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Multivariate calibration transfer applied to the routine polarographic determination of copper, lead, cadmium and zinc
ANAImcA ELSEVIER
CHIMICA ACTA Analytica
Chimica Acta 348 (1997) 5 1-59
Multivariate calibration transfer applied to the routine polarographic determination of copper, lead, cadmium and zinc A. Herrero*, M.C. Ortiz Dpto. de Quimica Analitica, Facultad de C.EZA. y C. Quimicas, Vniversidad de Burgos, Pza. Misael Bafiuelos s/n, 09001 Burgos, Spain Accepted
10 December
1996
Abstract The application of calibration transfer methods has been successful in combination with near-infrared spectroscopy or fluorescence spectroscopy for prediction of chemical composition. One of the methods developed that provides accurate performance is the piecewise direct standardization method (PDS), which in this paper is applied to transfer from one day to another the partial least squares (PLS) models built in the polarographic determination of copper, lead, cadmium and zinc. This is an electrochemical example in which interferences have been found, making necessary the use of soft calibration models because of their ability to model this phenomenon, which implies a large number of standard samples for the calibration. Once the PLS models are built, the calibration transfer is carried out to overcome the instrumental change over time in this routine analysis, allowing one to reduce from 28 to 8 the number of calibration standards necessary for later determinations. Standard errors of prediction (SEP) are found similar to those given by the complete recalibration. Keywords: Multivariate calibration; Partial least squares; Calibration transfer; Polarography;
1. Introduction Electroanalytical techniques applied to the determination of elements at trace levels provide an important alternative to the already traditional spectroscopic methods. The use of a multivariate methodology, through suitable regression techniques such as principal component regression (PCR) or partial least squares regression (PLS), allows one to extract in an adequate way the information contained in electrochemical data, greatly increasing the possibilities of the electroanalytical techniques.
*Correspondingauthor. OOO3-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII
SOOO3-2670(97)00154-2
Interferences;
Multicomponent
analysis
However, the applicability of these techniques has been limited by a problem that arises when a calibration model developed from a set of data in a particular situation cannot be applied to another, i.e. by a problem identified as ‘calibration transfer’. In this new situation some different responses are obtained, perhaps as the result of using other instruments, using the same instrument at a later time (drift, electronic fluctuations), changing the measurement conditions (temperature) or the physical constitution of the sample (particle size, surface texture), etc. Nevertheless, it is known that sometimes small variations in the analytical signals may be a sign of fundamental changes in the system under consideration. Therefore, small variations between the same
52
A. Herrero, M.C. OrtidAnalytica
measurements, carried out in different situations, could be sufficiently significant to prevent a calibration model being transferred, i.e. it could be applied to another situation providing different responses. One evident solution to the problems associated with a calibration transfer consists of performing a full recalibration in the new situation, which involves the repetition of all the calibration measurements with the aim of developing a similar calibration model. This procedure, traditionally used, requires considerable effort and cost, both in terms of experimental burden as in the time spent on the task, especially when the samples are numerous, chemically unstable, hazardous, etc. In any case, it would be desirable to calibrate one instrument in a new situation without feeling obliged to repeat the whole calibration procedure again. In spite of the importance of this question, there are not a lot of references in the bibliography on the matter, most of them in the field of NIR spectroscopy. Shenk and Westerhaus [l] proposed a univariate procedure to correct the experimental response, which algorithm has been modified and improved by Bouveresse et al. [2]. Wang et al. [3,4] compared five procedures, the Shenk procedure already cited and four multivariate standardization methods developed by themselves, in the determination of four analytes in gasoline samples by NIR spectroscopy, the piecewise direct standardization (PDS) method being that which gave the best standard errors of prediction (SEP) [5,6]. This method, based on PCR or PLS, has also been compared with a univariate slope correction of NIR spectra by Bouveresse et al. [7]. PDS method has been applied to the determination of two analytes in corn samples at different temperatures by NIR spectroscopy [8]. In the same way, a method has been developed to standardize second-order instruments together with an application to LC-UV data [9]. Forma et al. [lo], suggest a multivariate calibration transfer procedure that uses PLS regression twice: first to compute the relationship between the responses from two instruments and afterwards to compute the regression equation. Although most of the literature on multivariate calibration transfer comes from NIR spectroscopy, many of the strategies and methods described can also be applied to calibration transfer problems in other analytical fields [ 11,121. In this paper, the PDS
