Measurement 44 (2011) 1153–1165

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Measurement

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Design of experiments and data-fitting techniques applied to calibrationof high-frequency electromagnetic field probes

Massimo D’Apuzzo, Mauro D’Arco, Nicola Pasquino ⇑Dept. of Electrical Engineering, University of Naples Federico II, Via Claudio, 21 - 80125 Naples, Italy

a r t i c l e i n f o

Article history:Received 27 September 2010Received in revised form 16 February 2011Accepted 8 March 2011Available online 12 March 2011

Keywords:Sensor calibrationElectromagnetic field sensorsDesigned experiments

0263-2241/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.measurement.2011.03.007

⇑ Corresponding author. Tel.: +39 08176 83630; faE-mail addresses: massimo.dapuzzo@unina.it (M

darco@unina.it (M. D’Arco), nicola.pasquino@unina.

a b s t r a c t

Characterization of electromagnetic field sensors requires that the behaviour of each sen-sor is investigated over the whole functional space, i.e., the range in which the device isexpected to work. In most cases, for the sake of time reduction, experiments are run onlyover a subset of the functional space. The reduced extension of the experimental space lim-its the knowledge of the actual behaviour of the probe represented by the calibration factorK. Linear regression is a common technique to represent measurement data, but it is gen-erally not applied to sensor calibration. Furthermore, no comparative study of experimentalplans differing in the number and location of experimental points has ever been carried outnor has a methodology to determine the optimal degree of the regression polynomial beenoutlined. This paper compares the error variance between experimental points and fittedcurve obtained using different experimental plans and regression polynomials.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Measurements of electromagnetic fields involve corre-lating the output of the sensor to the incident wave bymeans of a calibration factor K defined as:

K ¼ Em

Einc; ð1Þ

where Einc is the amplitude, or field strength, of the inci-dent field and Em is the probe reading. According to theIEEE standard 1309:2005 [1], calibration can be performedusing one of three different methods. In the transfer stan-dard method, the field used for calibrating the sensor ismeasured using a similar probe that had been previouslycalibrated. In the reference field method, a standard devicesuch as a TEM [2] or GTEM cell [3], or a semi-anechoicroom, is used to generate a calibrated field. In the referencesensor method, the Einc calculated from the voltage output

. All rights reserved.

x: +39 08123 96897.. D’Apuzzo), mauro.

it (N. Pasquino).

of a sensor of accurately known shape and size, and with aresponse that can be mathematically calculated from Max-well’s equations, is used as a reference quantity value [4]for calibration. The calibration procedure is used to gener-ate a table of K values. A table consists of a set of discretepoints within the experimental space so the K values thatare not in the table can be linearly interpolated fromneighboring tabulated values.

Variables that are commonly assumed to mostinfluence K are frequency f and Einc [5]. In theory, otherinfluence quantities, such as temperature, should be takeninto account for their effect on the device; however,modern design techniques typically limit the effects oftemperature on electronic components. If the device doesnot have temperature correction then it must be calibratedimmediately before use so that temperature variations canbe neglected [6].

There are two limitations to commonly-used calibrationmethods. The first limitation, particularly for broadbandsensors, is that many measurements would need to be per-formed to completely characterize how all primary influ-ence quantities impact K. In the practice, calibration timeis reduced by running experiments over an experimental

1154 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

space comprising only few test points, even for systemsthat operate over many carrier frequencies within a rela-tively narrow frequency range.1 The consequence of suchreduction is that linearity tests, which involve measuring Kversus Einc, are performed at only one frequency, while band-width tests, which involve measuring K versus f, are per-formed at only one level of Einc. Unfortunately, theseapproaches provide no information about the probe’s behav-iour at all Einc and f combinations within the experimentalspace. Therefore, it is necessary to sample from a largerexperimental space to assess possible interactions betweenEinc and f, and provide the user with accurate calibrationdata rather than having him make use of measurement re-sults obtained for different experimental points, that wouldcause an increase in measurement uncertainty. In fact, fewcalibration services—except those that perform calibrationat higher levels of the metrological chain—perform calibra-tion at a variety of Einc and f combinations so that the outputof the calibration procedure are the values of K(f, Einc)tabulated at various combinations of Einc and f within theexperimental space. In this case, the calibration proceduregenerates a large amount of data that has to be managed.The approach presented in the following sections will reducethe problem of managing large amounts to data by reducingthe amount of data generated by the calibration procedure.

