Variance of errors and elimination of ou

Electrochimica Acta 44 (1999) 4163±4174

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
Institut fuÈr Technische Elektrochemie, TU Wien, Getreidemarkt 9/158, A-1060, Wien, Austria

b

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 systematic 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-

* Corresponding author. Fax: 00-48-22-6282171.
E-mail address: jrdygas@mp.pw.edu.pl (J.R. Dygas)

mates of the model parameters when errors are randomly distributed with zero mean. Each squared
deviation should be weighted by a factor inversely proportional 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 repeating measurements had been rare [9,10] until Orazem
and co-workers developed a method to ascertain the
variance of random errors when the successive impedance 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 variance of residual deviations was taken as the variance

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

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J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

of random errors. They observed that the variances of
the real and imaginary components had the same magnitude 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 circuit 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 evaluated 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 comparison 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 proportional 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 spectrum.

2. Experimental
The impedance spectra were obtained by an automated setup for the measurement of large impedances
that combines the Solartron SI-1260 Impedance/GainPhase 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 external I/V converter with
the maximum gain of 1011 V/A [5]. In order to avoid
large random and systematic errors, which are inevitable when the sensitivity of the current input is too low,
the I/V converters 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:
ÿ


ZM ˆ ZX 1 ‡ Z1 jo ÿ Z2 o 2 ÿ Z3 jo 3
…1†

Fig. 1. (a) Phase angle and absolute value of impedance of
the te¯on platelet as a function of frequency. Symbols represent data measured with di€erent gains of the I/V converters: 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, .Ð106 V/A, qÐ107 V/A, xÐ108 V/A.
Continuous lines represent the ®tted impedance of the capacitor, C = 1.059 pF, with distortions caused by the I/V converters modeled according to Eq. (1). (b) Evaluation of variance
for the 15 replicate spectra of the te¯on platelet. The variances of the measured values of real and imaginary components, divided by the squared absolute value of the
admittance (symbols), and the ®tted variance model (continuous lines).

where j=(ÿ1)1/2. Values of the coecients Z1, Z2, Z3,
characteristic for a given gain of the I/V converter,
were estimated by the CNLS ®tting as additional parameters of the model. The details of the correction
procedure have been described in [5].

The ac signal was 30 mV rms. Auto integration of

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

4165

Fig. 2. Complex plane plot of impedance of the polycrystalline sample of BICUVOX at 311 K. Symbols represent data measured
with di€erent gains of the I/V converters as in Fig. 3a. Plotted data were corrected according to Eq. (1). Continuous line represents
the ®tted impedance of the equivalent circuit shown in the insert. Cross hatched capacitors denote constant phase elements.

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 demonstrated on two series of experimental impedance spectra:

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
(9.8  4.7  0.7 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 polycrystalline sample of the oxygen ion conductor
BICUVOX with sputtered platinum electrodes. The
sample was in the high conductivity state [14]. No systematic 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,
DY'i, and imaginary, DY0i, parts of the complex deviation, DYi=YiÿFY ( fi, X), of the admittance, Yi,

measured at a frequency, fi, from the response, FY ,calculated for the model described by r parameters
X={x1,x2 . . .xr }:
Qˆ
m 
X

w11, i …DY 0 i †2 ‡ w22, i …DY0i †2 ‡ 2w12, i …DY 0 i †

iˆ0

 …DY0i †



…2†

Summation is over m frequencies of the spectrum. For
each frequency fi, the weights: w11,i, w22,i, w12,i=w21,i
form a 2  2 symmetric matrix, which should be the
inverse of the covariance matrix of errors of the real

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J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

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

used:
w11, i ˆ w22, i ˆ D2 =jYi j2 ,

w12, i ˆ 0

…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:

1=2
Rmsr ˆ Qmin =…2m ÿ r†

…4†

If the weights reproduce the proportions between
the variances of errors: s 2(Y'i)=d 2/w11,i and
s 2(Y0i)=d 2/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:


