Variance of errors and elimination of ou

Variance of errors and elimination of outliers in the least

squares analysis of impedance spectra

J.R. Dygas

a,

*, M.W. Breiter

b

a_{Instytut Fizyk, Politechnika Warszawska, Chodkiewicza 8, 02-524, Warszawa, Poland} b_{Institut fuÈr Technische Elektrochemie, TU Wien, Getreidemarkt 9/158, A-1060, Wien, Austria}

Received 7 August 1998; received in revised form 28 January 1999

Abstract

The variance of errors in the impedance spectra is evaluated from replicate measurements. The variance of residual deviations from the ®t of equivalent circuit is calculated in the case of semi-replicate measurements. A model of variance, which describes departures of the variance from being proportional to the squared absolute value of admittance, is ®tted to the evaluated variance. Weights inversely proportional to the variance model are used in the CNLS ®tting of the impedance spectra. The variance model weights are compared with the modulus weights in the proposed procedure for elimination of outliers in which the residual deviations are compared with standard deviations estimated in accordance with the applied weighting scheme and the root-mean-square residual of the ®t. It is shown that the estimates of model parameters, using the variance model weights, are less sensitive to random errors of measurements than those using modulus weights.#1999 Elsevier Science Ltd. All rights reserved.

Keywords:Impedance spectroscopy; Variance of errors; Weighted least squares; Modulus weights; Elimination of outliers

1. Introduction

Complex nonlinear least squares (CNLS) ®tting of model response functions to the experimental data enables quantitative analysis of the impedance spectra [1±4]. The di€erences between the ®tted impedance and the dataÐthe residual deviationsÐare composed of systematic and random contributions. When large sys-tematic errors are identi®ed, a correction formula may be incorporated into the model function, which is ®tted to the data. Such procedure has been used by us for correcting distortions produced by current to voltage converters in measurements of large impedances [5].

The method of least squares supplies unbiased

esti-mates of the model parameters when errors are ran-domly distributed with zero mean. Each squared deviation should be weighted by a factor inversely pro-portional to the variance of random errors of the measurement [6±8]. The majority of the published CNLS analyses of impedance spectra employ modulus weights, which provide scaling of the data and ensure that results of the ®t are approximately equal when the computations are made for a given spectrum expressed as impedance or as admittance [9].

Experimental assessment of the variance by repeat-ing measurements had been rare [9,10] until Orazem and co-workers developed a method to ascertain the variance of random errors when the successive impe-dance scans are not strictly replicate [11±13]. A measurement model, ®tted to each impedance scan, was used by them to ®lter out the drift, while the var-iance of residual deviations was taken as the varvar-iance

0013-4686/99/$ - see front matter#1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 9 ) 0 0 1 3 1 - 0

of random errors. They observed that the variances of the real and imaginary components had the same mag-nitude and proposed a model of the error structure. With the weights corresponding to the error structure, more information about the studied system was gained by ®tting the equivalent circuit [11].

We report an investigation of random errors in measurements of large impedances performed using a frequency response analyzer and current to voltage converters [5]. Instead of a measurement model used by Orazem and co-workers [11±13], an equivalent cir-cuit model of the system under study is ®tted to each of the consecutive impedance scans in order to ®lter out the drift. Based on the experimental observations, a new model of error variance is proposed to account for departures of the variance from being proportional to the squared absolute value of the admittance. The model is ®tted to the variance of the admittance evalu-ated for a series of replicate or semi-replicate scans. We describe a novel algorithm for the elimination of outlying measurements which is based on a compari-son between the residual deviation from the ®t and the standard deviation estimated in accordance with the applied weighting scheme. Comparison of results of the least squares analysis with weights inversely pro-portional to the variance model and with modulus weights demonstrates advantages of the former, when the ratio of the error variance to the squared absolute value of the admittance varies signi®cantly in the spec-trum.