The second limitation of the commonly-used calibra-tion methods is the lack of consistent rules regardinghow to use calibration data. Calibration factors are usedto adjust the sensor’s indication according to the specificcombination of Einc and f. Problems arise when measure-ments are carried out at a combination of frequency andamplitude where a K value is not available. Some manufac-turers allow the user or calibration service to store K valueson the sensor, which then automatically performs a linearinterpolation between the closest K values; however, thisapproach is viable only when a small set of data has beenobtained. If a denser experimental plan is adopted, thenit becomes unpractical to load the entire table of K valuesonto the sensor. The proposed approach will reduce theamount of data that needs to be managed by storing onlythe coefficients of the fitted function on the sensor so thatK values can be automatically generated using a polyno-mial model rather than using linear interpolation.

Data reduction techniques can be used to reduce theamount of data generated for calibration and K value tabu-lation. A polynomial model is used to fit measured quantityvalues so that the behaviour of the sensor can be character-ized. A step in this direction has been made in [6,7], whichare both focused on applying regression techniques to sen-sors’ calibration. However, the first paper does not provideguidelines on how to choose the number and position oftest points within the experimental space to be used dur-ing calibration, nor does it provide criteria for choosingthe degree of the interpolating polynomial required tointerpolate calibration data. On the other hand, the focusof the second paper is on the optimization of the calibra-tion procedure, in terms of number of calibration points,

1 In the 1–2 GHz frequency range only a few points are tested while theDCS-1800 mobile system has more than 500 channels.

their location, and number of repetitions at each point,once the calibration curve is given.

The study presented in this paper extends the outcomesof the research previously conducted by applying a proce-dure in two steps. In the first step, a set of experiments isdesigned as a compromise between the specifications ofthe calibration standard [1] and the requirements of thedesign of experiment (DOE) methodology [8], that selectsthe number and distribution of test points over the func-tional space based on the test objective and the expectedbehaviour of the sensor. The second step involves usinglinear regression procedures for obtaining a least-squaresestimate of the coefficients of a polynomial function [9],which is used to approximate the response of the sensoras accounted by calibration data.

The study involves the comparison of different modelsusing the residual error of the fit between the measuredquantity values and corresponding values generated bythe polynomial, and its dependence on the number ofpoints in the calibration data set. By providing the userwith the residual error, they can select the best combina-tion of experimental plan and polynomial model: a modelis accepted if there is a significant reduction in the modelfit error. The coefficient of multiple determination is alsoused to compare the capability of each model to followthe variations of the input.

The paper is organized as follows: Section 2 consists of abrief review of probe and sensor calibration, design of theexperimental plans, and linear regression techniques. InSection 3, the results obtained from applying the experi-mental plans are presented. Finally, some conclusionsand proposals for future work are discussed in Section 4.

2. Experimental methodology and data analysistechniques

2.1. Field calibration

According to (1), the calculation of K requires that theincident field Einc be known. The amplitude of the incidentfield can be obtained using the transfer standard methodgiven in Section 1 with the experimental setup shown inFig. 1. The incident field has been generated inside theGTEM cell available at the Department of ElectricalEngineering of the University of Naples, Italy. Unlike aTEM cell, which normally operates up to approximately200 MHz (or 500 MHz for high-quality cells), the GTEM iscapable of extending the frequency range up to a fewGHz. The reference sensor method was not consideredapplicable because the calibrator is so expensive that it isusually owned only by a primary laboratory so is oftennot available. The reference field method, also discussedin Section 1, could not be used because the GTEM cell,though accepted as a working measurement standard, isnot the preferred method to generate a reference fieldaccording to the usage guidelines in Table 3 in [1]; how-ever, the simplified expression in the reference field methodfor Einc generated at a height d/2:

Eincðd=2Þ ¼ffiffiffiffiffiffiffiffiffiffiffi

ZGPNp

d; ð2Þ

Fig. 2. HI-6105 and EP-330 probes setup during field calibration (left) and probe calibration (right).

