Estimating bias error distributionsTianshu Liu and Tom D. Finley Citation: Review of Scientific Instruments 72, 3561 (2001); doi: 10.1063/1.1394188 View online: http://dx.doi.org/10.1063/1.1394188 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/72/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurement of total ultrasonic power using thermal expansion and change in buoyancy of an absorbing target Rev. Sci. Instrum. 85, 054905 (2014); 10.1063/1.4878625 Automation of the resistance bridge calibrator AIP Conf. Proc. 1552, 392 (2013); 10.1063/1.4819572 Computed tomography dose measurements with radiochromic films and a flatbed scanner Med. Phys. 37, 189 (2010); 10.1118/1.3271584 Optimal two-point static calibration of measurement systems with quadratic response Rev. Sci. Instrum. 75, 5106 (2004); 10.1063/1.1818531 Nondestructive microwave permittivity characterization of ferroelectric thin film using microstrip dual resonator Rev. Sci. Instrum. 75, 136 (2004); 10.1063/1.1632999

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Estimating bias error distributionsTianshu Liua) and Tom D. Finleyb)

Model Systems Branch, MS 238, NASA Langley Research Center, Hampton, Virginia 23681-2199

~Received 1 May 2001; accepted for publication 18 June 2001!

In this article we formulate a general methodology for estimating the bias error distribution of adevice in a measurement domain from less accurate measurements when a minimal number ofstandard values~typically no more than two values! are available. A new perspective is that the biaserror distribution can be found as a solution of an intrinsic functional equation in a domain. Basedon this theory, the scaling- and translation-based methods for determining the bias error distributionare developed. These methods are applicable to virtually any device as long as the bias errordistribution of the device can be sufficiently described by a power series~a polynomial! or a Fourierseries in a domain. These methods were validated through computational simulations and laboratorycalibration experiments for a number of different devices.@DOI: 10.1063/1.1394188#

I. INTRODUCTION

The measurement of an instrument always has error de-fined as the difference between the measured value and thetrue value. The total error is the sum of the bias error and therandom error. In most books and articles on uncertaintyanalysis detailed statistical estimates of the random errorhave been studied.1–3 However, the bias error, which is thefixed, systematic component of the total error, is not suffi-ciently discussed because it is often assumed that all biaserrors have been eliminated by calibration. Indeed, the biaserror can be determined by comparison with a standard hav-ing accuracy much better than the device being tested. Thestandard device is ultimately traceable to a national or inter-national standards laboratory. If the standard values over thewhole range of measurements are known, the determinationof the bias error distribution of an instrument is extremelytrivial. Unfortunately, it is not always possible to have a stan-dard device available for calibration. Also, the standard de-vice itself has limited accuracy. In this article we attempt toformulate a general methodology for determining the biaserror distribution. Finley4 proposed a unique idea for extract-ing the absolute bias error of an angular measurement deviceby comparing it with another instrument of comparable qual-ity. The differences in readings from the two devices areobtained with two different initial angles of the devices.Next, a Fourier analysis of the two sets of the differencesrecovered the bias errors for both devices. Finley’s methodwas further discussed by Snow5 from a standpoint of theFourier transform. Finley’s work shows that in angular mea-surements the absolute bias error distributions can be ob-tained without the use of a more accurate device~standard!.Finley’s method is actually the translation-based method thatwill be discussed in this article. Naturally, a legitimate ques-tion is whether there is a general and universal method forestimating the bias error distribution of a device. This ques-tion will be explored in this article.

A general methodology is formulated for estimating theabsolute bias error distribution of a device in a measurementdomain in which no more than two standard values areknown. In the proposed approach, the bias error distributionis sought as a solution of an intrinsic functional equation in adomain. The scaling-based method and translation-basedmethod are developed, in which the analytical solutions tothe functional equations for the bias error distributions are,respectively, expressed as a power series and a Fourier seriesin a given domain. The scaling-based method is effective fora large class of the bias error distributions that can be suffi-ciently described by a power series or a polynomial. Theapproximate scaling-based method for practical implementa-tion is developed, and it requires no more than two standardvalues to determine the complete bias error distribution. Anempirical rule for selecting an appropriate order of a polyno-mial is suggested to achieve the good accuracy of calculatingthe bias error distribution. Although it is assumed that therandom error has been removed by averaging repeated mea-surement data, effects of the random error on calculation ofthe bias error distribution are studied through Monte Carlosimulations. The translation-based method is particularlyuseful for angular measurements since the bias error distri-bution can be naturally expressed as a Fourier series. Thescaling- and translation-based methods for simultaneouslyestimating the bias error distributions of two devices are alsodescribed. Computational simulations and laboratory experi-ments for calibrating a number of different devices are con-ducted to validate the proposed methodology to recover thebias error distribution. For more complicated measurementswhere extensive data analysis and application of many cor-rections are used, the present methods are still applicablesince they only deal with the final output of a measuredphysical quantity. In this case, the recovered bias error dis-tribution includes all the contributions from the intermediatedata-reduction procedures in the measurements. Although inthis article we mainly study the measurement of a singlephysical quantity, the proposed methodology can be ex-tended to recover the multidimensional bias error distributionof a number of coupling measurement variables. Further-

a!Electronic mail: t.liu@larc.nasa.govb!Electronic mail: t.d.finley@larc.nasa.gov

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 9 SEPTEMBER 2001

3561

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more, the methodology may be potentially useful for deter-mining the ‘‘bias error distribution’’ of theoretical models if atheoretical model can be treated as a virtual instrument.

