Piecewise Direct Standardization Method Applied to theSimultaneous Determination of Pb(II), Sn(IV) and Cd(II) byDifferential Pulse Polarography

Ana Herrero*andM. Cruz Ortiz

Dpto. Quımica, Fac. de Ciencias, Universidad de Burgos, Pza. Misael Ban˜uelos s/n, E-09001 Burgos, Spain

Received: March 2, 1998Final version: April 20, 1998

AbstractA partial least squares (PLS) regression has been used to carry out the simultaneous determination of PbII, SnIV and CdII by differential pulsepolarography (DPP) because overlapping peaks exist. But this multivariate regression requires the measurement of a large number of calibrationsamples, so a standardization procedure, the piecewise direct standardization (PDS) method, has been used in order to reduce the experimentaleffort that will be necessary in later analysis. The number of calibration samples has been reduced by about 60 %, the results being comparable tothose obtained with a full recalibration.

Keywords: Piecewise direct standardization, Partial least squares regression, Multianalyte determination, Overlapping signals, Differential pulsepolarography

1. Introduction

Several standardization procedures [1–4] have been developed inthe last few years for solving the problem due to the fact that asample measured in two different situations (different instruments,different days, etc.) usually gives a different analytical response,which means that a regression model calculated in a specificsituation could not be used in a new situation. The aim of thesemethods is, in general, to reduce the number of calibration samplesthat will be needed to carry out the analytical determination in thenew situation. This is of great interest when several analytes aresimultaneously determined since the calibration sets are normallylarge if a wide concentration range is analysed.

Among the standardization procedures, the piecewise directstandardization (PDS) method [1, 5] is based on establishing arelationship between a subset of samples measured in bothsituations through a transformation matrix,F. Next, this matrixcan be used to correct the signals measured in the new situation insuch a way that the calibration model calculated with the samplesmeasured in the first situation (i.e., calculated with the wholetraining set) can be used for determining the concentrations of theanalytes from the corrected signals. This implies that it is possibleto carry out determinations in the new situation from a smaller set ofcalibration samples.

The PDS method has been widely used in the field of NIRspectroscopy [6] and is beginning to be used in electrochemistry. Infact, this method has been successfully applied in electrochemicalproblems where interferences such as intermetallic compoundformation or matrix effect exist [7, 8]. In this article, the PDSmethod has been applied to an electrochemical case where thereduction potentials of several metals are so close together thatoverlapping signals occur. Such is the case of the simultaneousdetermination of PbII, SnIV and CdII by differential pulsepolarography (DPP).

The overlapping of the polarographic signals of SnIV and PbII andthe tendency of SnIV to hydrolyze and polymerize are seriousobstacles in the simultaneous determination of these metals by DPP.The use of acidic media in the analysis avoids the hydrolysis of theSnIV, and with hydrochloric acid usually being used as supportingelectrolyte SnIV becomes stabilized by formation of a chlorocom-plex. However, these experimental conditions lead to the tin peak

completely overlapping the lead peak and very near to the cadmiumpeak.

Several instrumental and experimental approaches have beenproposed to overcome this problem of overlapping, including theuse of separation [9, 10] and matrix exchange [11] techniques, theaddition of complexing agents [12], suitable electrolytes [13, 14] orsurfactants [15], and the use of adsorptive stripping voltammetry[16, 17]. Moreover, other methods such as those based on the use ofthe Kalman filter [18,19] or several multivariate methods [20, 21]have been widely applied to solve overlapping signals, the partialleast squares (PLS) regression, a multivariate regression, beingamong the techniques which give the more successful results.

In this article, a PLS regression model has been built for eachmetal, PbII, SnIV and CdII, respectively, in order to simultaneouslydetermine these three metals. This determination has been carriedout in different situations, concretely on different days, in such away that, through the PDS method, a standardization procedureover time has been made in order to reduce the experimental effortthat will be necessary in the second and later determinations.

2. Experimental

2.1. Apparatus

The polarographic measurements were carried out using aMetrohm 646 VA processor with a 647 VA stand in conjunctionwith a Metrohm multimode electrode (MME) used in the staticmercury drop electrode (SMDE) mode. The three-electrode systemwas completed by means of a platinum auxiliary electrode andan Ag/AgCl/KCl (3 mol dm¹3) reference electrode. The analysisof data was done with PARVUS [22], MATLAB [23] andSTATGRAPHICS [24].