Fig. 1. Experimental setup.

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1155

where d is the distance between the septum and theground plane at the measurement point and ZG is the cell’scharacteristic impedance (nominally ZG = 50 X), has beenused to obtain an initial estimate of the net power PN, thedifference between the forward power Pf and reversepower Pr, to be injected into the cell in order to establishthe desired Einc value.2 Equation (2) assumes that a TEMwave is propagating along the longitudinal axis of the cellwith linear polarization along the axis perpendicular to theground plane. The field generated is then measured by anHI-6105 probe by ETS-Lindgren (Fig. 2a), which is used asthe transfer standard, and corrected by applying the valuesof the calibration factors Kt published in its calibration cer-tificate. In there, it is also stated that the transfer standardhas been calibrated with the reference field method, wherethe generation uncertainty is 0.7 dB over the frequencyrange used in the experiments presented in the following

2 Because the transfer standard and the probe being calibrated load thecell differently, the variations in the power actually delivered to the cell iscountered by focusing on net power instead of forward power.

sections. That value can be considered as the lowest uncer-tainty level of the calibration field’s amplitude that can bereached in the calibration procedure. The power deliveredto the GTEM cell is then varied until the corrected readingmatches the desired value Einc for the incident field. The cor-responding PN value is then stored and the procedure is iter-ated for all f and Einc values indicated in the calibrationcertificate of the transfer standard. The set of PN values isthen used to generate the calibration field strength whenthe transfer standard is replaced by probe to be calibrated,which in our experiment is an EP-330 probe by PMM(Fig. 2b). Probes are steady during the experiment, as theisotropy characteristic has not been measured. The experi-mental plan used for the probe to be calibrated may includevalues of f and Einc that are not included in the calibrationcertificate of the transfer standard. Therefore, at thoseexperimental points the PN values are not available fromthe field calibration procedure, but they can be calculatedusing a bilinear interpolation of the nearest square of Einc

and f calibration points available from the previous step.The uncertainty involved in the generation of the calibration

Table 1Regression polynomials used in the 28-point experiment.

Model Regression polynomial

1 b1 + b2f2 b1 + b2Einc

3 b1 + b2f + b3Einc

4 b1 + b2f + b3Einc + b4f � Einc

5 b1 + b2f + b3Einc + b4f2

6 b1 þ b2f þ b3Einc þ b4E2inc

7 b1 þ b2f þ b3Einc þ b4f 2 þ b5E2inc

8 b1 þ b2f þ b3Einc þ b4f 2 þ b5E2inc þ b6f 2 � E2

inc

9 b1 + b2f2

10 b1 þ b2E2inc

11 b1 þ b2f 2 þ b3E2inc

12 b1 þ b2f 2 þ b3E2inc þ b4f 2 � E2

inc

Table 2Regression model statistics for the 28-point factorial.

Model R2 (%) R2adj (%) Fe p value r2

e ð�10�3Þ

1 7.3 10.8 2.07 0.16 1.202 7.2 10.7 2.03 0.17 1.203 14.6 20.9 2.14 0.14 1.154 17.1 26.3 1.65 0.20 1.165 15.7 25.1 1.49 0.24 1.236 17.3 26.5 1.67 0.19 1.187 18.4 30.5 1.29 0.30 1.218 19.3 34.2 1.05 0.41 1.259 6.6 10.1 1.83 0.18 1.21

10 5.1 8.6 1.4 0.24 1.2211 11.7 18.2 1.66 0.21 1.1912 12.6 22.3 1.15 0.34 1.23

1156 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

procedure at interpolated points can be obtained from theuncertainty of the power meter used to measure Pf and Pr.

