 gtaken from the calibration certificate

kB Air buoyancy factor for ρair ∈ [1.1kgm−3, 1.2kgm−3] Rectangular 0.999 8543 0.000 0047and ρweight ∈ [7800 kg m−3, 8000kg m−3].

δW CPL(P ) Imperfect coupling: maximum deviation: Rectangular 0 0.115 g±0.2 g (EN 45501)

δW M(t a) Ambient temperature: maximum deviation within Rectangular 0 0.012 g±4 ◦C: ±0.02 g (EN 45501)

evaluating all involved input quantities that appearon theright-

hand side of equation (4.2) by assigning appropriate pdfs to

them (see section 2).

As to the knowledge about the input quantities: the(best estimated) values of the standard weight W S and its

error W S, respectively, should be given together with their

associated uncertainties in the calibration certificate issued

for the standard. The nominal value W S0 will be clearly

indicated at the weight. The values for the deviation

δW IND can be derived from the instrument’s resolution. The

knowledge about the deviations δW CPL(P ) and δW M(t a)

may be taken up from the manufacturer’s manual or from

requirements set up in the European Standard EN 45

501 [14]. The air buoyancy factor kB might be estimated

from the knowledge about the ambient conditions and the

density of the standard used. Table 1 exemplifies thequantitative evaluation of the input quantities for the example

given.

5. Model structures and measurement methods

Almost all measurements and calibrations can be reduced

to only a few generic model structures of the cause-and-

effect relationships. The chaining sequence of the modelling

components and the structure of the chain are completely

determined by the method of measurement employed [9].

Direct measurements result in an unbranched chain of the

components utilized. Figure 9 shows the generic structure of

the respective cause-and-effect relationship.

Other measurement methods are used to achieve higher

accuracies and to ensure proper traceability of calibration

results. These methods mostly result in branched cause-and-

effect relationships. The direct comparison of two indicating

measuring instruments and the substitution method may serve

as examples. Figures 10 and 11 show the generic structures

of their cause-and-effect relationships. When derivingthe mathematical cause-and-effect relationship from block

diagrams having branched structures, such as the methods

S206 Metrologia, 43 (2006) S200–S210


8/11


9/11


Figure 10. (a) Generic structure of the cause-and-effect relationship of a calibration by means of substitution (comparison with a standard,conjoining chains). (b) Example: calibration of a weight piece. Symbols: P CAL—calibration conditions; XX0, XS0—nominal values for thematerial measures SRCX (UUT) and SRCS (Standard); XSRCX(P CAL), XSRCS(P CAL)—instrumental errors of material measures SRCXand SRCS at the calibration conditions (operating point) P CAL; for other symbols see figure 9.

Figure 11. (Left) generic structure of the cause-and-effect relationship of a calibration by direct comparison of two indicating measuringinstruments (forking chain) and (right) example: calibration of a liquid-in-glass thermometer. Symbols: P CAL—calibration conditions; forother symbols see figures 9 and 10.

. . .X T1 ∆X INSTR(P M)

X IND

INDTRANS1

X SRC

SRC

X Tm . . .

TRANS2

Setvariable

Indication

0 0 0 0

Measurand

F G = m . g

Arm of balance

δS = 0F COMP

Set variable

COMPI

Coil

(a)

(b)

Figure 12. (a) Generic structure of the cause-and-effect relationship of the compensation method (closed loop). (b) Example:electromagnetic force compensated scale. Symbols: F G—force; m—mass; g—acceleration due to gravity; F COMP—compensating force;I COMP—compensating current and δS —deflection; for other symbols see figures 9, 10 and 11.

S208 Metrologia, 43 (2006) S200–S210


10/11


Figure 13. Example of splitting a measurement or calibration intosub-tasks for separate modelling: calibration of a pressure sensor bymeans of a piston gauge. Model 1: piston gauge; Model 2: pressuresensor; Model 3: indicating unit (voltmeter).

SRC1

W S01

W S02

∆W S2

W S2

W S1

W SSRC2

TRANS

−∆W S1

Patched standards

W S1

Scale

W S2

(a) (b)

Figure 14. (a) Depiction of the application of ‘patched standards’for the calibration of a scale (see example given in figure 9).(b) Respective cause-and-effect relationship (cut-out). Symbols:W S01, W S02—nominal values of the partial standards used; W S1,W S2—instrumental errors of the standards used (correlatedquantities because they have been determined within the sameexperiment or calibration).