2.2. Reagents

Analytical-reagent grade (Merck) chemicals were used withoutfurther purification. All the solutions were prepared in acidicmedium with deionized water obtained in a Barnstead NANO PureII system. Successive additions of 100mL of PbII 5.00× 10¹4 M,

717

Electroanalysis1998, 10, No. 10 q WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1998 1040-0397/98/1008-0717 $ 17.50þ.50/0

SnII 8.55× 10¹4 M and CdII 7.84×10¹4M standard solutions weremade to a volume of 25 mL in order to obtain the mixture samplesindicated in the design showed in Figure 1; i.e., five different levelsof concentration for each metal. The voltammetric measurementswere carried out in an oxalic acid (0.1 mol dm¹3) and hydrochloricacid (0.1 mol dm¹3) supporting electrolyte medium (pH 1.08).Nitrogen (99.997 %) was used to remove dissolved oxygen.

2.3. Procedure

The solution was placed in the polarographic cell and purgedwith nitrogen for 10 min. Once the solution had been deoxygenated,polarograms were recorded from¹0.266 V to¹0.656 V. All theresults presented were obtained using differential-pulse polarogra-phy (DPP) with a pulse amplitude of¹50 mV, drop time 0.6 s, droparea 0.40 mm2 and scan rate¹10 mV s–1. After each addition, thesolution was stirred and deoxygenated for 15 s before applying thepolarographic procedure again.

3. Results and Discussion

The experimental design used to carry out the analysis, which isvery near to a central composite design [25], has five levels for eachanalyte (each level indicates an addition of 100mL of metalstandard solution). This experimental strategy has been used inorder to obtain the chemical information necessary to developuseful multivariate models since if the experimental design used isnot sufficiently descriptive of the chemical phenomena, theregression models built would not have any prediction ability.The polarograms recorded in this way constitute calibration A.Figure 2 shows those signals corresponding to the five additions(the concentration range analysed) of SnIV made when the level forthe rest of the metals remains constant. The figure shows that aunivariate analysis would be possible for PbII and CdII, but theoverlapping that exists between PbII and SnIV signals does not allowthe determination of SnIV with a univariate regression. So it isnecessary to use a multivariate technique that simultaneously

considers not only the peak current but the current recorded at allthe potentials useful for determining the concentration of the SnIV

in a sample and, simultaneously, the concentration of PbII and CdII.The PLS regression allows one to use the whole polarogram forextracting the information related to each metal and for successfullydetermining their respective concentrations.

A PLS model has been built independently for each metal withthe original data (once the blank signal was subtracted) by using thefull crossvalidation method [26]. The raw data contain the objectsshowed in Figure 1 (66 potentials have been digitalized in eachsample), the test set being formed by samples 15, 17, 20 and 22,whereas the rest of the samples constituted the training set. The PLSmodels will be developed by using only the training set. In this way,the test set is useful for determining the real prediction ability of themodels built.

The quality of the PLS models can be evaluated from theexplained and crossvalidated variance values. The explainedvariance of the PLS models built (the crossvalidated varianceappears between brackets) was as follows: for PbII 99.65 %(99.21 %) with 4 latent variables, for SnIV 99.43 % (98.53 %) with5 latent variables, and for CdII 99.97 % (99.95 %) with 6 latentvariables. The number of latent variables which constitutes eachmodel has been chosen in such a way that the crossvalidatedvariance is maximum. All the explained variance values are higherthan 99 %, which implies the adequacy of the fit; moreover, thecrossvalidated variance values are very near to those correspondingto the explained variance, pointing out the stability and goodprediction ability of the models built. In spite of the nonselectivityof the SnIV peak, the variance values obtained for its PLS model areof the same order as those corresponding to PbII and CdII, whosesignals appeared very well defined. This shows the ability of thePLS regression to extract the useful information, excluding thosevariability sources not related to the concentration of the analyte ofinterest.

Moreover, it is important to find a chemical sense of the latentvariables that take part in each regression model. This can be madefrom the corresponding scores that each object has in the new spaceformed by these latent variables. Figure 3 shows the scorescorresponding to the first three latent variables of the PLS modelbuilt for determining SnIV, which is similar to those correspondingto PbII and CdII. It is clear that the first and second latent variablesare mainly related to the concentration of CdII and PbII, which is

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Electroanalysis1998, 10, No. 10

Fig. 1. Experimental design followed, where the concentration ranges foreach analyte were: PbII : 1.95–9.58mM, SnIV: 3.32–16.37mM, and CdII:3.05–15.01mM. The replicates of the central point of the design correspondto samples 11, 16, and 21, respectively.