II. BIAS ERROR AND INTRINSIC FUNCTIONALEQUATION

Let m(x) be the measurement value of a ‘‘true’’ physicalquantityx by a device having a deterministic bias error«(x),then

m~x!5x1«~x!. ~1!

The symbolm(x) can be interpreted as a measurement op-erator of the variablex. In order to obtain an additional in-dependent equation for«(x), a quantityax1b that is alinear transformation ofx is measured by the same device,wherea is a scaling constant andb is a translation constant.The measurement value ofax1b is expressed as

m~ax1b!5ax1b1«~ax1b!. ~2!

Eliminating x in Eqs. ~1! and ~2!, one obtains a functionalequation for«(x)

«~x!2a21«~ax1b!5d~x!, ~3!

where the differenced(x) is a known function that can bemeasured by the device for givena andb, i.e.,

d~x!5m~x!2a21@m~ax1b!2b#. ~4!

Equation~3! is an intrinsic functional equation governing thebias error distribution«(x). Because no assumption has beenmade, this functional equation for the bias error distributionis applicable to virtually any measurement instrument. Al-though more complicated functional equations can be simi-larly constructed as a model of the bias error, Eq.~3! enjoysits simplicity without loss of generality. Now, the fundamen-tal problem is to find a solution to Eq.~3! for «(x) in a givendomainD5@x1 ,x2# and a class of admissible functions~in-cluding domains and ranges!. The discussions on functionalequations from a mathematical perspective can be found inRefs. 6 and 7. Although a general solution to Eq.~3! inpower-series form can be found, two special but very usefulcases are considered here; they are a pure scaling case~aÞ0,1 andb50! and a pure translation case~a51 and bÞ0!. Not only do the special cases lead to simpler solutions,but they are also more easily implemented into the real mea-surements.

A. Scaling-based method

In the pure scaling case~aÞ0,1 andb50!, the intrinsicfunctional equation is

«~x!2a21«~ax!5d~x!, ~5!

whered(x) is a known function,

d~x!5m~x!2a21m~ax!. ~6!

Assume that the functionsd(x) and«(x) can be expanded asa power series in a given domainD5@x1 ,x2#, that is,

d~x!5 (n50

N

dnxn

and

«~x!5 (n50

N

enxn. ~7!

For a given set of measurement data points ofd(x), thecoefficients dn can be obtained using the least-squaresmethod~see the Appendix!. By substituting Eq.~7! into Eq.~5!, one can determine all the coefficientsen except forn51,

en5dn /~12an21! ~n50,2,3,...,N!. ~8!

Note that whenm(x) in Eq. ~6! can be exactly expressed asa power series,d1 is zero and thereforee1 is undetermined.For n51, to determine the remaining unknown coefficiente1 , the measurement valuem(x) is rewritten as

m~x!5 (n50nÞ1

N

enxn1~11e1!x. ~9!

Integrating Eq.~9! over a given domainDs5@x1s ,x2s##D5@x1 ,x2#, one obtains an expression for the coefficiente1 ,

e152

x2s2 2x1s

2 Ex1s

x2sFm~x!2 (n50nÞ1

N

enxnGdx21, ~10!

where a conditionx1s2 Þx2s

2 must hold. We often assign@x1s ,x2s#5@x1 ,x2# if x1

2Þx22. Thus, all the coefficientsen

are obtained and the bias error distribution«(x) is, in prin-ciple, determined. The above method is called as the scaling-based method for recovering the bias error distribution.

An underlying assumption for the scaling-based methodis that the functionsd(x) and «(x) can be sufficiently ex-pressed as a power series or a polynomial inD5@x1 ,x2#. Toillustrate this method, we consider a hypothetical bias errordistribution of a device inD5@1,10#,

«~x!50.320.01Ax10.03x2431024x221024x326

31026x510.5 exp~20.3x2!. ~11!

As shown in Fig. 1, the bias error distribution computed byusing the scaling-based method fora52 and N513 is inexcellent agreement with one given by Eq.~11!. The differ-ence between the given and computed bias errors is alsoplotted in Fig. 1.

In an ideal case, it seems likely that the bias error distri-bution can be recovered even without using any standardvalue. However, careful readers may notice that bothd(x)and «(x) in Eq. ~7! are expanded as a function of the truevariablex which is not knowna priori in real measurements.It will be pointed out in Sec. III that the approximate scaling-based method for actual measurements still needs no morethan two standard values to recover the bias error distributionin the domainD5@x1 ,x2#.

B. Translation-based method

In the pure translation case~a51 andbÞ0!, instead ofusing a power series, a Fourier series is employed because ofthe translation property of a complex exponential function.Since a Fourier series has the periodic property, the variable

3562 Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 T. Liu and T. D. Finley

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x and the translation constantb in Eqs. ~3! and ~4! are re-placed by the angular variablesu andu0 , respectively. In adomainDu5@0,2p#, the intrinsic functional equation for thebias error distribution«~u! is

«~u!2«~u1u0!5d~u!, ~12!

whered~u! is a known function for a givenu0 ,

d~u!5m~u!2m~u1u0!1u0 . ~13!

Assume that the functionsd~u! and«~u! in Du5@0,2p# canbe expanded as a Fourier series,

d~u!5 (n51

N

@an~d! cos~nu!1bn

~d! sin~nu!#,

~14!

«~u!5a0~«!/21 (

n51

N

@an~«! cos~nu!1bn

~«! sin~nu!#.