Qmin
s…Y i † ˆ
…2m ÿ r†w11, i

0

1=2

,
…5†



Qmin
s…Y0i † ˆ
…2m ÿ r†w22, i

1=2

The above estimates can be used for testing the consistency of the measured admittance with the ®tted response 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
than k-times the estimated standard deviation:
jDY 0 i j > k
s…Y 0 i †

or

jDY0i j > k
s…Y0i †

…6†

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

Fig. 3. Evaluation of variance for the 33 semi-replicate spectra
of the polycrystalline BICUVOX. (a) Phase angle and absolute value of the impedance as a function of frequencyÐone
of the spectra. Symbols represent data measured with di€erent
gains of the I/V converters: QÐ50 V/A, wÐ5000 V/A,
+Ð105 V/A, qÐ107 V/A, xÐ108 V/A. Continuous lines represent the ®tted impedance of the equivalent circuit of Fig. 2,
with distortions caused by the I/V converters modeled according to Eq. (1). (b) Variances of the measured values of the
real and imaginary components of admittance at each frequency, divided by the squared absolute value of the average
admittance. (c) Variances of the real and imaginary components of the residual deviation of the measured admittance
from the ®t of equivalent circuit of Fig. 2 to individual spectra, divided by the squared absolute value of the admittance
(symbols) and the ®tted variance model (continuous lines).

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

gram. After classifying measurements at certain frequencies 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 additional 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 outlying measurements is based on the examination of residual 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 with k35, proves to
be useful for the elimination of coarse outliers which
signi®cantly spoil the ®t. With k33, it may be applied
to test the consistency of the weights employed in the
CNLS ®tting with the distribution of residual deviations of the ®t.

4. Investigation of variance
For a stable system, when the admittance at the
same frequency, fi, is measured n times, 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 platelet, variances of the real and imaginary component,
divided by the square of the absolute value of admittance are plotted in a logarithmic scale in Fig. 1b. The
relative variances were dependent on the gain of the
I/V converter 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 absolute value of the admittance and would be represented 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 calculate 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-

4167

mated for each spectrum. The stochastic errors randomly contribute to the measurements at di€erent
frequencies and remain after subtraction of the model
response from the data. The above procedure, originally 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 variance. Similar values of the variance were obtained
when the evaluation was repeated using weights inversely 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 capacitor C0, and the geometric capacitance Cg reproduced the high frequency dispersion in
the measured spectra. The two parallel cpe Pgp and Pgr
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 reproduce the measured spectrum. The cpe Pe with exponent
Ne30.55 was used to reproduce the impedance of platinum electrodes. The impedance of electrodes was visible 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 variation 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 variance were similar for the real and imaginary components. 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, respectively. 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.
The slope of the logarithmic plots of the relative var-

4168

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

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 measurements 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

Coecient b0
s 2(Y')
s 2(Y0)

5000 V/A
b0

bZ

105 V/A

106 V/A

107 V/A

108 V/A

109 V/A

1010 V/A

b0

b0

b0

b0

bF/Hz

b0

bF/Hz

1.9  10ÿ5 0.07  10ÿ7 1.3  10ÿ10 1.2  10ÿ7 1.9  10ÿ7 0.5  10ÿ7 0.8  10ÿ7 2.5  10ÿ5
0
1.3  10ÿ5
(12%)
(58%)
(8%)
(20%)
(53%)
(21%)
(23%)
(11%)
(8%)
1.2  10ÿ5 2.10  10ÿ7 1.1  10ÿ10 2.9  10ÿ7 2.6  10ÿ7 1.8  10ÿ7 2.0  10ÿ7 4.5  10ÿ5 9.0  10ÿ6 1.4  10ÿ5
(45%)
(10%)
(9%)
(8%)
(38%)
(15%)
(16%)
(13%)
(34%)
(10%)