2. Experimental

The impedance spectra were obtained by an auto-mated setup for the measurement of large impedances that combines the Solartron SI-1260 Impedance/Gain-Phase Analyzer and the Keithley 428 Current Ampli®er. Since the sensitivity of the current channel of the Solartron SI-1260 is limited by the gain of the internal current to voltage (I/V) converter, the Keithley 428 is used as the externalI/Vconverter with the maximum gain of 1011 _{V/A [5]. In order to avoid}

large random and systematic errors, which are inevita-ble when the sensitivity of the current input is too low, theI/Vconverters were used at frequencies higher than the ¯at response bandwidth. The measured impedance spectrum was distorted and discontinuous upon change of the I/V conversion gain. A correction formula, expressed as a polynomial of the frequency, o=2pf, was used to reproduce the deviation of the measured impedance, ZM, from the actual impedance of the

sample,ZX:

characteristic for a given gain of the I/V converter, were estimated by the CNLS ®tting as additional par-ameters of the model. The details of the correction procedure have been described in [5].

The ac signal was 30 mV rms. Auto integration of Fig. 1. (a) Phase angle and absolute value of impedance of the te¯on platelet as a function of frequency. Symbols rep-resent data measured with di€erent gains of theI/V conver-ters:QÐ50 V/A above 5 MHz and 109_{V/A below 100 Hz,} wÐ5000 V/A above 30 kHz and 1010 _{V/A below 10 Hz,} +Ð105 _V/A,

the SI-1260 was used for the analyzer, which measured the current. Evaluation of error variance, elimination of outliers and usage of di€erent weights is demon-strated on two series of experimental impedance spec-tra:

2.1. Example 1

15 replicate spectra (125 frequencies from 10 MHz to 0.01 Hz) measured at 370 K for a te¯on platelet (9.84.70.7 mm3_{) placed between contact plates in}

the sample holder (Fig. 1a).

2.2. Example 2

33 semi-replicate spectra (127 frequencies from 10 MHz to 0.003 Hz) measured at 311 K for a poly-crystalline sample of the oxygen ion conductor BICUVOX with sputtered platinum electrodes. The sample was in the high conductivity state [14]. No sys-tematic change of the impedance with time was observed. Small variation of the measured impedance was related to ¯uctuations of temperature within 20.4 K (Fig. 2 and Fig. 3a).

3. Fitting and elimination of outliers

In the case of large impedances, the measurement of the low current is critical for the generation of errors. During analysis, the data are represented as complex admittance, thus leaving the current in the numerator and avoiding the transformation of errors. The CNLS ®tting of the admittance of an equivalent circuit to the data is performed with the aid of a new version of the program FIRDARM [4]. The minimized objective function, Q, is a weighted sum of products of real, DYi', and imaginary, DYi0, parts of the complex

devi-ation, DYi=YiÿFY (fi, X), of the admittance, Yi,

measured at a frequency,fi, from the response,FY

,cal-culated for the model described by r parameters

X={x1,x2...xr}:

Qˆ

Xm

iˆ0

w11,i…DY0i†2‡w22,i…DY0i†2‡2w12,i…DY0i†

…DY0_i† …2†

Summation is overmfrequencies of the spectrum. For each frequency fi, the weights: w11,i, w22,i, w12,i=w21,i

and imaginary parts of the admittance [6±8]. When the variance of errors is not known, modulus weights are

used:

w11,iˆw22,iˆD2=jYij2, w12,iˆ0 …3†

In this work the constant is D= 100, supposing that errors are about 1% of the absolute value.