Chimica Acta 348 (1997) 51-59
method has been applied to electrochemical data provided by a single instrument on different days. To carry out the work, two calibrations were developed with the aim of determining simultaneously copper, lead, cadmium and zinc by differential pulse polarography. Frequently, when several metals are simultaneously on a mercury electrode intermetallic compounds are formed between them [13]. Numerous intermetallic compounds have been reported (Au-Zn, Cu-Cd, CuZn, Cu-Sn, Co-Zn, Ni-Zn,...), but most of the works on intermetallics have focused on the Cu-Zn compound because of the presence of these metals in several analytical samples. The formation of intermetallic compounds on the electrode may be reflected in the electrochemical response recorded, causing its severe depression or shift, which generates large errors in the determinations made by stripping analysis [14,15]. Several methods have been suggested for minimizing or eliminating errors caused by intermetallic compounds formation, such as the addition of a third element [16], the use of hyphenated methods [17], or usually, if the concentration of the metals in the sample is above 1 ppm, the analysis can be carried out by differential pulse polarography [ 181. But even in this last case, interferences are found in the analysis, suggesting the suitability of using multivariate calibration techniques capable of modelling phenomena of interferences, which allows one to analyse the problem in a different way [19]. In this paper, the simultaneous polarographic determination of copper, lead, cadmium and zinc in aqueous solution has been carried out, using a technique of multivariate regression (PLS) that has been successfully applied to the resolution of electrochemical problems in which interferences exist [20,21]. The determination of these four metals is usually made in water and food routine analysis [22,23] and it is common to find in the bibliography references to the interferences found in their electrochemical analysis [24,25].
2. Materials and equipment Analytical-reagent grade chemicals were used without further purification. All the solutions were pre-
53
A. Herrero, M.C. Ortiz/Analytica Chimica Acta 348 (1997) 51-59
pared with deionised water obtained with a Barnstead NAN0 Pure II system. Nitrogen (99.99%) was used to remove dissolved oxygen. The buffer solution (pH = 4.335) was acetic acid 2 M and ammonium hydroxide 1 M. Polarographic measurements were carried out using a Metrohm 646 VA processor with a 647 VA stand in conjunction with a Metrohm multimode electrode (MME) used in the static mercury drop electrode (SMDE) mode. The three-electrode system was completed by means of a platinum auxiliary electrode and an Ag-AgCl-KCl(3 M) reference electrode. Analysis of the data was done with PARVUS [26] STATGRAPHICS [27] and MATLAB [28].
This method uses a moving window so each predictor variable from situation A is related to the variables in this window from situation B. The window Zi used to calculate the correction of the variable xi is given according to
3. Experimental
F = diag (br, bl, . . . . b:, . . . . b;)
procedure
Polarographic measurements were carried out by the following procedure. The solution was placed in a polarographic cell and purged with nitrogen for 10 min. Once the solution had been deoxygenated a polarogram was recorded from 0.095 V to - 1.117 V, using the differential-pulse mode with a pulse amplitude of -50 mV. The drop time was 0.6 s, the drop area was 0.40 mm2 and the scan rate was - 10 mV s-i. After each addition the solution was stirred and purged for 15 s.
4. Results
and discussion
4.1. Piecewise
direct standardization
(PDS) method
Piecewise direct standardization (PDS), a method developed by Wang et al. [3], is based on establishing a relationship between each experimental point from a situation, A, with a small window of predictor variables close to itself in a different situation, B, by means of a transfer matrix F, itA = XnF
(1)
where XA and iin are the transfer subset (small series of measurements, usually a subset of the training set) of situations A and B respectively, and F is a square matrix dimensioned p x p, p being the number of predictor variables measured.