2.2. Experimental plans

Once the field has been calibrated using the HI-6105transfer device in the first step of the transfer standardmethod, the second step of the procedure involves calcu-lating K for the EP-330, that is the probe being calibrated.Different choices can be made about the number of calibra-tion points and the method of selecting and distributingthem over the experimental space. Each plan has bothadvantages and disadvantages with respect to some fea-tures, the most relevant usually being the variance of theprediction error and the degree of reliability of the regres-sion model. We employ two different factorial plans, whichwill be briefly reviewed in the following subsections.

2.2.1. Full factorial plansThe full factorial plan is probably one of the simplest, so

most widely used, design after the One-Factor-At-a-Time(OFAT) plan [10]. The OFAT plan has not been used in theresearch carried out by the authors because it lacks theability to determine whether the variations in the outputdue to variations in one factor also depend on the levelof the other factors (interactions). As a matter of fact, theprocedure adopted by calibration laboratories is usuallybased on OFAT plans because they measure linearity orbandwidth for only one level of the other input, thusexcluding the existence of an interaction effect in thebehaviour of the sensor.

Strictly speaking, there is no design in a full factorialplan other than the choice of input factors, their range,and the spacing criterion. The experimental space is simplythe cartesian product of the vectors of the test points cho-sen for each variable. The only planning involves designinghow dense the plan must be and how to choose the levelswithin the experimental space. Standard [1] presents threedifferent approaches: linear frequency spacing (clauseA.1.2.1), logarithmic frequency spacing (clause A.1.2.2),and percentage frequency spacing (clause A.1.2.3). Overlarge frequency ranges (usually more than one decade) lin-ear spacing is not advisable because the same Df could betoo large to test the system at the low-frequency end andtoo small to test the system at the high-frequency end.The latter two approaches determine the next frequencywith a logarithmic increment based on the total numberof test points to be contained within a given interval(e.g., 10 frequencies per octave or decade), or with a per-centage increment of the current frequency. The numberof points used to test the effect of Einc on K has been chosenas a trade-off between acquiring a large number of pointsfor the best possible insight into linearity features andthe resolution of the probe, which has a limit of0.01 V m�1. The experimental range of Einc is from8 V m�1 to 125 V m�1, the same interval for which calibra-tion factors are available for the transfer standard, and con-tains more points than required by the standard.

Two full factorial plans have been designed. In the firstplan, we have chosen the same calibration points used forthe transfer standard so that we can use the K values

available in the calibration certificate. As a result, theexperimental space is a 28-point set given by Einc � f = [8,20, 70, 125] V m�1 � [0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1] GHz. Inthe second plan, we have designed a denser plan using a5% increment in the f axis and a 14.7% increment in the Einc

axis. The initial idea was indeed to have approximately thesame number of points along each axis as the total numberof points in the first experimental plan, i.e. 28 points forboth Einc and f, with a total of 784 points. However, sucha sampling of the experimental space was too dense forthe relatively narrow intervals of variation of the field’samplitude and frequency. So we set for a plan containinga total of 20 � 20 = 400 experimental points. However, thisdoes not limit applicability of the proposed calibration pro-cedure, that can be implemented with any number ofexperimental points as long as execution time and costsare not an issue. In Section 2.2.2 we will show how the per-formance of the experimental plan changes with the num-ber of experimental points. The PN values required toestablish the target field have been chosen following thebilinear interpolation procedure highlighted at the end ofSection 2.1. The performance of each plan has been evalu-ated using the variance of the error between the valuespredicted by the regression function and the measured val-ues (prediction error).