When errors are randomly distributed with zero mean and the inverse of the covariance matrix of errors is used as the weight matrix, the expected value of the objective function after regression is equal to the number of degrees of freedom:E(Qmin)=2mÿr[6±

8]. The root-mean-square residual of the ®t is:

Rmsrˆ

Qmin=…2mÿr†

1=2

…4†

If the weights reproduce the proportions between the variances of errors: s2(Yi')=d2/w11,i and s2(Y0_i)=d2/w22,i, then the proportionality constant, d,

can be estimated by the residual of the ®t,Rmsr[6] and

one obtains estimates for standard deviations of measurement errors:

s…Y0i† ˆ

Qmin

…2mÿr†w11,i 1=2

,

s…Y0_i† ˆ

_Q

min

…2mÿr†w22,i 1=2

…5†

The above estimates can be used for testing the con-sistency of the measured admittance with the ®tted re-sponse function. Let us classify the measurement, Yi, as an outlier from the ®t at the level of k-times the standard deviation, when the real or imaginary part of the residual deviation, DYi=YiÿFY(fi, X), is greater

thank-times the estimated standard deviation:

jDY0

ij>ks…Y0i† or jDY0ij>ks…Y0i† …6†

An iterative procedure for elimination of outlying measurements is implemented in the CNLS ®tting

gram. After classifying measurements at certain fre-quencies as outliers, these data are removed from the spectrum and the ®t is repeated. A lower value of the objective function, Qmin, is obtained, the estimates of

standard deviations, Eq. (5), become smaller and ad-ditional data may be classi®ed as outliers at the same level of k-times the standard deviation. The above steps are iterated as long as outliers are found.

The proposed procedure for identi®cation of outly-ing measurements is based on the examination of re-sidual deviations, similar as most of the tests for discordancy used in statistics [15]. Since the statistical distribution of errors is not known and the ®tted model is nonlinear, critical values of the multiplier k

for rejection of outlier at a given con®dence limit are not available. Our procedure, used withk3_{5, proves to} be useful for the elimination of coarse outliers which signi®cantly spoil the ®t. Withk3_{3, it may be applied} to test the consistency of the weights employed in the CNLS ®tting with the distribution of residual devi-ations of the ®t.

4. Investigation of variance

For a stable system, when the admittance at the same frequency,fi, is measuredntimes, the variance of a statistical sample:Yi,1,Yi,2...Yi,n, is the estimate of the variance of random errors [6,7]. One can either repeat the measurement at each frequency several times before going to the next frequency or repeat the acquisition of the entire spectrum several times. The latter method was used in this work.

For the series of 15 replicate scans of the te¯on pla-telet, variances of the real and imaginary component, divided by the square of the absolute value of admit-tance are plotted in a logarithmic scale in Fig. 1b. The relative variances were dependent on the gain of the

I/Vconverter and increased with decreasing frequency within the band covered with a given gain. In contrast, the variances which are assumed in constructing the modulus weights are proportional to the squared ab-solute value of the admittance and would be rep-resented by a horizontal line in this plot.

When the impedance of the studied system changes with time, the random errors must be singled out from the drift. If the rate of change is suciently slow to allow ®tting of the measured spectrum with a response function, which is not dependent on time, one can cal-culate the variance of the residual deviations from ®ts of the same model to the consecutive impedance scans DYi,k=Yi,kÿFY(fi, Xk). The systematic changes of the

admittance, which occur between the subsequent measurements of the spectrum, are followed by the model response function with parameters, Xk,

esti-mated for each spectrum. The stochastic errors ran-domly contribute to the measurements at di€erent frequencies and remain after subtraction of the model response from the data. The above procedure, orig-inally developed by Orazem and co-workers [11,12], was here modi®ed by using an equivalent circuit model of the electrochemical system under study instead of a measurement model. Modulus weights were used to ®t the spectra in the procedure for evaluation of the var-iance. Similar values of the variance were obtained when the evaluation was repeated using weights inver-sely proportional to the variance model.