The coefficientsan(d) andbn

(d) in d~u! can be determined us-ing the least-squares method for a given set of data points~see the Appendix!. For nÞ0, a relation between (an

(«) ,bn(«))

and (an(d) ,bn

(d)) (n51,2,...,N) is derived by substituting Eq.~14! into Eq. ~12!, that is,

~an~«! ,bn

~«!!T5G21~an~d! ,bn

~d!!T ~n51,2,3,...,N!, ~15!

where

G5S 12cos~nu0! 2sin~nu0!

sin~nu0! 12cos~nu0!D .

Because the determinant det(G)52@12cos(nu0)# should notbe zero for the existence of a unique solution, the necessaryconditions for a unique solution arenu0Þkp1p/2 (k50,1,2,...,) andnÞ0. For n50, similar to the pure scalingcase, the coefficienta0

(«) can be determined by the followingintegral overDu5@0,2p#:

a0~«!5p21E

0

2pH m~u!2 (n51

N

@an~«! cos~nu!

1bn~«! sin~nu!#J du22p. ~16!

Therefore, all the coefficients (an(«) ,bn

(«)) are known and thebias error distribution«~u! is readily calculated. We refer tothis method as the translation-based method. Note that theconstant term in the Fourier series ford~u! in Eq. ~14! iszero. This is a restrictive assumption for the translation-based method. Another implicit assumption embedded in Eq.~14! is that the bias error distribution must be periodic.

As an example, consider a hypothetical bias error«~u!defined inDu5@0,2p#,

«~u!50.0810.02 cos~u!10.03 cos~2u!20.06 sin~2u!

10.04Au sin~4u!. ~17!

Figure 2 shows a comparison between the given biaserror distribution, Eq.~17!, and the distribution recovered byusing the translation-based method~u05p/20 and N56!.Simulations indicate that the translation-based method isgood for recovering the bias error distribution whose behav-ior is dominated by trigonometric functions. The Fourier se-ries solution is particularly useful for angular measurementsin the domainDu5@0,2p#. Compared to the scaling-basedmethod, the translation-based method has the implicit as-sumption of periodicityu(0)5u(2p). When the periodicityconditionu(0)5u(2p) is imbedded in data, the translation-based method does not explicitly require any standard valueto recover the bias error distribution. In contrast, the scaling-based method typically needs no more than two standardvalues in the real measurements~see Sec. III!.

III. APPROXIMATE SCALING-BASED METHOD

In Sec. II, we described the scaling-based method forrecovering the bias error distribution as a solution of thefunctional equation. However, the method in the ideal con-dition cannot be directly applied to the real measurementsbecause the true variablex in Eq. ~7! is not knowna priori.Therefore, an approximate approach is developed, in whichthe true variablex is replaced by a known approximate ref-erence valuexappr. The approximate reference valuexappr isoften a measurement value given by a reference device witha comparable accuracy. Due to the substitution ofxappr for x,the scaling-based method gives an approximate bias errordistribution «(x)appr that deviates from the true bias error

FIG. 1. Given bias error distribution and computed results using the scaling-based method~a52, N513!.

FIG. 2. Given bias error distribution and calculated results using thetranslation-based method~u05p/20, N56!.

3563Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 Estimating bias error distribution

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distribution«(x). The deviation of«(x)appr from «(x) can bereasonably modeled by a linear function ofxappr. Thus, thisapproximate method requires two standard values to deter-mine the unknown coefficients in the linear function of de-viation for recovering the whole bias error distribution.

Replacing the true variablex in Eq. ~7! by a knownapproximate reference variablexappr, one obtains an alterna-tive formulation,

d~x!appr5 (n50

N

dn~xappr!n

and

«~x!appr5 (n50

N

en~xappr!n, ~18!

where the functiond(x)appr is an approximate form ofd(x)in Eq. ~6!, i.e.,

d~x!appr5m~x!2a21m@x~axappr!#. ~19!

In real experiments,m@x(axappr)# is the measurement valueby the device being tested when the approximate referencevalue is set at a scaled valueaxappr. From the viewpoint ofcomputational simulation,x(axappr) can be determined bysolving an algebraic equationaxappr5x1«8(x) for x, where«8(x) is the bias error distribution of the approximate refer-ence valuexappr. The coefficientsdn in the power series ford(x)appr can be obtained by using the least-squares methodand then the coefficientsen are given by Eqs.~8! and~10! inSec. II. Therefore, the approximate bias error distribution«(x)appr is determined.

The difference between«(x) and«(x)appr is estimated inthe sense of the mean square. Assumingxappr5x1«8(x),one knows

«~x!appr5 (n50

N

en~xappr!n

5«~x!1 (n51

N

nen«8~x!xn211o~«8!, ~20!

where«8(x) is the bias error distribution of the approximatereference valuexappr ando(«8) is a higher-order small termof «8(x). Based on the triangle inequality and the Cauchy–Schwarz inequality, furthermore, one obtains the followingestimate for the difference between«(x) and«(x)appr in thedomainD5@x1 ,x2#,

i«2«appri /i«i<~ i«8i /i«i !

3 (n51

N

nuenuS x22n212xl

2n21

2n21 D 1/2

1c,

~21!

wherec is a positive constant andi•i is theL2 norm definedasi f i5@*x1

x2f 2(x)dx#1/2. Equation~21! indicates that the up-

per bound of the normi«2«appri is simply proportional toi«8i/i«i, but it is related to the size of the domainD5@x1 ,x2# in a nonlinear fashion. Fori«8i /i«i!1, «(x)appr

approaches«(x).

Computational simulations indicate that«(x)appr is oftena shifted, rotated, sheared representation of«(x). Eventhough there is no rigorous mathematical proof, we find thatthe difference between«(x) and«(x)appr can be reasonablymodeled by a linear function ofxappr for a class of the dis-tributions that can be described by a power series. Thus, thebias error distribution«(x) is related to«(x)appr by

«~x!5«~x!appr2~C01C1xappr!. ~22!