Fig. 2. Polarograms recorded for samples 14 to 18 in Figure 1, correspond-ing to the five levels of concentration of SnIV (3.3–16.4mM). Thepolarographic peaks correspond to, from left to right, PbII (5.8mM), SnIV

and CdII (9.1mM), respectively.

reasonable taking into account that the data have not beenautoscaled. Therefore, the directions of maximum variability inthe predictors block (currents) depend on the size of the signal (sizefactor). In the plane formed by the scores of the first two latentvariables, see Figure 3a, it is possible to draw both lines fromsamples 19 to 23 and 9 to 13 which cover the five levels ofconcentration of CdII and PbII, respectively.

In the plane formed by the first and third latent variables,Figure 3b, a new line can be drawn from samples 14 to 18, but inthis case the position of the rest of the samples with reference to thisaxis is less clear than in Figure 3a, which shows the high influenceof the other two peaks in the model. This PLS model must, forinstance, subtract the current related to the reduction of PbII thatappears at the same potentials of that corresponding to SnIV. Therest of the latent variables, which explain 0.36 % of the variance, arerelated to other minor variability sources that contribute with verylow percentages to the explained variance.

Finally, to evaluate the real prediction ability of the PLS models,the relative errors corresponding to the test set samples have been

calculated, see Table 1. Most of these errors are smaller or verynear to 1 %, which leads one to conclude that the PLS models builtallow one to simultaneously and accurately determine theconcentrations of the three metals, with a high prediction ability.

However, in spite of the good features shown by the PLS modelsbuilt, the fact that 17 calibration samples are needed each time thatthese determinations are carried out is a big disadvantage. With theaim of reducing the number of samples needed in later determina-tions by means of multivariate regression, several standardizationapproaches have been developed [1–4]. The PDS method has givensuccessful results when applied to spectroscopic and electro-chemical data [5, 7, 8].

The PDS [1] method is based on relating a subset ofmeasurements (standardization subset) carried out on two differentsituations, A and B, by means of a transformation matrix,F,

XAs ¼ XBsF

XAs and XBs are the predictor variables corresponding to thestandardization subsets of situations A and B, respectively, i.e.,those samples measured in both situations. To calculateF, a movingwindow,Zi , is used in such a way that each variable from situationA, xi , is related to the variables in this window from situation B.

Z i ¼ xB;i¹j ; xB;i¹jþ1;…; xB;iþk¹1; xB;iþk

� �Next, a local multivariate regression (PCR or PLS) is established inorder to obtain the regression vectorsbi

xA;i ¼ Z ibi

which are arranged along the mean diagonal of the matrixF, the restof the elements being zero. Then, this matrix of transformation isused to correct the polarograms recorded in situation B to the formatthey would have had if they had been measured in situation A,xA;est,by

xTA;est ¼ xT

BF

In this way, the PLS models already built for situation A using thewhole training set can be used to calculate the concentrationscorresponding to these corrected signals, even when in situation Bonly a subset of calibration samples has necessarily been measured.

With the aim of selecting a suitable standardization subset forcarrying out the standardization procedure, all the calibrationsamples have been measured again after some days, constitutingcalibration B. Several subsets of the training set have been used asstandardization subsets, see Table 2, and in the same way severalwindow sizes (3, 5, 7, 9, 11 and 13), to study the influence of thesetwo parameters in the standardization procedure. In this way, a totalof 42 different standardization procedures have been carried out.This implies that the concentrations of the samples measured oncalibrate B have been calculated by using the PLS models built forcalibrate A, with prior correction of the polarograms recorded insituation B by using different transformation matrices obtained forthe different standardization subsets and window sizes.

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Electroanalysis1998, 10, No. 10

Fig. 3. Scores corresponding to the first-second (a) and first-third (b) latentvariables (L. V.) of the PLS model built for SnIV.

Table 1. True concentrations, relative errors [%] obtained with the PLS models corresponding to the test set samples for calibrates A and B, and relative errors[%] related to the standardization procedure for standardization subset 4 and window size 13.