Such procedure was applied to the 33 spectra of BICUVOX. The equivalent circuit, shown as insert in Fig. 2, was used to model the measured impedance. The circuit was composed of resistors, capacitors and constant phase elements, cpe, whose admittance was expressed as: Y(o)=A(jo)1ÿN. The branch parallel to the bulk resistance, Rb, composed of the cpe Pb in

series with the capacitorC0, and the geometric

capaci-tance Cg reproduced the high frequency dispersion in

the measured spectra. The two parallelcpe PgpandPgr

modeled the grain boundary polarization: the ®rst was nearly capacitive Ngp30.15, the second nearly resistive Ngr30.96. At higher temperatures, the dispersion of the

grain boundary resistance was not observed and a resistor could be used in place of the cpe Pgr, but at

311 K the cpe with Ngr< 1 was necessary to

repro-duce the measured spectrum. Thecpe Pewith exponent Ne30.55 was used to reproduce the impedance of

plati-num electrodes. The impedance of electrodes was vis-ible in the complex plane diagram at higher temperatures as a low-frequency inclined spur. At 311 K the electrode impedance was not evident in the plot, see Fig. 2, and could only be estimated by ®tting.

The variance of the measured admittance data had large values with very little scatter at frequencies below 1 kHz, especially in the case of the real component, see Fig. 3b. The correlation coecient between the real and imaginary components showed a systematic vari-ation with frequency and had values close to 1 below 1 kHz. Those were signs of systematic changes of the impedance. The variance of the residual deviations from the ®ts of the equivalent circuit did not show this anomaly and was much smaller below 1 kHz than the variance of data, see Fig. 3c. The values of the var-iance were similar for the real and imaginary com-ponents. The model parameters were adjusted during ®tting: 11 parameters of the equivalent circuit and 9 coecients of the instrumental correction, Eq. (1) applied with 2, 3, 2, 1 and 1 parameters for the I/V

conversion gain: 50, 5000, 105, 107 and 108 V/A, re-spectively. Similar values of the variance were obtained when the coecients of correction were ®xed at their respective average values for the series of spectra.

var-iance, see Figs. 1b and 3c, has two distinct values. In the frequency range from 40 to 400 kHz, the relative variance is inversely proportional to the square of fre-quency, while at frequencies below 100 Hz in Fig 1b and below 0.1 Hz in Fig. 3c it is inversely proportional to the frequency. One has to consider the variation of the measured impedance with the frequency and the re-lation of the absolute value of the impedance to the gain of the I/V converter. The absolute value of the impedance of the te¯on platelet is inversely pro-portional to the frequency, see Fig. 1a. The same is true for the BICUVOX sample at frequencies above 1 kHz. The 1/f2 dependence, observed with the 5000 V/A gain of the Solartron SI-1260 internalI/V conver-ter, may be expressed by the ratio of the high measured impedance to the low feedback resistance of the converter (RF=5 kO): s2(Ys)/vYv20(vZv/RF)2. The

inferred model of the relative variance of errors is:

n…Ys† ˆs

whereYsdenotes the real or imaginary component:Y' orY0.

This model of the variance is justi®ed when one con-siders errors of the frequency response analyzer and noise in the measuring system. If one assumes,

simi-larly as Spinolo et al. [16], that the linearity error is proportional to the absolute value of the measured sig-nal, while the resolution error adds a constant com-ponent to the variance, then the variance of current may be approximated as:

s2…Is_{† c}

pI2‡ …dI0†2‡cSI‡cLI=f …8†

The absolute value of current I can be expressed by the voltage applied to the cell:I=VS/vZv, while the

cur-rent resolution error dI0 by the resolution of the

vol-tage analyzer dV0 and the feedback resistance RF: dI0=dV0/RF. The third and the fourth terms are

con-tributions from the shot noise and the low frequency 1/f noise respectively [17]. If one neglects errors of measurement of theacvoltage, then

s2…Ys†=jYj2s2…Is†=I2

cp‡ …dV0†2jZj2=…VSRF†2‡cSjZj=VS

‡cLjZj=…VSf † …9†

The 1/f term in Eq. (7) may originate either from the third term of Eq. (9), when vZv01/f, or from the fourth term, when the absolute value of impedance nearly does not depend on frequency.