The linear term serves as a correction for the approximatesolution «(x)appr. Clearly, two standard values are requiredto determine the constantsC0 andC1 . When two true valuesare known at two specific points, the constantsC0 and C1

can be determined. The linear model, Eq.~22!, is able to givea reasonable estimate for the bias error distribution. The ap-proximate reference valuexappr is provided by an indepen-dent reference device that has accuracy comparable to thedevice being tested.

The difference between«(x)appr and «(x) depends onthe behavior of the bias error distribution«8(x) of the refer-ence device. In a pathological case where the reference de-vice has the exactly same behavior as the tested device, i.e.,«8(x)5«(x), the bias error distribution«(x) cannot be re-covered since the solution of the functional equation«(x)appr

is trivially zero in the whole domain. When the behavior ofthe reference device is very close to that being tested, thebias error distribution cannot be accurately determined.Thus, a conjecture is that better recovery of the bias errordistribution could be achieved when the behavior of«8(x)considerably differs from«(x). From a theoretical stand-point, how the functional behavior of«8(x) affects the solu-tion of the intrinsic functional equation for«(x)appr is still anopen mathematical problem. It is challenging to identify aclass of the functions«8(x) maximally approaching«(x)that still lead to the nontrivial solution of the intrinsic func-tional equation.

As an example, consider a bias error distribution«(x) inD5@1,10#,

«~x!50.3– 0.08Ax2831023x21831025x4

2exp~20.3x2!. ~23!

The approximate reference value is given byxappr5x1«8(x), where

«8~x!5A@0.22531023Ax10.02x22331023x3

20.5 exp~20.2x!10.5 sech~0.18x!#. ~24!

The constantA in Eq. ~24! is used to adjust the magnitude of«8(x) in computational simulations. Figure 3 shows thegiven bias distribution, Eq.~23!, and the computed distribu-tions for a52 andN513 when the relative magnitudes of«8(x) arei«8i /i«i50.29, 1.17, 1.75, and 2.95. As indicatedin Fig. 3, the computed distribution is able to describe thebehavior of the bias error distribution«(x) even when theaccuracy of the approximate reference value is considerablyworse than that of the tested device. Figure 4 shows thedifference between the given and computed bias error distri-butions i«(x)2«(x)compi /i«(x)i as a function ofi«8(x)i /i«(x)i . The linear relation betweeni«(x)

3564 Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 T. Liu and T. D. Finley

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2«(x)compi /i«(x)i and i«8(x)i /i«(x)i shown in Fig. 4 isconsistent with the theoretical estimate, Eq.~21!.

The selection of the order of the polynomialN in Eq.~18! significantly affects the accuracy in calculation of thebias error distribution. A polynomial having a low order maylead to a poor fit to the true values, while a polynomialhaving an excessively high order may produce a large vari-ance due to the overfitting problem. A question is whetherthere is an optimal order of the polynomial to achieve thehighest accuracy of calculation. An answer to this problem isnot available in a strictly mathematical sense. Nevertheless,based on computational simulations for various bias errordistributions, an empirical rule is proposed here for deter-mining an appropriate order of the polynomial. We denote«(x,N) as the bias error distribution calculated using aNth-order polynomial and define the normi«(x,N11)2«(x,N)i as the distance between the bias error distribu-tions calculated using the (N11)th-order andNth-orderpolynomials. Computational simulations show that the dis-tancei«(x,N11)2«(x,N)i usually becomes small in a cer-

tain range of the orderN when the polynomial correctly de-scribes the true bias error distribution. This phenomenon canbe clearly seen wheni«(x,N11)2«(x,N)i is plotted as afunction of the orderN. Figure 5 shows a typical semiloga-rithmic plot of i«(x,N11)2«(x,N)i as a function of theorderN for the given bias error distribution, Eq.~23!, indi-cating a characteristic valley in a range ofN510– 15. In thiscase, a polynomial having the order inN510– 15 can pro-vide a reasonable estimate for the bias error distribution.

IV. EFFECTS OF RANDOM PERTURBATIONS

In actual calibration experiments, a measurementm(x)has the random error in addition to the bias error. In theaforementioned discussion, it is assumeda priori that therandom error has been removed by averaging repeated mea-surement data. However, the random error affects the solu-tion of the intrinsic functional equation for recovering thebias error distribution. To simulate effects of the random er-ror, we consider a perturbed measurementm(x)@11p(x)#,wherep(x) is a random perturbation with the normal distri-bution. Monte Carlo simulations can give the probabilitydensity distribution of the relative differencei «̃(x)2«(x)i /i«(x)i that is a random variable, where«̃(x) is theperturbed bias error distribution calculated using the scaling-based method. Forp(x) having the standard deviation of0.005, the scaling-based method witha52 and N513 isused to recover the bias error distribution, Eq.~11!, for 500samples. Figure 6 shows the probability density distributionof the relative differencei «̃(x)2«(x)i /i«(x)i . The ex-pected value ofi «̃(x)2«(x)i /i«(x)i is plotted in Fig. 7 as afunction of the standard deviation of the random perturbationp(x).

The scaling constanta may be also changed by a smallrandom perturbation in certain measurements. The randomlyperturbed scaling constant is expressed asa(11pa), wherepa obeys the normal distribution. Whenpa has the standarddeviation of 0.005, the disturbed bias error distribution«̃(x)is calculated using the scaling-based method~a52 andN513! for 500 samples for the given bias error distribution,

FIG. 3. Given bias error distribution and calculated results using the ap-proximate scaling-based method~a52, N513! for four magnitudes of thebias error of the approximate reference value.