2.2.2. Randomly-reduced factorial planIn addition to the approach specified by the standard,

test frequencies and strengths can be selected randomly

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1157

within the experimental space. By adopting a very denseplan, there may be an overestimation of the number ofpoints needed to efficiently describe the behaviour of theprobe. In fact, the same performance of the calibrationprocedure in terms of, for example, prediction error couldbe obtained using a smaller number of points. Therefore,

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0.99

Resid

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abilit

y

Normal Prob

Fig. 3. Regression model and analysis of residual

starting from the 400-point factorial plan, we haveexcluded up to 98% of the points from the analysis. Eachsubset has been created by picking points uniformly fromthe full factorial plan. Regression coefficients have been cal-culated for each subset and the associated error variance isevaluated as a performance comparison parameter.

500600

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9001000

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.02 0.04 0.06 0.08

ue

ability Plot

s for the fitted polynomial b1 + b2f + b3Einc.

1158 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

2.3. Data analysis and linear regression

The amount of data obtained from the experimentsneeds to be reduced so that it can be more easily managedwhen the calibration factors are used in practice. The eas-iest way to reduce the amount of data is to fit the K valuesto a polynomial model. The selection of the polynomialmodel with the best performance is made by progressively

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Fig. 4. Regression model and analysis of residuals for

increasing the order of the polynomial. First, all first orderpolynomial models are tested and the polynomial modelwith the best fit, i.e., the one that presents the smallest pre-diction error variance, is chosen. This model is then used asthe starting function for the second order polynomial model.Once again, all combinations of the polynomial modelare tested and the polynomial model with the smallestprediction error variance is selected. This procedure can

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0.02 0.04 0.06 0.08ue

bability Plot

the fitted polynomial b1 þ b2f þ b3Einc þ b4E2inc .

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1159

be iterated for as many steps as needed to make the predic-tion error variance smaller than a given threshold valuethat is chosen such that its combination with theuncertainties due to other sources like, for example, theuncertainty in the calibration factors Kt of the transferstandard (see Section 2.1) will sum up to a value lowerthan the uncertainty budget that the calibration and

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abilit

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Fig. 5. Regression model and analysis of residuals

regression procedure is allowed reach. In the followingsection an example is presented in which the procedureis iterated up to a second order polynomial model.

The coefficients of the polynomial model can be storedon an EEPROM instead of the tabulated values, as is thecurrent practise. Storing calibration coefficients on anEEPROM would require much less memory usage than a

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2 0.04 0.06 0.08ue

bability Plot

for the fitted polynomial b1 þ b2f 2 þ b3E2inc .

Table 3Regression polynomials in the 400-point experiment.

Model Regression polynomial

5 b1 + b2f + b3Einc + b4f � Einc + b5f2

6 b1 þ b2f þ b3Einc þ b4f � Einc þ b5E2inc

7 b1 þ b2f þ b3Einc þ b4f � Einc þ b5f 2 þ b6E2inc

8 b1 þ b2f þ b3Einc þ b4f � Einc þ b5f 2 þ b6E2inc þ b7f 2 � E2

inc

Table 4Regression model statistics for the 400-point factorial.

Model R2 [%] R2adj [%] Fe p value r2

e ð�10�3Þ

1 23.3 23.5 120.6 0 2.972 0.8 1.0 3.1 0.08 3.843 24.1 24.4 62.8 0 2.954 24.6 25.2 43.1 0 2.935 31.3 32.0 45.1 0 2.686 24.6 25.4 32.3 0 2.947 31.3 31.2 35.9 0 2.698 31.6 32.8 30.2 0 2.689 26.2 26.4 141.5 0 2.86

10 0.7 0.9 2.8 0.09 3.8411 26.9 27.3 73.2 0 2.8412 27.3 27.9 49.7 0 2.83

3 For interpretation of color in Figs. 3–5, the reader is referred to the webversion of this article.

1160 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

full set of experimentally-obtained calibration data. Theexample presented in the next section will show that forthe procedure iterated up to the second order, the numberof coefficients for the best-fit polynomial model in a 28-point experiment is 4, with a reduction of 85.7% over thetabulated K values; while for the 400-point experimentthe number of coefficients to store is at most 7, with areduction of 98.3%.