Eq. (7) was ®tted to the relative variance evaluated Table 1

Coecients estimated for the model of error variance, Eq. (7), by ®tting to variances of data for the series of 15 replicate measure-ments of the te¯on platelet. Numbers in parenthesis are con®dence limits of the estimates expressed in percent of the value

I/V 50 V/A 5000 V/A 105V/A 106V/A 107V/A 108V/A 109V/A 1010V/A

for the two series of spectra. The coecients: b0, bZ, bF, for variances of the real and imaginary components

were treated as independent parameters. Estimates of the coecients were obtained for each gain of theI/V

conversion. At the most two of the three terms of Eq. (7) were used for any given gain. Good agreement between variances and model in the logarithmic plots as well as reasonable con®dence limits of the

wÿ_11,1_iˆ

hn…Y0_i†i ÿn…Y0_i†2‡an…Y0_i†2,

wÿ_22,1_iˆ

hn…Y0_i†i ÿn…Y0_i†2‡an…Y0_i†2

…10†

where hn(Y_i')i denotes the average over neighbor fre-quencies. The ®ts of the variance model presented in Figs. 1b and 3c were obtained using the average over three neighbor frequencies and a= 0.1. Estimated values of the coecients are given in Tables 1 and 2.

Table 3

Parameters of the equivalent circuit of Fig. 2 estimated for the 33 semi-replicate spectra of the BICUVOX sample by the CNLS ®tting: (i) with modulus weights to all data in each spectrum, (ii) with modulus weights after elimination of outliers at the level of 3-times the standard deviation, (iii) with variance model weights to all data in each spectrum. For each parameter, the average of values estimated for the series of spectra is followed by the standard deviation of the series of estimated values and the average value of the con®dence limit for the CNLS estimate, both expressed in percent of the average value of the parameter

Modulus weights, all data Modulus weights, outliers eliminated Variance model weights, all data

Average value Standard dev. Conf. limit Average value Standard dev. Conf. limit Average value Standard dev. Conf. limit

Cg 0.713 pF 0.24% 0.38% 0.716 pF 0.26% 0.12% 0.716 pF 0.11% 0.20% Ab 105.0 pS 7.8% 4.7% 103.6 pS 1.8% 1.7% 105.0 pS 1.4% 2.0% Nb 0.557 0.9% 1.2% 0.556 0.3% 0.36% 0.557 0.27% 0.40% C0 2.34 pF 2.8% 5.2% 2.35 pF 1.2% 0.9% 2.33 pF 0.8% 0.9% Rb 291.2 MO 0.9% 0.7% 291.3 MO 0.8% 0.1% 290.9 MO 0.8% 0.1% Agp 806 pS 1.7% 7.2% 795 pS 1.1% 1.0% 810 pS 0.7% 1.0% Ngp 0.156 3.8% 17% 0.150 2.1% 2.7% 0.156 1.2% 2.1% Agr 5.31 nS 0.6% 2.0% 5.34 nS 0.4% 0.3% 5.34 nS 0.37% 0.21% Ngr 0.956 0.4% 1.0% 0.954 0.5% 0.17% 0.957 0.46% 0.20%

Ae 450 nS 24% 54% 368 nS 23% 10% 295 nS 21% 12%

Ne 0.537 3.6% 8.1% 0.610 2.4% 1.0% 0.593 1.8% 1.2%

Rmsr 0.806 5.8% 0.115 10% 2.64 2.2%

Dygas,

M.W.

Breiter

/

Electrochimica

Acta

44

(1999)

4163±4174

Kronig transform. Large di€erences between the var-iances of the real and imaginary components, re¯ected in signi®cantly di€erent values of the respective model coecients, are observed when systematic errors are present and the Kramers±Kronig transform may not be obeyed, e.g., around 106Hz in Figs. 1b and 3c.