FIG. 4. Difference between the given and computed bias errors as a functionof i«8(x)i /i«(x)i .

FIG. 5. Difference normi«(x,N11)2«(x,N)i as a function of the order ofa polynomialN.

3565Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 Estimating bias error distribution

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Eq. ~11!. Figure 8 shows the resulting probability densitydistribution of i «̃(x)2«(x)i /i«(x)i , indicating a roughlyuniform distribution.

V. METHODS FOR SIMULTANEOUSLY ESTIMATINGBIAS ERRORS OF TWO DEVICES

A. Scaling-based method

Here in Sec. V A, we describe the scaling-based methodfor simultaneously determining the bias error distributions oftwo devices. Consider devicesA and B that have measure-ments of

mA~x!5x1«A~x!

and

mB~x!5x1«B~x!. ~25!

The intrinsic functional equations for the bias error distribu-tions «A(x) and«B(x) are

«A~x!2«B~x!5d1~x!,~26!

«A~x!2a21«B~ax!5d2~x!,

where d1(x)5mA(x)2mB(x) and d2(x)5mA(x)2a21mB(ax) are known functions obtained from measure-ments. Assume that the functionsd1(x), d2(x), «A(x), and«B(x) can be expanded as a power series in a given domainD5@x1 ,x2#, that is,

d1~x!5 (n50

N

dn~1!xn

and

d2~x!5 (n50

N

dn~2!xn,

«A~x!5 (n50

N

en~A!xn

and

«B~x!5 (n50

N

en~B!xn. ~27!

For a given set of data points ofd1(x) andd2(x), the coef-ficientsdn

(1) anddn(2) can be obtained using the least-squares

method. Except forn51, the coefficientsen(A) and en

(B) canbe determined by

en~A!5

dn~2!2an21dn

~1!

12an21

and

en~B!5

dn~2!2dn

~1!

12an21 ~n50,2,3,...,N!. ~28!

For n51, since functional Eqs.~26! are reduced to one in-dependent equation, there is only a single equationd1

(1)

5d1(2)5e1

(A)2e1(B) for the two unknowns,e1

(A) ande1(B) . To

determine the remaining unknown coefficients,e1(A) and

e1(B) , we use an integral expression

FIG. 6. Probability density function ofi «̃(x)2«(x)i /i«(x)i for a randomlyperturbed measurement.

FIG. 7. Expected value ofi «̃(x)2«(x)i /i«(x)i as a function of the stan-dard deviation of the random perturbation.

FIG. 8. Probability density function ofi «̃(x)2«(x)i /i«(x)i for a randomlyperturbed scaling constant.

3566 Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 T. Liu and T. D. Finley

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e1~A!5

2

x2s2 2x1s

2 Ex1s

x2sS mA~x!2 (n50nÞ1

N

en~A!xnD dx21, ~29!

and the relatione1(B)5e1

(A)2d1(1) . The given domainDs

5@x1s ,x2s#(x1s2 Þx2s

2 ) for integration is a subdomain of themeasurement domainD5@x1 ,x2#.

As discussed previously, since the true variablex is notknown a priori in practical applications, approximationshould be used in which the true variablex is replaced by theknown approximate variablemA(x). The approximate for-mulations are

d1~x!appr5 (n50

N

dn~1!@mA~x!#n,

d2~x!appr5 (n50

N

dn~2!@mA~x!#n,

~30!

«A~x!appr5 (n50

N

en~A!@mA~x!#n,

«B~x!appr5 (n50

N

en~B!@mA~x!#n,

where d1(x)appr5mA(x)2mB(x) and d2(x)appr5mA(x)2a21mB(x(amA(x))) are known functions. The approxi-mate bias errors«A(x)appr and «B(x)appr can be determinedusing Eqs.~28! and ~29!. The approximations«A(x)appr and«B(x)appr describe the general behavior of the bias error dis-tributions, but deviate from the true ones. The linear modelfor correcting«A(x)appr and«B(x)appr is given by

«A~x!5«A~x!appr2@CA01CA1mA~x!#,~31!

«B~x!5«B~x!appr2@CB01CB1mA~x!#.

When two true values are known at two specific points, theconstantsCA0 , CA1 , CB0 , andCB1 can be determined.

Simulations indicate that Eqs.~31! are able to describe aclass of bias error distributions«A(x) and «B(x) that arereasonably represented by a power series. As an example, weconsider the bias error distributions«A(x) and «B(x) in D5@1,10#

«A~x!50.32431022Ax2231024x22831025x4,~32!

«B~x!50.12531023Ax1231022x22331024x3.

Figure 9 shows the given and calculated bias error distribu-tions «A(x) and «B(x) for a52 andN510. The empiricalrule for selecting the order of the polynomials in Sec. III isalso applicable to the two-device case.

B. Translation-based method

In a domainDu5@0,2p#, the intrinsic functional equa-tions for bias errors«A(u) and«B(u) are

«A~u!2«B~u!5d1~u!,~33!

«A~u!2«B~u1u0!5d2~u!,

where u0 is a constant translation in radians,d1(u)5mA(u)2mB(u), andd2(u)5mA(u)2mB(u1u0)1u0 are

known functions. Assume that inDu5@0,2p# the functionsd1(u), d2(u), «A(u), and«B(u) can be expanded as

d1~u!5a0~1!/21 (

n51

N

@an~1! cos~nu!1bn

~1! sin~nu!#,

d2~u!5a0~2!/21 (

n51

N

@an~2! cos~nu!1bn

~2! sin~nu!#,

~34!