& = [xB,i-j> XS,i-j+l>
“‘> XE,i+k-13
XB,i+k]
(2)
where a window from index i-j to i+k is used (not necessarily symmetric around i). Then it is possible to establish a local multivariate regression given by xA,i = Zibi
(3)
where each regression vector, bi, can be calculated by means of PCR or PLS regression. These regression vectors bi are arranged along the main diagonal of the transformation matrix F while the rest of the elements are zero, which results in a banded diagonal matrix, (4)
Using the transfer matrix F, the predictor variables of a new sample measured in situation B (row vector XL) can be transformed to the A format: i;
= x;F
(5)
Each predictor variable Xi from situation A is obtained by multiplying the predictor variables from situation B in a window Zi by a transformation matrix F, and then predictions can be made using the original model. The use of a moving window to establish the relationship between the experimental variables from two different situations, instead of using all the predictor variables, avoids coming up against an illconditioned problem when calculating the transfer matrix because the number of variables is much larger than the number of standard samples, which is very common in electrochemical data. 4.2. Calibration
standards
Electroanalytical techniques are frequently used to determine several analytes in the same sample, the number of experimental samples necessary to carry out a multicomponent calibration being variable. If the number of standard measurements required for a successful calibration is quite high, their influence on time and cost of the experience (preparation of samples, cleaning of polarographic cell and material,
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A. Herrero, M.C. Ortiz/Analytica
Chimica Acta 348 (1997) 51-59
Table 1 Experimental design for calibrates A and B. Each unit in the table represents an addition of 100 pl of standard solutions, 5.0 x 10e4 M copper, 5.0 x 10m4 M lead, 7.9 x 10e4 M zinc and 5.0 x 10m4 M cadmium respectively. First column lists the index of the samples and second column indicates that the additions were carried out on the same initial solution, using a different symbol for each initial solution used Run
1 2 6 3 9 10 7 4 12 13 15 14 16 11 8 5
Experimental
X
; X
A ;, 0 w
0 _ A X
design
Run
cu
Pb
Cd
Zn
2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4
2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4
2 2 4 4 2 2 4 4 2 2 4 4 2 2 4 4
2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4
deoxygenation and scan time, etc.) is negative. This fact indicates the advisability of a tool that allows the optimization of the number of experimental runs in order to obtain the most useful information from the analytical signal recorded with reasonable experimental effort. Therefore, an experimental design has been used, in which calibration samples are distributed approximately like a central composite design [29,30] but making equal volume additions of each analyte, as is shown in Table 1, so each factor (metal) takes five different levels of concentration. Following this design two complete calibrations have been carried out with the aim of analysing the variability due to the day to day effect between them, and afterwards developing a calibration transfer. It is necessary to model all pertinent variations expected in the samples to be analysed. So there are two sets of data, calibration A (carried out first) and calibration B (carried out several days later). Fig. 1 shows the polarograms of the test set samples from calibration B. The peaks appearing in the polarograms are well defined and characteristic in the
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Experimental
v v v v v 0 0 0 0 0 0 0 0 0 0 0 V V V V
100 loo
design
CU
Pb
Cd
Zn
1 2 3 4 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 1 2 3 4 5 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 1 2 3 4 5 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2 3 4 5
-140
380
-620
-860
-1100
Potential / mV Fig. 1. Polarograms concentration levels 4.83 pM and 7.61 cadmium, 4.83 pM
of the test set samples in calibration B, whose are between 4.83 pM and 7.61 pM for copper, pM for lead, 6.OOpM and 11.92 pM for and 7.61 pM for zinc.
A. Herrero, M.C. Ortiz/Analytica
polarographic analysis of these metals, but in spite of the clear similarities found between the polarograms from the two calibrations they are not exactly equal for the same samples. After additions of the different analytes it was evident that there exists severe interference between metals, as was expected, which will probably render difficult a univariate analysis. 4.3. Development
and evaluation
of PLS models