5. Applications of the variance model

The weights used in CNLS ®tting, Eq. (2), were taken equal to the inverse of the variance model. For each coecient of Eq. (7), the larger of two values, estimated for the real or the imaginary component, see Tables 1 and 2, was used during computation of weights,w11,i=w22,i.

In the case of the BICUVOX data, values of the cor-relation coecient between the real and imaginary parts of the residual deviations of the admittance from ®t of the equivalent circuit were scattered around zero. The correlation coecient between the real and ima-ginary components of the measured admittance of the te¯on platelet was also scattered around zero. Since no signi®cant correlation between the random errors of the real and imaginary components was observed, the products of the real and imaginary parts of deviation in the objective function, Eq. (2), could be neglected,

w12,i=0.

The CNLS ®tting of capacitance and instrumental corrections to the admittance spectrum of the te¯on platelet resulted in the ®t residual Rmsr=11.7. This

value is much larger than the valueRmsr=1.4 obtained

using modulus weights of Eq. (3) withD= 100. The modulus weights correspond to the variances of Eq. (7) with the coecients:b0=10ÿ4,bZ=0,bF=0, which

in Figs. 1b and 3c would be represented by horizontal line at the level: log[s2(Ys)/vYv2]=ÿ4. The majority of experimental variances and the ®tted variance model lie below this level. The large value of Rmsr indicates

that the standard deviations of replicate measurements underestimate the residual deviations of the ®t, com-pare Fig. 4. The systematic errors at certain frequen-cies are larger than random errors.

Outliers larger than 3-times the standard deviation, Eq. (6), were eliminated. With the variance model weights, after 5 iterations, the ®t residual was reduced toRmsr=3.9 and 18 outlying measurements were

elimi-nated: 14 in the frequency range from 31.6 to 383 Hz a€ected by interference with the 50 Hz frequency of the power line, three at the high frequency end (27.8± 36 kHz) of the range covered with the Keithley 428 and one at 10 MHz. The residual deviations of the remaining data are scattered around zero, see Fig. 4d. The data points with large random errors, which are retained in the spectrum, are weighted signi®cantly less

than other data points, as can be deduced by compar-ing the residual deviations with the limit of 3 standard deviations represented by the dashed lines in Fig. 4. With the modulus weights, 15 outliers were eliminated after 4 iterations and the ®t residual was reduced to

Rmsr=0.7. The outlying measurements were: 6 at

fre-quencies below 0.6 Hz with large random errors, 7 at frequencies around 50 Hz with large systematic errors and at the 2 highest frequencies of the spectrum. The residual deviations of several of the remaining data exhibited systematic errors, see Fig. 4c.

The CNLS ®tting with variance model weights of the equivalent circuit of Fig. 2 and instrumental cor-rections to the admittance spectrum of the BICUVOX sample gave the ®t residual Rmsr=2.64. Examination

of the residual deviations, see Fig. 5b, indicates good agreement between the scatter of the data and limits of 3-times the standard deviation, Eq. (5). Only 3 outliers were eliminated at the level of 3-times the standard de-viation, see Fig. 5d, and the ®t residual was reduced to

Rmsr=2.24. When modulus weights were applied for

®tting of the same data, a total of 31 outliers at the level of 3-times the standard deviations were eliminated after 8 iterations of the procedure, see Fig. 5a±c. The ®t residual was reduced from Rmsr=0.82 to Rmsr=0.11. All data contaminated by large random

errors were eliminated.

Thus, with the aid of the procedure for elimination of outliers, when the variance model weights are employed, measurements with large systematic errors can be selectively eliminated, compare Figs. 4b and d, while those containing large random errors are retained with smaller weights, see Figs. 5b and d. In contrast, when the modulus weights are employed, measurements a€ected by the largest relative errors, both random and systematic, are eliminated as outliers, see Figs. 4a±c and 5a±c.