«A~u!5a0~A!/21 (

n51

N

@an~A! cos~nu!1bn

~A! sin~nu!#,

«B~u!5a0~B!/21 (

n51

N

@an~B! cos~nu!1bn

~B! sin~nu!#.

The coefficients (an(1) ,bn

(1) ,an(2) ,bn

(2)) in d1(u) and d2(u)can be determined using the least-squares method. FornÞ0, a relation between (an

(A) ,bn(A) ,an

(B) ,bn(B)) and

(an(1) ,bn

(1) ,an(2) ,bn

(2)) is

~an~A!bn

~A!an~B!bn

~B!!T5G21~an~1!bn

~1!an~2!bn

~2!!T

~n51,2,3,...,N!, ~35!

where

G5S 1 0 21 0

0 1 0 21

1 0 2cos~nu0! 2sin~nu0!

0 1 sin~nu0! 2cos~nu0!

D .

Since the determinant ofG is det(G)52@12cos(nu0)#, anecessary condition for the existence of a unique solution isnu0Þkp1p/2 (k50,1,2,...,). Forn50, only one equation,a0

(1)5a0(2)5a0

(A)2a0(B) , is available for two unknowns,a0

(A)

and a0(B) . Using a similar method to that used in the pure

scaling case, the coefficienta0(A) can be determined by an

integral overDu5@0,2p#

FIG. 9. Given bias error distributions and computed results using the ap-proximate two-device scaling-based method~a52, N510! for devicesAandB.

3567Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 Estimating bias error distribution

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a0~A!5p21E

0

2pH mA~u!2 (n51

N

@an~A! cos~nu!

1bn~A! sin~nu!#J du22p. ~36!

Therefore, the coefficienta0(B)5a0

(A)2a0(1) is readily known.

As an example, we consider the following bias errors«A(u)and«B(u) in Du5@0,2p#:

«A~u!50.031531024u10.02 cos~u!13

31023 cos~2u!2631023 sin~u!14

31024 sin~3u!,~37!

«B~u!50.051331024u10.05 cos~u!26

31023 cos~3u!11024 sin~u!13

31023 sin~4u!.

Figure 10 shows a comparison between the given bias errorsand those computed by the translation-based method~u0

5p/35 andN56!.

VI. APPLICATIONS

In principle, the aforementioned methods are applicableto any device or instrument as long as the bias error distri-bution can be sufficiently described by a power series or aFourier series in the measurement domain. Here, to illustratediverse applications of these methods, we present severaltypical examples in voltage measurement, angular measure-ment, optical measurement, and roundness measurement.

A. Voltage calibrations

An analog-to-digital~A/D! converter is calibrated usingthe scaling-based method. A voltage divider was constructedusing stable resistors to obtain a nominal voltage ratio of 0.5~the scaling constant 0.5 in the scaling-based method!. Theinput to the voltage divider was connected to a stable voltagesource and the output was connected to one channel of a 24

bit multiplexing A/D converter~Lawson model 201!. An-other channel was connected directly to the voltage source.The voltage source was varied from25 to 15 V ~the fullrange of the A/D converter!. The 100 readings on both chan-nels were recorded. No accuracy was assumed for the volt-age source or the voltage divider ratio. Calculation for thebias error distribution was performed by the approximatescaling-based method. Two reference readings were taken toprovide the standard values, one with the input shorted andthe other with the input voltage set to exactly 3 V@as mea-sured on a precision voltmeter~HP 3458A! with accuracy of8 ppm#. This is all that is required to find the calibration ofthe A/D converter using the approximate scaling-basedmethod. Eleven additional points were taken using the pre-cision voltmeter~HP 3458A! over the full range to verify themathematical solution. The voltage calibration plots in Fig.11 show the verification points and the calibration resultscomputed by the scaling-based method fora50.5 andN515. The calibration curve computed by the scaling-basedmethod for scaling by the voltage divider is in good agree-ment with the verification data.

Instead of using a voltage divider for scaling, nominalscaling can be achieved based on the manually set dials of avoltage source ~e.g., Wavetek model 220 differentialvoltmeter/calibrator!. An advantage of this approach is that aphysical device like the divider is no longer required forscaling. Figure 11 shows the bias error distribution computedby the scaling-based method in which scaling is controlledbased on the dial readings. Compared with the results givenusing the divider, the distribution obtained using the dialreadings shows larger local variations. This is because themechanical voltage setting device of the power supplycaused abrupt voltage variations over a certain range of op-eration. It is expected that a more stable reference devicewould produce a smoother bias error distribution. In fact, thebias error distribution of the power supply itself was alsorecovered by the scaling-based method, indicating the abruptvoltage variations.

FIG. 10. Given bias error distributions and computed results using the two-device translation-based method~u05p/35, N56! for devicesA andB.

FIG. 11. Measured bias error distribution of the voltage and computed re-sults using the scaling-based method~a50.5,N515! for an A/D converter.