3. Results

In this section we present the results obtained by apply-ing the methodology described in Section 2 to differentexperimental plans.

3.1. Full factorial plans

3.1.1. 28-point planTable 1 shows the regression models used in the first

experiment, in which the 28 points published in thetransfer probe calibration document have been used.Models are grouped into three subsets: the first subsetcontains three polynomial models with Einc and f maineffects and an f � Einc interaction; the second subsetcontains three polynomial models with E2

inc and f2 maineffects and an f 2 � E2

inc interaction, each effect beingadded to the polynomial of the first subset that resultedin the smallest error variance r2

e ; and the third subsetthat only contains second order terms and is used to ver-ify how the completeness of the polynomial model affectsthe error variance r2

e . This quantity is reported in Table 2together with the other statistics obtained for all thetested regression models [8]. The Fe-value is used to testthe hypothesis that all regression coefficients are zero(H0) versus the alternative hypothesis (H1) that at leastone coefficient is not zero, and is computed as the ratioof the regression mean squares to the error mean squares.If Fe is larger than the value tabulated for the Fisher dis-tribution Fa,l,N�l�1 (where l is the number of regressorsused and N is the total number of experimental points)then we can reject the null hypothesis H0 and claim theregression significant. Usually the significance level isset as a = 95%. The p value indicates the probability thata value can occur by chance, based on the assumptionthat the frequency histogram of values associated withthe variable follows a normal, or Gaussian, distribution:it is the probability that Fa,l,N�l�1 > Fe. The coefficient ofmultiple determination R2 indicates the amount of variancein the observed K values that is explained by the variationin the predictors. It is defined as:

R2 ¼ SSR

SST¼ 1� SSE

SST; ð3Þ

where SST is the total sum of squares, SSR is the regressionsum of squares and SSE is the error (i.e. residual) sum ofsquares. The higher the value of R2, the better the regres-sion model describes the underlying behaviour of the cali-bration factor. Because R2 can increase as a result of addingnew variables, the adjusted R2, written as R2

adj, is often usedinstead of R2. The R2

adj value is obtained using [8]:

R2adj ¼ 1� SSE=ðN � l� 1Þ

SST=ðN � 1Þ u1� SSE=ðN � lÞSST=ðN � 1Þ ; ð4Þ

where l is the number of regressors used and N is the totalnumber of experimental points. The approximated equal-ity holds when N is large and l is small, such as in our case.The R2

adj value will increase only if the variable added issignificant because it has reduced the mean square error.The small R2

adj values indicate that the regressors chosen forthe different models cannot fully explain the variationin the measured values, even if the error variance r2

e remainsconsistently small. In this case, the R2

adj values agree withthe large p values that indicate a large probability thatthe regression models based on the 28-point data set areindeed not significant enough to describe the behaviourof the K values adequately. Additional considerations aboutthe interpretation of both parameters will be given later onin this section. Figs. 3a, 4a, 5a show the calibration factorsobtained using the 28-point data set (blue dots)3 and thefitted curves obtained for the models 3, 6 and 11, thatare those with the smallest r2

e in each of the three subset.For the same models, Figs. 3b, 4b, 5b show the normalprobability plot of the errors (residuals) of the predictingcurve measured at the experimental points. The plot showsthe experimental cumulative frequency for each predictionerror (blue cross) whereas the y-axis is scaled so that thered line obtained by extrapolating the segment connectingthe first and third quartile to the ends of the sample repre-sents a normal distribution. Therefore, if the residuals lieon the dashed-dotted line then they are normally distrib-uted, which implies that the regression model has beencorrectly chosen because the differences between the mea-sured and fitted points are due only to experimental noise.