Unit under test(UUT)

Standard

Comparator

SubstitutionW S1

W S2

∆W INSTR1 ∆W INSTR2

W IND1, W IND2

(∆W IND)

Figure 15. Example: graphical depiction of the cause-and-effectrelationship (simplified) of a substitution measurement. Symbols:W S1, W S2—quantities provided by the material measures; W INSTR1,W INSTR2—errors of the comparator used separately for themeasurement of the standard and the unit under test; W IND1,

W IND2—indicated quantities.

In the example discussed in section 4, the quantities

kB, δW CPL(P ) and δW M(t a) dependon temperature. In order to

compute the corresponding correlation coefficient, one needs

to model their dependence on temperature. However, in view

of the given information this is only straightforward for kB.

Using equation (4.2) and the values for the uncertainties in

table 1 one can compute the combined uncertainty. Its value is

0.130 g upon neglecting correlation and0.178g upon assuming

the value +1 for all correlation coefficients (worst case). In

practical calibration one would accept the worst case, and

definitely so in the example discussed, since for scales inthat class a maximum permissible deviation of 0.5 g is to be

certified, only.

Standard

resistance

Resistance

decade

1000R DEC =

R 1

R 2

R 10

R S

100 Ω

100 Ω

R S = (100 ± 10.10−3) Ω

R i = a . R s

a ≅ 1

100 Ω

Figure 16. Illustration of the calibration of a resistance decade with

a standard resistance (see GUM [1] 5.2.2 and F.1.2.3).

7.3. Methods for taking correlation into consideration in

the uncertainty evaluation

Generally, there are three possible ways of taking the

correlation into consideration when evaluating the combined

measurement uncertainty:

(1) If therelationshipbetween thecorrelated quantities canbe

unambiguouslyexpressed,e.g.in thecaseofknowingtheir

dependences on another (third) quantity, this relationship

should be introduced in the model equation explicitly.

This will result in resolving the correlation.

(2) If the correlation coefficient(s) or the respectivecovariance(s) are sufficiently well known, they can be

taken into account andpropagated as recommended by the

GUM [1], i.e. using the Gaussian uncertainty propagation.

(3) Since logical correlation is alwaysrelated to an influencing

(third) quantity, be it explicitly stated or not, one can

formally introduce an auxiliary quantity that represents

the correlated fraction of that quantity. In some cases, if

the physics of the measurement is well known, this allows

us to straightforwardly resolve correlation by modelling

explicitly the dependence of two or more input quantities

on the same systematic effect expressed by the auxiliary

quantity.

Special care is needed with simultaneous repeated

measurement of more than onequantity because here statistical

correlation may either mask or pretend correlation.

7.4. Effects of correlation

It can easily be derived from the law of Gaussian uncertainty

propagation that, in the case of identical signs of the correlated

quantities (see examples depicted in figures 14 and 16),

correlation yields an enhanced total uncertainty contribution.

For example, for two correlated quantities X1 and X2:

[u2x1 + u2x2]

1/2 uxTOTAL ux1 + ux2, (7.2)

where ux1 and ux2 are the individual uncertainty contributions

and uxTOTAL is the total uncertainty contribution associated

Metrologia, 43 (2006) S200–S210 S209


11/11


with the combined expectation value of the quantities

X1 and X2.

In the case of different signs of two correlated quantities

(see thesubstitution example depicted in figure 15), correlation

yields a decreased total uncertainty contribution:

0 uxTOTAL [u2x1 + u

2x2]

1/2. (7.3)

In the case of the example depicted in figure 15, the uncertainty

would completely disappear in the case when the instrumental

error of the comparator is of systematic nature and absolutely

stable. This, by the way, is exactly what substitution aims at.

In non-linear models, a positive correlation coefficient

enhances the uncertainty for a product of correlated quantities

and decreases the uncertainty for the ratio of correlated

quantities, and vice versa for the negative correlation

coefficient.