3568 Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 T. Liu and T. D. Finley

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B. Angle calibrations

The two-device translation-based method was used forangle calibration on a dividing head with an encoder at-tached. The secondary device was an indexing table with aresolution of 1°. Both devices have a nominal accuracyspecification of 1 arcsec. The axis of rotation was horizontalso that a precision servo accelerometer could be used as anindicator of the level. The procedure involves rotating thedividing head clockwise in nominal 10° increments, rotatingthe indexer counterclockwise in nominal 10° increments, andreading the level indicated by the precision accelerometer.The second set of data was taken with the indexer and accel-erometer translated260° with respect to the encoder. Torecover the bias error distributions, the first set of data andthe second set of260°-translated data were processed by thetwo-device translation-based method with a fourth-orderFourier series. The calibration curves for both the encoderand indexing table can be determined simultaneously, asshown in Fig. 12. The black closed circles in Fig. 12 are theresults of a conventional calibration using a device with ac-curacy four times better than that of the dividing head withthe encoder. The bias error distribution measured by the pre-cision accelerometer is in good agreement with the computeddistribution for the encoder. It is worth noting that we do notexplicitly use any standard value to recover the bias errordistribution. Nevertheless, the periodicity conditionu(0)5u(2p), which is automatically satisfied in the Fourier se-ries solution, can be considered to be an imposed constraint.The two-device translation-based method, originally pro-posed by Finley,4 has been used regularly by Wyle Labora-tories to calibrate precision angle-measuring devices in theirfacility and at NASA Langley since 1991.

C. Radial lens distortion

An interesting example is application of the scaling-based method to determination of the radial lens distortion.In reality, a lens used for imaging is not perfect and theimperfect lens distorts an image. Thus, camera calibration to

determine the camera parameters including the lens distor-tion parameters is crucial for accurate image-basedmeasurements.8 The most dominant lens distortion is the ra-dial lens distortion that is symmetric about the principalpoint ~close to the geometric center of an image!. The radiallens distortion is described by a simple modeldr 5K1r 3,wheredr is a bias error in the radial distance due to the lensdistortion,K1 is the radial distortion parameter, andr is theradial distance from the principal point. The radial distortionparameterK1 in addition to other camera parameters can beobtained in comprehensive camera calibrations by analyticalphotogrammetric techniques. For an 8 mm Computar TVlens used in this test, an optimization camera calibrationmethod8 givesK150.001 297 mm22.

Unlike analytical photogrammetric techniques that use aspecial mathematical model for the lens distortion, thescaling-based method determines the radial lens distortionunder a general theoretical framework of the bias error. Dur-ing tests, a 22 in.317 in. target plate with 121 retro-reflectivetargets of 1/2 in. diameter was used to provide a planar targetfield. A charge coupled device~CCD! video camera~HitachiKP-F1U! with an 8 mm Computar TV lens, viewing the tar-get plate perpendicularly, was used to take images of theplate placed at two different distances from the camera. Fig-ure 13 shows typical images of the target plate at two differ-ent distances from the camera. The digitized image has 6403480 pixels and the nominal pixel spacing is 9.9mm in boththe horizontal and vertical directions. The centroids of thetargets and the radial distances of the targets from the geo-metric center in these images were computed. Two images ofthe target plate at two different distances from the cameranaturally provide scaling in the radial distance in imageplane. The scaling constanta50.8262 was obtained by av-eraging ratios between the radial distances of the correspond-ing targets from the geometric center in two images. The biaserror distribution«(r ) was calculated by the approximatescaling-based method when the order of the polynomial issuitably chosen inN520– 28. By the definition of«(r ), oneknowsdr (r )52«(r ). The conditiondr (0)50 must be sat-isfied. Thus, only one standard value is required to determinethe unknown constants in Eq.~22!, which is given by theoptimization camera calibration method. Figure 14 shows theradial lens distortion distributions obtained by the scaling-based method (N525) and the optimization camera calibra-tion method for an 8 mm Computar TV lens.

D. Roundness measurement

The methodology for determining the bias error can beextended to measure the roundness of an object that is nomi-nally cylindrical. The measurement value of the radius of anear-cylindrical object is

mR~u!5R1«R~u!, ~38!

whereR is a nominal radius,mR(u) is the measurement ofR,and«R(u) is the variation in the radius as a function of theangle. Without loss of generality, the nominal radius can beloosely defined as an average radius or an effective radius ofthe near-cylindrical object. We consider an angular transla-

FIG. 12. Measured bias error distribution of the angle and computed resultsusing the two-device translation-based method~u052p/3, N54! for anencoder and an indexing table.

3569Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 Estimating bias error distribution

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tion u0 and a position shift (Dx,Dy) of the rotational centerassociated with linear motion as the object is rotated. In thiscase, the measurement of the radius ismR(u1u0)5R81«R(u1u0), where the effect of the position shift (Dx,Dy)is characterized byR85@(x1Dx)21(y1Dy)2#1/2. Whenthe shift is relatively small, i.e.,uDxu/R!1 anduDyu/R!1,one has

mR~u1u0!5R1«R~u1u0!1Dx cosu1Dy sinu. ~39!

From Eqs.~38! and~39!, we obtain a functional equationfor «R(u),

«R~u!2«R~u1u0!5d~u!, ~40!

where

d~u!5mR~u!2mR~u1u0!1Dx cosu1Dy sinu. ~41!

Equation~40! has the same form as Eq.~12! although thefunction d~u! is different. Hence, the Fourier series solutionto Eq.~40! can be also described by Eqs.~14! in which mostof the Fourier coefficients are given by Eq.~15!. However,the constant coefficienta0

(«) is different, and is given by

a0~«!5p21E

0

2pH mR~u!2 (n51

N

@an~«! cos~nu!

1bn~«! sin~nu!#J du22R. ~42!

In this stage, the Fourier series solution for«R(u) is only aformal solution since the solution contains the unknown shift(Dx,Dy).

In order to determine the center position shift (Dx,Dy),we first rotate mathematically the contourmR(u) by u0

whose coordinates are given by

Fx1

y1G5F cos~u! sin~u!

2sin~u! cos~u!G FmR~u!cos~u!

mR~u!sin~u! G . ~43!