First, the prediction ability of univariate methods has been evaluated by means of a classical method of univariate standard addition procedure using the peak currents of calibration B, which will be used in subsequent analyses to evaluate the multivariate models. A statistically evaluated univariate model has been built for each metal with samples whose levels go from 2 to 5 in this metal (3 being the level in the rest of metals), see Table 1, and the concentration of the metal in the first of these samples is then calculated (samples l&23,28 and 33, for copper, lead, cadmium and zinc, respectively). The concentration of these samples will also be calculated in multivariate analysis to allow comparisons. In the analyses carried out the results yielded by lead and cadmium can be considered acceptable (~2% of relative error), but those of copper and zinc (~40%) are unacceptably high, showing the inviability of the univariate analysis. So it is necessary to apply another kind of mathematical technique able to consider the existent interferences. In this sense, multivariate data analysis allows one to find underlying or latent information in electrochemical data, where the partial least squares [31,32] regression is increasingly used [21,33]. This multivariate regression method has been applied to both calibrations, A and B, developing one different regression model for each metal, so that eight PLS models have been built. Every one of these models has been developed taking the concentration of one of the metals as response variable and the currents recorded at 203 equally spaced potentials from 0.095 V to - 1.117 V as predictor variables. The data sets have been divided in training set (28 objects) and test set (samples 1820, 23, 25, 28, 30,33 and 35) to build and evaluate the regression models. For calibration A 5,4,4 and 5 latent variables were required for copper, lead, cadmium and zinc models
Chimica Acta 348 (1997) 51-59
55
respectively, the explained variance for each model being 99.66,99.83,99.79 and 99.14%, while the cross-validated variance was 99.55,99.77, 99.73%, and 99.11% respectively. However, for calibration B 8, 8, 5 and 6 latent variables were necessary for copper, lead, cadmium and zinc models respectively, obtaining in this case explained variance values of 99.87, 99.90,99.55 and 99.42%, and cross-validated variance values for each model of 99.45,99.54,99.51 and 99.19% respectively. Although there are similarities between both data sets, their internal structures do not coincide so much, as the different latent structures of the models demonstrate. Every model developed gives satisfactory explained variance values, while the proximity of the corresponding cross-validated variance values to them is a sign of the stability of the models built and of their prediction ability. Moreover, the calculated concentration values for test set samples provide another way to evaluate the real prediction ability of these models. The standard deviation (between brackets) and the mean of the absolute values of the relative prediction errors obtained for each metal with calibrations A and B are respectively: 1.34% (1.66) and 2.11% (2.64) for copper, 3.09% (5.44) and 2.50% (4.00) for lead, 1.57% (1.54) and 2.60% (3.04) for cadmium, and 3.49% (5.40) and 2.41% (2.94) for zinc. These errors are much smaller than those obtained in the univariate analysis (around 40% for copper and cadmium); in fact, 27 out of 32 error values are lower than 3% in calibration A, indicating the suitability of the multivariate models built. 4.4. Calibration
transfer
The calibration transfer has been carried out from situation A (standard situation) to B, which implies a calibration transfer from one day to another. The purpose of this procedure is to achieve a reduction of the experimental effort that will be necessary in subsequent calibrations, by means of reducing the number of samples required to calibrate. A m-function, pdsgen, implemented in the PLS Toolbox by Wise [34] to spectroscopic instrument standardization has been used to obtain the calibration transfer matrix. The inputs were as follows:
56
A. Herrero, M.C. Ortiz/Analytica
Table 2 Subsets of samples from training set used to calculate matrix Transfer subject (no)
Samples
Transfer subset (no)
Samples
1 2 3 4 5
all 1 to 5 1 to 8 1to11 1to14
7 8 9 10 11
6
1 to 15
12
1 to 16 6to 11 6 7 8 12 9 to 14 17 19 21 31 32 34 17 21 22
the transfer
13 14 22 24 26 27 29 36 26 27 31 32 36
A is the matrix of predictor variables referring to situation A, dimensioned 36 x 203. B is the matrix of predictor variables referring to situation B, dimensioned 36 x 203. Samples are in the same order in A and B. w stands for the number of predictor variables to consider in the calculation of the transformation, i.e. moving window size. The following values have been used: 3, 5, 7, 9, 11 and 13. no is the vector indicating which samples will be used in the calibration transfer subset, dimensioned 1 x d, where d depends on the number of transfer subset samples. Table 2 shows the different transfer subsets that have been used in the calibration transfer procedures. Each one of the raw data, A and B, is considered as divided into three groups. One of them, the test set, is formed in every case by the same eight test samples that were in the PLS models. The second is formed in each case by the transfer subset samples, and the third contains the rest of the samples. As result of applying the pdsgen function two outputs were generated, the matrix which transforms the later day polarograms to the first, i.e. transfer matrix F, and a vector with the index of the samples used in its calculation, sub. The transfer matrix is a banded diagonal matrix, dimensioned 203 x 203, which changes when the parameters w or ~to vary. Different values of these two parameters have been used to evaluate their effect on the calibration transfer procedure. Once the transfer matrix has been calculated, it is possible to transform the polarograms from situation
Chimica Acta 348 (1997) 51-59