8. Conclusions

Although it does not seem possible to develop a theory

that allows for designing a model stringently, this paper

demonstrates that it is, nevertheless, possible to achieve

systematic modelling for uncertainty evaluation based on the

idea of reflecting the cause–effect relation by a measuring

chain. The concept and the procedure presented can be

implemented using only three different kinds of generic

standard modelling components—source, transmission and

indication. Furthermore, it leads quite naturally to the

described step-wise procedure and allows derivation of

basic generic models for the few existing methods of

measurement. Thus a versatile basis for systematic modelling

of measurements and calibrations is provided. The modelling

concept is applicable to most areas of uncertainty evaluation

of measurements performed in the steady state. However, an

extension to dynamic measurements or measurements with

more than one measurand or more than one indication is

possible!

In conclusion, the step-wise procedure described in this

paper as derived from theconcept of reflecting thecause–effect

relation by a measuring chain is well suited for systematic

modelling of measurement and for deriving the model for the

evaluation of uncertainty from it.

AcknowledgmentsAngelika Poziemski provided substantial technical assistance.

Stefan Heidenblut and a referee made valuable comments on

drafts of this paper.

References

[1] ISO 1993 Guide to the Expression of Uncertainty in Measurement (GUM) 1st edn, corrected and reprinted 1995(Geneva: International Organization for Standardization(ISO))

[2] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIMLEvaluation of measurement data—Supplement 1 to theGuide to the Expression of Uncertainty in

Measurement —Propagation of distributions using a MonteCarlo method (Draft)

[3] Sivia D S 1996 Data Analysis—A Bayesian Tutorial (Oxford:Clarendon)

[4] Wöger W 2001 Zu den modernen Grundlagen derDatenauswertung in der Metrologie PTB Mitteilungen 111210–25

[5] Sommer K D, Kochsiek M and Siebert B R L 2005 A Bayesianapproach to the modelling and uncertainty evaluation of measurement 3rd Int. Conf. on Systems, Signals and

Devices SSD-05 (Sousse, Tunisia, 21–24 March 2005)[6] Siebert B R L and Sommer K D 2004 Weiterentwicklung des

GUM und Monte-Carlo-Techniken tm—Technisches Messen 71 (2) 67–80

[7] Sommer K D, Kochsiek M, Siebert B R L and Weckenmann A2003 A generalized procedure for modelling of measurement for evaluating the measurement uncertaintyProc. 17th IMEKO World Congr. 2003 (Dubrovnik, IMEKOand Croatian Metrology Society, Zagreb) (ISBN953-7124-00-2)

[8] Sommer K D, Siebert B R L, Kochsiek M and Weckenmann A2005 A systematic approach to the modelling of measurements for uncertainty evaluation 7th Int. Symp. on

Measurement Technology and Intelligent Instruments, ISMTII 2005 (University of Huddersfield (UK), 6–8September 2005)

[9] Internationales W¨ orterbuch der Grund- und Allgemeinbegriffein der Metrologie (VIM) 1993 2nd edn (Geneva:

International Organization for Standardization (ISO))[10] Kiencke U and Eger R 1995 Messtechnik—Systemtheorie f¨ ur Elektrotechniker (Berlin: Springer)

[11] Bachmair H 1999 A simplified method to calculate theuncertainty of measurement for electrical calibration andtest equipment Proc. 9th Int. Metrology Congress (Bordeaux 18–21 October 1999) (Nanterre Cedex: MouvementFrançais pour la Qualité) (ISBN 2-909430-88X)

[12] Kessel W 2001 Messunsicherheit—einige Begriffe undFolgerungen f ̈ur die messtechnische Praxis PTB

Mitteilungen 111 226–44[13] Kind D 2001 Die Kettenschaltung als Modell zur

Berechnung von Messunsicherheiten PTB Mitteilungen 111338–41

[14] ISO 1992 Metrological aspects of non-automatic weighinginstruments Standard EN 45501 (Geneva: International

Organization for Standardization (ISO))[15] Sommer K D, Kochsiek M and Siebert B R L 2004 A

consideration of correlation in modelling and uncertaintyevaluation of measurement Proc. NCSL Int. WorkshopSymp. (Salt Lake City) (Boulder, CO: NCSL International)

S210 Metrologia, 43 (2006) S200–S210

modelling of measurement for uncertanty evaluation (1)

Documents