The e€ect of using either modulus weights or var-iance model weights on the estimated values of par-ameter of the equivalent circuit was tested on the series of impedance measurements of BICUVOX. Results of ®tting each of the 33 spectra were recorded, the aver-age value and the standard deviation of the estimates of each parameter were computed. The procedure was repeated with iterative elimination of outliers at the level of 3-times the standard deviation. The results are presented in Table 3, where the con®dence limits of the parameter estimates obtained from the CNLS ®tting are also listed. In the case of variance model weights, only results obtained with all data in the spectrum are presented because the elimination of maximum 4 out-liers did not bring signi®cant changes. The estimated values of parameters are nearly the same in the three columns of Table 3, except for the cpe Pe, which was

weights indicates that the equivalent circuit of Fig. 2 is indeed a proper model of the investigated system.

The important advantage of the variance model weights, demonstrated by the results in Table 3 with all data points taken into account, is that the standard deviations of the parameters estimated for a series of semi-replicate spectra are lower than in the case of modulus weights. The di€erence between standard de-viations is considered to be statistically signi®cant when the opposite hypothesis (that the variances are equal) can be rejected based on the Fisher'sF-test. For two samples, each having 32 degrees of freedom, the hypothesis of equal variances can be rejected at a sig-ni®cance level, a=0.02, when the ratio of sample var-iances is larger than 2.35 [7]. This condition is ful®lled for the following parameters:Cg,Ab,Nb,C0,Agp,Ngp, Ne, see Table 3. Another advantage of the variance

model weights is a good agreement between the con®-dence limits of parameter estimates and the corre-sponding standard deviations of the series. Only for the bulk resistance, Rb, the standard deviation of the

series is larger than the con®dence limit, as expected for the parameter which varied as a result of tempera-ture ¯uctuations. With modulus weights, signi®cant re-duction of the standard deviations of the parameter estimates is obtained after elimination of outliers. This demonstrates applicability of the elimination of out-liers, when the variance model is not available and modulus weights are used, despite contamination of data by large random errors.

6. Conclusions

The variance of errors can be evaluated for a series of semi-replicate impedance scans taking into account residual deviations of individual spectra from the ®t of an equivalent circuit model of the electrochemical sys-tem. Such procedure is an alternative to using a measurement model [12].

The model of variance, proposed for the admittance measured using a frequency response analyzer equipped with a current to voltage converter, contains terms which describe the deviation of the variances of random errors of the real and imaginary components from being simply proportional to the squared absol-ute value of the measured admittance. The coecients of Eq. (7) for the variances of real and imaginary com-ponents have similar values. The observation of Orazem and co-workers [12,18], that the standard devi-ations of the real and imaginary components of the complex immittance are equal, is con®rmed by the pre-sent result.

The proposed procedure for elimination of outliers is a valuable tool for improving the quality of ®t. When the variance model weights are employed,

outly-ing measurements with large systematic errors can be selectively eliminated, while those containing large ran-dom errors are retained but are weighted respectively less during ®tting. When the modulus weights are used, data whose relative variance is signi®cantly larger than in the rest of the spectrum, are eliminated as out-liers, leading to reduction of the ®t residual but also to possible loss of information.

When the relative variance of errors is signi®cantly di€erent in various frequency ranges, as in the case of measurement of large impedances, the improvement of the least squares analysis associated with the variance model weights is worth the e€ort involved in the inves-tigation of the error variance. The parameters of the equivalent circuit, estimated by the CNLS ®tting with the variance model weights, have narrower con®dence limits and are much less prone to be biased by measurements contaminated by large random errors than the estimates of parameters obtained using the modulus weights. However, in common experimental situations, when the relative variance of errors is ap-proximately constant over the entire frequency range, ®tting using the modulus weights is both convenient and satisfactory.

Acknowledgements

This work has been supported in part by Grant of the Rector of Politechnika Warszawska. Discussion with Mark Orazem (University of Florida), that stimu-lated the investigation of error variance, is gratefully acknowledged.

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