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1161

3.1.2. 400-Point planThe same procedure has been applied using 400-point

(i.e., the 20 � 20 experimental space presented in Section2.2.1) data set. Starting from the PN values obtained by cal-ibrating the field at the points contained in the 28-pointplan, the additional PN values required to establish the de-sired Einc at the additional 372 points have been calculatedby applying the procedure outlined at the end of Section

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Fig. 6. Regression model and analysis of residuals for

2.1. The 400 Einc values obtained by the set of 400 PN valuesare measured by the probe under calibration and (1) isused to determine the K values. The regression procedureis then applied to these new experimentally-obtained Kvalues. Table 3 shows only the set of polynomials used inthe second iteration of the selection algorithm. They appar-ently differ from those obtained in the case of 28 calibra-tion points because they are based on a different seed

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0.05 0.1due

bability Plot

the fitted polynomial b1 + b2f + b3Einc + b4f � Einc.

1162 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

resulting from the selection made by the algorithm in thefirst iteration. In fact, while Table 2 shows that the polyno-mial model with the smallest error variance in the firstsubset is model 3, Table 4, that reports the statistics foreach model, shows that for the 400-point experiment thebest model is number 4 because it results in the smallestr2

e . The best-fit model for each subset is shown in Figs.

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Fig. 7. Regression model and analysis of residuals for the

6–8. The large residues shown in the normal probabilityplots are due to the lack of fit at the boundaries of theexperimental space between measured and fit data.

By comparing Tables 2 and 4, the clearest difference be-tween the two experimental plans are the p values which,for the 400-point plan, indicate that the sets of regressioncoefficients are significant for all models, yet the adjusted

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fitted polynomial b1 + b2f + b3Einc + b4f � Einc + b5f2.

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1

1.05

1.1

1.15

1.2

Frequency [MHz]Field [V/m]

K

−0.1 −0.05 0 0.05 0.10.001

0.003

0.010.02

0.05

0.10

0.25

0.50

0.75

0.90

0.95

0.980.99

0.997

0.999

Residue

Prob

abilit

y

Normal Probability Plot

Fig. 8. Regression model and analysis of residuals for the fitted polynomial b1 þ b2f 2 þ b3E2inc þ b4f 2 � E2

inc .

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1163

R2adj statistics still indicates that variations observed in the

experimental data have not been fully accounted for bythe models. The apparent contrast between the p valueand the R2 (or R2

adj) statistics can be explained by consider-ing that the latter may not serve as a good indicator for atypical resonant behaviour like that of an electromagneticfield sensor, which is characterized by localized steep vari-ations in the responses which can be fully described onlyby very high degree polynomials.

Table 4 also shows that models 2 and 10, which containonly the terms Einc and E2

inc respectively, return a value forthe error variance that is much larger than the typical val-ues of other polynomials in the same subset. The incre-ment is about 31% for model 2 and 35% for model 10, asopposed to a variation evaluated for the other models inthe same sets of about 1.3% and 1.1% respectively. Also,model 6, in which E2

inc is the only second-order term in-cluded, has an error variance which is comparable to the

4

4.5

5x 10−3

e2

σe2(150) ± 2 uσ

e

2(150)

150 points50 points

1164 M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165

values associated with the subset containing only first-order terms (with a variation of about 8% as opposed to0.3% for the remaining three models). These considerationslead to the conclusion that the field strength is significantin the regression only when it is combined in a joint-effectterm like f � Einc.

1 2 3 4 5 6 7 8 9 10 11 122

2.5

3

3.5

Model

Erro

r var

ianc

e σ

Fig. 10. Comparison of error variance r2e .