B to the format expected for them in situation A simply applying Eq. (1) having as a result a new matrix C, dimensioned 36 x 203, with the corrected signals. Next, PLS models built for calibration A can be used to predict concentration values from polarograms recorded in situation B by using the regression coefficients, calculated in the PLSC programm [26] by the procedure proposed by Marengo and Todeschini [35], taking into account the optimum number of latent variables indicated above for each model. Then, it is only necessary to multiply the corrected signals, C, by the coefficients of this closed form, saved in a matrix D (dimensioned 203 x 4, 4 being the number of analytes), to obtain the concentration values calculated for the later situation, Ycalc. Following this procedure, illustrated as a whole in Fig. 2, and taking the different values of w and no specified above, 72 different sets of concentration values have been calculated for each analyte from the corrected signals of calibration B, i.e. as many sets as transfer matrices have been computed. These results have been evaluated by means of the standard error of prediction (SEP), that is given for each metal by the squared root of
(6) where yj and jsj are the true concentrations and those computed by means of the PLS model built for situation B respectively, and J is the number of objects in the test set. To evaluate the effect that both transfer subset and window size have on the calibration transfers, an analysis for the SEP values has been made through the multiple range tests of Tukey and Newman-Keuls [29]. The analysis suggests that the window size used in the calibration transfer does not lead to statistically significant different SEP values, except for cadmium, which shows groups between certain levels. However, there is a major advantage of using the same window for all the metals in the calibration transfer procedure, since in this way it should be necessary to calculate only one transfer matrix, and the execution of the whole calibration transfer procedure should be carried out for the four metals at the same time. This fact indicates the selection of only one window size, 5,
A. Herrero, M.C. OrWAnalytica
5-l
Chimica Acta 348 (1997) 51-59
36x
203
3
pdwn 36
x
203
36 x 203
Ixd
IXd
I
J
36x
203 203x4
Fig. 2. Schematic
representation
of the calibration
which has been chosen on the basis of the results of the tests. With regard to the transfer subset, as was expected, both range tests found statistically significant differences of SEP between levels, pointing out which of them reach the best values. The lowest SEP values are related with transfer subsets 1, 11 and 12, the first of these subsets being the whole training set, while subset 11 is formed by 12 samples and subset 12 by only 8, as is shown in Table 2. As the aim of this calibration transfer procedure was to minimize the experimental effort necessary to carry out the determination, subset number 12 has been chosen as the best because of having two advantages: low number of samples that need to be measured in the new situation, and good prediction results in the calibration transfer procedure. The last affirmation is supported by the relative prediction errors obtained for the test set samples, which standard deviations (between brackets) and means of the absolute values are: 2.58% (3.71) for copper, 1.02% (1.42) for lead, 2.61% (3.15) for cadmium and 4.28% (3.91) for zinc. These results have been compared with those obtained in the full recalibration by means of a least squares regression between the concentration values calculated without and with calibration transfer,
transfer procedure
carried out.
Fig. 3. The parameters corresponding to this regression have been evaluated using a join confidence interval test [26] which accepts the hypothesis of zero intercept and unity slope for a level of significance o =5% (Fcalc < F, “-2, bil = 2.042), i.e. the concentra-
..
.,. . .?
4
P
’
*. 4
I 5
6 Conccnmtion
7 cahlated
10 8 9 with full redibmtion I pM
11
12
Fig. 3. Concentration values of the test set calculated with the standardization procedure versus full recalibration. Least squares regression y = b+ + bo, where bl = 1.0013 (sb, = 0.02), b,, = -0.0268 (sb = 0.12), p = 0.9906 and the standard deviation of regression is syx = 0.18.
58
A. Hewem, M.C. Ortiz/Analytica
tions calculated with the standardization procedure are statistically equal to those corresponding to the whole recalibration. With the calibration transfer procedure the number of samples necessary to effect the determination has been reduced from 28 to 8, which results after only two additions of each metal. This would be the minimum number of experiments necessary for a univariate analysis, that in this case would not be adequate because of the high errors reached. Whereas with the same experimental effort, the calibration transfer procedure applied achieves accurate results, assured by a multivariate methodology that has considered the internal relationships between metals.
5. Conclusions Partial least squares regression has been applied successfully to the resolution of the interference problem in the polarographic determination of copper, lead, cadmium and zinc. By means of a calibration transfer procedure it has been possible to use the experimental effort required to build a PLS model with a whole calibration, and also the information about the variability modelled (both experimental and internal), in later calibrations with less experimental effort (lower number of samples) and a certain guarantee in the results.
Acknowledgements
This work has been partially supported by Consejeria de Education y Cultura de la Junta de Castilla y Leon under project BU22/96.