For comparison, the Cartesian coordinates of the measuredrotated contourmR(u1u0) are given by

x25mR~u1u0!cos~u!,~44!

y25mR~u1u0!sin~u!.

Note that the contour (x2 ,y2) is translated by (Dx,Dy) rela-tive to the contour (x1 ,y1). Hence, we can determine(Dx,Dy) by matching the contours (x2 ,y2) with (x1 ,y1)using an optimization scheme. Thus, the translation-basedmethod has been developed for determining the bias errordistribution of roundness or the variation in the radius. As anexample, consider a simulated case in which the bias errordistribution of roundness is given by

«R~u!520.05 cos~2u!10.02 sin~4u!, ~45!

and the position shift is (Dx,Dy)5(0.03,20.02). Figure 15shows a comparison between the given and computed biaserror distributions of roundness.

FIG. 13. Images of the target plate at two different distances relative to thecamera.

FIG. 14. Radial lens distortion distribution computed by the scaling-basedmethod~a50.8262,N525! for an 8 mm Computar TV lens, along with thecamera calibration result obtained by an optimization method.

3570 Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 T. Liu and T. D. Finley

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An apparatus can be made to measure roundness of anear-cylindrical specimen or object by using the translation-based method. Multiple displacement sensors are mountedradially around the periphery of a holder and the specimen isinserted into the system such that all of the sensors monitorthe radius at spaced angular increments. The sensors are allreset to zero and the specimen is rotated at a given angle.Thus, the difference in readings prior to and after rotationgives d~u! in Eq. ~41! and then the bias error of roundness«R(u) can be obtained by the above-described translation-based method. This apparatus uses only the relative differ-ence d~u! rather than the absolute radius measurementsmR(u) andmR(u1u0). Hence, an iterative scheme is neededto determine the center position shift (Dx,Dy) in Eqs. ~43!and ~44!. The initial value formR(u) is mR

(0)(u)5R, whichrepresents a perfect circle with a nominal radiusR. Next, theiterative relationmR

(n11)(u)5R1«R(n)(u) is used to modify

the contour shape and to recalculate the center position shift(Dx,Dy) using the optimization scheme for matching con-tours, wheren is an iteration index. After a number of itera-tions ~for example, n520!, a reasonable estimate for(Dx,Dy) can be obtained.

The advantage of this technique is that the specimendoes not have to be mounted on a spindle which would causeerrors because of the imperfections in the bearings of therotation system. The position shift of the center that mayoccur as the specimen is rotated can be determined math-ematically along with the bias error distribution of round-ness. This sets the method apart from other techniques thatrely on coordinate measuring systems and single sensor sys-

tems that rely on the accuracy of a bearing. This methodcould be extended to a real-time measurement that could beused while the part is still being manufactured.

APPENDIX: LEAST-SQUARES ESTIMATION OF THECOEFFICIENTS

To determine the coefficientsdn (n50,1,2, . . . ,N) inthe power series or polynomiald(x) in Eq. ~7! for a set ofdata pointsxi ( i 51,2, . . . ,M ), a system of equations fordn

(n50,1,2, . . . ,N) is

Pd5d, ~A1!

whered5(d0d1d2¯ dN)T, d5@d(x1)d(x2)¯d(xM)#T, and

P5S 1 x1 x12 x1

2¯ x1

N

1 x2 x22 x2

2¯ x2

N

¯

1 xM xM2 xM

2¯ xM

N

D .

The least-squares solution to Eq.~A1! is

d5„PTP…À1PTd. ~A2!

Similarly, a system of equations for the coefficients(an

(d) ,bn(d)) in the Fourier series in Eq.~14! for a set of data

pointsu i ( i 51,2, . . . ,M ) is

FaÄd, ~A3!

where a5(a1(d)¯aN

(d)b1(d)¯bN

(d))T,d5@d(u1)d(u2)¯d(uM)#T, and

F5S cos~u1!¯cos~Nu1! sin~u1!¯sin~Nu1!

cos~u2!¯cos~Nu2! sin~u2!¯sin~Nu2!

]

cos~uM !¯cos~NuM ! sin~uM !¯sin~NuM !

D .

The least-squares solution to Eq.~A3! is

a5„FTF…À1FTd. ~A4!

1H. W. Coleman and W. G. Steele,Experimentation and UncertaintyAnalysis for Engineers~Wiley, New York, 1989!, pp. 7–16.

2P. R. Bevington and D. K. Robinson,Data Reduction and Error Analysisfor the Physical Sciences~McGraw–Hill, New York, 1992!, pp. 1–16.

3J. R. Taylor,An Introduction to Error Analysis, 2nd ed.~University Sci-ence, Sausalito, CA, 1997!, pp. 106–109.

4T. D. Finley, NASA Technical Memorandum 104148~August 1991!.5W. L. Snow, NASA Technical Memorandum 109048~November 1993!.6J. Aczel,Lectures on Functional Equations and Their Applications, Vol.19, Mathematics in Science and Engineering~Academic, New York,1966!, pp. 1–20.

7E. Castillo and M. R. Ruiz-Cobo,Functional Equations and Modeling inScience and Engineering, Vol. 161, Monographs and Textbooks in Pureand Applied Mathematics~Dekker, New York, 1992!, pp. 1–44.

8T. Liu, L. N. Cattafesta, R. H. Radeztsky, and A. W. Burner, AIAA J.38,964 ~2000!.

FIG. 15. Given bias error of the roundness along with the calculated one fora simulated case.

3571Rev. Sci. Instrum., Vol. 72, No. 9, September 2001 Estimating bias error distribution

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