3.2. Randomly-reduced factorial plans

To assess the performances of a factorial plan of Mpoints obtained through the random reduction procedureof Section 2.2.2, the error variance r2

e versus the numberof calibration points for each regression model is shownin Fig. 9. For each model, 30 observations of a set of M uni-formly distributed calibration points (for M varying be-tween 8 and 400) have been drawn from the completeset of 400 points, and as can be seen in Fig. 9, for each valueof M, the means and standard deviation of r2

e obtained forthe regression model based on the M points. Generallyspeaking, each subset is characterized by error variancesthat are very close to each other, besides those models thatshow a different behaviour as explained at the end of theprevious subsection. Moreover, we notice that small valuesof M may give small mean values, although the associatedstandard deviation is almost of the same magnitude as themean itself, proving that a regression model based only ona few points can, in principle, give small error variance butis also very dependent on the specific set of points used inthe regression. This is the reason why the error varianceobtained with the regression models based on the 28points summarized in Table 2 is smaller than that obtainedwith 400 points. From M = 100 onwards, the average valueof r2

e takes on its final value, and the associated variancedecreases constantly giving larger and larger significanceto the mean value. When M is close to 400 points the 30repeated measurements begin to show some correlationbecause two observations of approximately 400 pointsdrawn from a set of 400 points are very likely to be notindependent, and, therefore, the variance associated tothe 30 repeated measurements decreases.

0 50 100 150 200 250 300 350 4002

3

4

5 x 10−3

Calibration points

Mea

n of

σe2

0 50 100 150 200 250 300 350 40010−5

10−4

10−3

10−2

Calibration points

Stan

dard

de

viat

ion

of σ e

2

Subset 1 Subset 2 Subset 3

Fig. 9. Error variance r2e versus number of regression points.

3.3. Robustness

To determine the robustness of the regression proce-dure, for each model a set of M = 150 points out of the400 has been chosen to determine the regressioncoefficients, and for the same M the mean r2

e ð150Þ andthe associated variance ur2

e ð150Þ has been obtained fromthe procedure shown in the preceding subsection andshown in Fig. 9. Of the remaining 250 calibration factorsavailable, a set of 50 has been drawn and the variance ofthe error between these 50 points and the fitted polyno-mial based on the 150 points has been calculated. FromFig. 10, while it is clear that, as expected, the error varianceof the polynomial fitted to the 150 points falls within theuncertainty interval given by r2

e ð150Þ � 2ur2e ð150Þ for all

models, it is remarkable that the r2e calculated for the 50

points also falls within that interval only for some models,proving that the regression can provide a good fit for pointsnot included in the regression procedure only for somemodels.

4. Conclusion

This paper shows the application of experimental de-sign and data-fitting techniques to the calibration ofhigh-frequency electromagnetic field probes. The proposedapproach has the advantage of reducing the amount ofdata needed to represent the calibration procedure. More-over, the outputs of the calibration procedure are theregression coefficients that, being obtained from the wholeset of experimental K values, describe the probe globallyand not only at the specific calibration points. The mainlimitation is inherent in the use of regression that canintroduce an error between the predicted K values andthose experimentally determined; however, this error issignificant only when calibration is performed with fewpoints or in the presence of sensors with extreme variabil-ity in frequency response and linearity. Finally, the methodprovides no a priori information about the prediction errorfor points not included in the experimental design,

M. D’Apuzzo et al. / Measurement 44 (2011) 1153–1165 1165

although the proposed polynomial model is thought torepresent the physical behaviour of a probe better thanthe linear interpolation between adjacent point suggestedby the current standards.

The proposed procedure selects the polynomial modelthat returns the least error variance and is based on arecursive selection of lower order polynomial models.The results show that higher order polynomial models usu-ally are better, particularly, if the polynomial model in-cludes terms of lower order. It has also been shown thatif we employ a calibration procedure in which the calibra-tion points are chosen randomly over the experimentalspace then when the number of points exceeds a thresholdvalue the prediction error variance reaches a steady state.A further increase in the number of points results in a re-duced variation in the prediction error variance, thusincreasing its significance. In addition, this procedureproves robust in terms of prediction error evaluated atpoints that were not included in the regression procedure,even though the results depend on the specific modeladopted and possibly on the number of experimentalpoints used in the regression.

Based on the results presented in this paper, a revisionof the current calibration standard can be proposed to in-clude the possibility for calibration laboratories to performa regression procedure for the calibration data and tochoose the degree of the regression polynomial and the

density of the experimental plan according to the expectedprediction error variance.

References

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