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 frequency, 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 relation 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 proportional to the frequency, see Fig. 1a. The same is
true for the BICUVOX sample at frequencies above 1
kHz. The 1/f 2 dependence, observed with the 5000
V/A gain of the Solartron SI-1260 internal I/V converter, may be expressed by the ratio of the high
measured impedance to the low feedback resistance of
the converter (RF=5 kO): s 2(Y s)/vYv2 0 (vZv/RF)2. The
inferred model of the relative variance of errors is:

larly as Spinolo et al. [16], that the linearity error is
proportional to the absolute value of the measured signal, while the resolution error adds a constant component to the variance, then the variance of current
may be approximated as:
s2 …I s †  cp I 2 ‡ …dI0 †2 ‡ cS I ‡ cL I=f

…8†

The absolute value of current I can be expressed by
the voltage applied to the cell: I=VS/vZv, while the current resolution error dI0 by the resolution of the voltage analyzer dV0 and the feedback resistance RF:
dI0=dV0/RF. The third and the fourth terms are contributions from the shot noise and the low frequency
1/f noise respectively [17]. If one neglects errors of
measurement of the ac voltage, then
s2 …Y s †=jYj2  s2 …I s †=I 2
 cp ‡ …dV0 †2 jZj2 =…VS RF †2 ‡ cS jZj=VS

s2 …Y s †
jZj2
1
n…Y s † ˆ
ˆ b0 ‡ bZ 2 ‡ bF
2
f
RF
jYj

…7†

where Y s denotes the real or imaginary component: Y '
or Y0.
This model of the variance is justi®ed when one considers errors of the frequency response analyzer and
noise in the measuring system. If one assumes, simi-

‡ cL jZj=…VS f †

…9†

The 1/f term in Eq. (7) may originate either from the
third term of Eq. (9), when vZv 0 1/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 2
Coecients estimated for the model of variance, Eq. (7), by ®tting to variances of the residual deviations for the series of 33 semireplicate measurements of the BICUVOX sample. Con®dence limits of the estimates expressed in percent of the value are given in
parenthesis
105 V/A

I/V

50 V/A

5000 V/A

Coecient

b0

b0

bZ

b0

s 2(DY ')

2.1  10ÿ5
(22%)
0.8  10ÿ5
(23%)

1.9  10ÿ7
(22%)
3.7  10ÿ7
(12%)

2.0  10ÿ10
(14%)
1.3  10ÿ10
(10%)

4.5  10ÿ7
(32%)
3.1  10ÿ7
(40%)

s 2(DY0)

107 V/A

108 V/A

bZ

b0

b0

bF/Hz

0.8  10ÿ11
(36%)
2.3  10ÿ11
(20%)

0.5  10ÿ6
(12%)
1.3  10ÿ6
(10%)

8.1  10ÿ8
(7%)
7.9  10ÿ8
(7%)

3.2  10ÿ9
(20%)
5.4  10ÿ9
(19%)

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

4169

Fig. 4. Residual deviations of the admittance of the te¯on platelet (data of Fig. 1a) from ®t of capacitance with the instrumental
corrections: (a) and (c) employing modulus weights, (b) and (d) using the variance model weights. The dashed lines represent the
limit of 3 standard deviations from the ®t, Eq. (5). In (a) and (b) the ®t was to all measured data, in (c) and (d) outlying measurements, whose residual deviations were larger than 3-times the standard deviation, were eliminated.

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 the I/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 coe-

cients were obtained with weights based on the deviation from the moving average, similar to the
weighting scheme used for ®tting a model of the standard deviations by Agarwal et al. [12]. In this scheme,
the weight factors are inversely proportional to the
sum of the squared deviation from the moving average
and the squared value of the relative variance of real
or imaginary components multiplied by a constant:

4170

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

Fig. 5. Residual deviations of the admittance of the BICUVOX sample (data of Fig. 3a) from ®t of the equivalent circuit of Fig. 2
with the instrumental corrections: (a) and (c) employing modulus weights, (b) and (d) using the variance model weights. The dashed
lines represent the limit of 3-times the standard deviation from the ®t, Eq. (5). In (a) and (b) the ®t was to all measured data, in (c)
and (d) outlying measurements, with residual deviations larger than 3-times the standard deviation, were eliminated.