Sample True cocentrations PLS models for Standardizationmethod

[mM] Calibrate A Calibrate B for Calibrate B

PbII SnIV CdII PbII SnIV CdII PbII SnIV CdII PbII SnIV CdII

15 5.81 6.62 9.12 0.48 1.39 0.83 2.04 2.98 0.01 1.50 1.52 0.4017 5.77 13.15 9.05 ¹1.30 ¹0.35 0.25 ¹3.90 ¹1.04 0.47 ¹0.61 ¹2.12 0.6820 5.81 9.94 6.08 ¹0.56 ¹3.41 ¹0.30 ¹1.12 ¹1.30 ¹3.48 ¹1.05 ¹4.81 ¹2.0622 5.77 9.86 12.06 0.27 1.48 ¹0.17 1.93 ¹3.63 ¹2.15 ¹0.32 ¹2.76 ¹1.45

To calculate the transformation matrices, the m-functionpdsgen[27], implemented in Matlab [23], has been used. This functionrequires as inputs the signals corresponding to both calibration sets,the standardization subset and the window size, giving as output thetransformation matrix,F. This is a squared matrix, dimensioned66×66, that is then used to correct the polarograms of the test setsamples of calibrate B in order to then apply to the corrected signalsthe closed form (regression coefficients) [28] of the PLS modelsbuilt with the data on calibrate A for determining the concentrationof these samples or others from situation B.

Figure 4 shows the SEP values, calculated from the test setsamples, obtained for the different standardization subsets andwindow sizes used in the analysis. The influence of both parameterson the errors is clear, the window size is important for some of thestandardization subsets, whereas the different standardizationsubsets have a significant influence on the SEP, greater than thatof the window size. To analyze the SEP values the multiple rangetests of Tukey and Newman-Keuls [24] are used, the SEP values

obtained for the different window sizes being considered asreplicates when the standardization subsets are analyzed and viceversa. Both tests conclude, at a significance levela¼ 0.05, that thewindow sizes used do not lead to statistically significant differentSEP values, except the sizes 3 and 5 which are related to the highesterrors. With reference to the standardization subsets, both testsreveal the differences that exist between the standardizationsubsets, shown clearly in Figure 4, and point out the standardiza-tion subset 4 as that related to the best results when the three metalsare considered.

But, since the principal aim of the standardization procedure is toreduce the number of standardization set samples to minimize theexperimental effort that will be necessary in the later calibrations,this fact should be taken into account for analyzing the results. Inthis way, the same standardization set (a single set) would bedesirable for the three metals, and also the same window size inorder to calculate a single transformation matrix that corrects eachpolarogram just one time. In this way, both the experimental effortand the computing time are notably reduced. So, taking into accountthe advantage of using a single window size, the size 13 has beenchosen as the bets for subset 4. This standardization subset has only7 samples and, when the window size used was 13, the errors relatedto the analysis are those shown in Table 1.

These errors are of the same order as those obtained for the testset samples of calibrate A (Table 1), but in this case, only 7standardization samples have been necessary, as opposed to the 17that form a full recalibration, to reach similar results. In this way,the experimental effort necessary for simultaneously determiningPbII, SnIV and CdII has been reduced to 59 %, in spite of the nonspecificity of the signal of the SnIV, without a significant reductionin the quality of the predictions.

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Table 2. Standardization samples subsets from the training set used forcalculating the transformation matrix.

Subset (No) Samples of training set

1 all2 1 to 83 9 11 13 14 16 18 19 21 234 9 14 16 18 19 21 235 9 14 18 19 236 1 2 3 4 7 87 9 13 14 18 19 23

Fig. 4. Surfaces for SEP vs. window size (w) and standardization subset (no) for PbII (a), SnIV (b) and CdII (c).

4. Conclusions

The PLS regression has successfully been applied (mean globalerrors about 0.80 %) to carry out the resolution of overlappingsignals in the determination of PbII, SnIV and CdII by DPP in theconcentration range analyzed. This regression technique is able toextract the useful information overlapped by other sources ofvariability not related to the analyte of interest. The use of thepiecewise standardization method in order to minimize the numberof calibration samples in the analysis has lead to a reduction in theexperimental effort near 60 % (from 17 to 7).

5. Acknowledgement

This work has been partially supported by Consejerı´a deEducacion y Cultura de la Junta de Castilla y Leo´n under ProjectBU16/98.

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