2
2


0
0
0
wÿ1
11, i ˆ hn…Y i †i ÿ n…Y i † ‡a
n…Y i † ,

…10†


2

2
wÿ1
22, i ˆ hn…Y0i †i ÿ n…Y0i † ‡a
n…Y0i †

where hn(Y'i)i denotes the average over neighbor frequencies. 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.

These coecients of the variance model, which are
estimated with narrow con®dence limits, have similar
values for the real and imaginary components. The observation of Agarwal et al. [12] that the standard deviations for the real and imaginary components of
impedance are equal is con®rmed by the present result.
It has been shown by Durbha et al. [18], that equal
variance of the real and imaginary components of the
complex immittance is a consequence of the Kramers±

Modulus weights, all data

Cg
Ab
Nb
C0
Rb
Agp
Ngp
Agr
Ngr
Ae
Ne
Rmsr

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

0.713 pF
105.0 pS
0.557
2.34 pF
291.2 MO
806 pS
0.156
5.31 nS
0.956
450 nS
0.537
0.806

0.24%
7.8%
0.9%
2.8%
0.9%
1.7%
3.8%
0.6%
0.4%
24%
3.6%
5.8%

0.38%
4.7%
1.2%
5.2%
0.7%
7.2%
17%
2.0%
1.0%
54%
8.1%

0.716 pF
103.6 pS
0.556
2.35 pF
291.3 MO
795 pS
0.150
5.34 nS
0.954
368 nS
0.610
0.115

0.26%
1.8%
0.3%
1.2%
0.8%
1.1%
2.1%
0.4%
0.5%
23%
2.4%
10%

0.12%
1.7%
0.36%
0.9%
0.1%
1.0%
2.7%
0.3%
0.17%
10%
1.0%

0.716 pF
105.0 pS
0.557
2.33 pF
290.9 MO
810 pS
0.156
5.34 nS
0.957
295 nS
0.593
2.64

0.11%
1.4%
0.27%
0.8%
0.8%
0.7%
1.2%
0.37%
0.46%
21%
1.8%
2.2%

0.20%
2.0%
0.40%
0.9%
0.1%
1.0%
2.1%
0.21%
0.20%
12%
1.2%

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

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

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Kronig transform. Large di€erences between the variances 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 106 Hz 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 correlation 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 imaginary 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 value Rmsr=1.4 obtained
using modulus weights of Eq. (3) with D = 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[s 2(Y s)/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, compare Fig. 4. The systematic errors at certain frequencies 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
to Rmsr=3.9 and 18 outlying measurements were eliminated: 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 comparing 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 frequencies 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 corrections 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 deviation, 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 variance model weights on the estimated values of parameter 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 average 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 outliers 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
not well resolved in the measured spectra. Good agreement of the parameters values estimated with di€erent

J.R. Dygas, M.W. Breiter / Electrochimica Acta 44 (1999) 4163±4174

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 deviations is considered to be statistically signi®cant
when the opposite hypothesis (that the variances are
equal) can be rejected based on the Fisher's F-test. For
two samples, each having 32 degrees of freedom, the
hypothesis of equal variances can be rejected at a signi®cance level, a=0.02, when the ratio of sample variances 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 corresponding 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 temperature ¯uctuations. With modulus weights, signi®cant reduction of the standard deviations of the parameter
estimates is obtained after elimination of outliers. This
demonstrates applicability of the elimination of outliers, 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 system. 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 absolute value of the measured admittance. The coecients
of Eq. (7) for the variances of real and imaginary components have similar values. The observation of
Orazem and co-workers [12,18], that the standard deviations of the real and imaginary components of the
complex immittance are equal, is con®rmed by the present 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-

4173

ing measurements with large systematic errors can be
selectively eliminated, while those containing large random 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 outliers, 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 investigation 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 approximately 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 stimulated the investigation of error variance, is gratefully
acknowledged.

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