References [l] J.S. Shenk and M.O. Westerhaus, Crop. Science, 31 (1991) 1694. [2] E. Bouveresse, D.L. Massart and P. Dardenne, Anal. Chem., 67 (1995) 1381. [3] Y. Wang, D.J. Veltkamp and B.R. Kowalski, Anal. Chem., 63 (1991) 2750. [4] Z. Wang, T. Dean and B.R. Kowalski, Anal. Chem., 67 (1995) 2379. [5] Y. Wang and B.R. Kowalski, Appl. Spectrosc., 46 (1992) 764.
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K51Y. Wang,
M.J. Lysaght and B.R. Kowalski, Anal. Chem., 64 (1992) 562. [71 E. Bouveresse, C. Hartmann, D.L. Massart, I.R. Last and K.A. Prebble, Anal. Chem., 68 (1996) 982. PI Y. Wang and B.R. Kowalski, Anal. Chem., 65 (1993) 1301. [91 Y. Wang and B.R. Kowalski, Anal. Chem., 65 (1993) 1174. [lOI M. Forma, G. Drava, C. Armanino, R. Boggia, S. Lanteri, R. Leardi, P Corti, P Conti, R. Giangiacomo, C. Galliena, R. Bigoni, I. Quartari, C. Serra, D. Ferri, 0. Leoni and L. Lazzeri, Chemom. Intell. Lab. Syst., 27 (1995) 189. Ull O.E. de Noord, Chemom. Intell. Lab. Syst., 25 (1994) 85. [W L. Nergaard, Chemom. Intell. Lab. Syst., 29 (1995) 283. u31 F. Vydra, K. Stulik and E. JuHkov$ Electrochemical Stripping Analysis, Ellis Horwood, Chichester, UK, 1976. H41 T.R. Copeland, R.A. Osteryoung and R.K. Skogerboe, Anal. Chem., 46 (1974) 2093. 1151 S.T. Crosmun, J.A. Dean and J.R. Stokely, Anal. Chim. Acta, 75 (1975) 421. V.I. Ignatov and G.M. H61 E.Y. Neiman, L.G. Petrova, Dolgopolova, Anal. Chim. Acta, 113 (1980) 277. u71 J.H. Aldstadt and H.D. Dewald, Anal. Chem., 64 (1992) 3176. WI J. Wang, Stripping Analysis: principles,instrumentation and applications, VCH Publishers, Deerlield Beach, FL, 1985. H91 S.D. Brown and R.S. Bear, J. Crit. Rew. in Anal. Chem., 24 (1993) 99. WI A. Herrero, MC. Ortiz, J. Amos, J. L6pez-Palacios and L.A. Sarabia, Anal. Chim. Acta, 293 (1994) 277. WI D. Jagner, L. Renman and S.H. Stefansdottir, Electroanalysis, 6 (1994) 201. Lw P. Bruno, M. Caselli, A. di Fano and C. Fragale, Analyst, 103 (1978) 868. ~231 M. Oehme and W. Lund, Fresenuis J. Anal. Chem., 294 (1979) 391. ~241 D. Jagner and L. Kryger, Computerized electroanalysis. Part III. Multiple scanning anodic stripping and its application to sea water, Analytica Chimica Acta, 80 (1975) 255266. ~251 M.I. Abdullah, B.R. Berg and R. Klimek, Anal. Chim. Acta, 84 (1976) 307. WI M. Forina, R. Leardi, C. Armanino and S. Lanteri, PARVUS: An Extendable Package of Programs for Data Exploration, Classification and Correlation, Ver 1.1, Elsevier Scientific Software, 1990. Ver. 5, STSC, Inc., Rockville, MD, 1271 STATGRAPHICS, 1991. Numeric Computation and WI MATLAB, High-Performance Visualization Software, Ver. 4.0, The MathWorks, Natick, MA, 1992. ~291 V.L. Anderson and R.A. McLean, Design of experiments: a realistic approach, in Statistics: textbooks and monographs, Marcel Dekker, New York, ~01.5, 1974. [301 G.E.P. Box, W.G. Hunter and J.S. Hunter, Estadistica para investigadores: introduccidn al disefio de experimentos,
A. Herrem, M.C. OrtizJAnalytica Chimica Acta 348 (1997) 51-59 an&lisis
de datos
y constroccidn
de modelos,
Revert&
Barcelona, Spain, 1989 [31] H. Weld, Soft modelling in systems under indirect observation II, North-Holland, New York, 1982. [32] H. Wold, Partial least squares, Encyclopedia of Statistical Science, Wiley, New York, 1989.
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F. Scholz,
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CHIMICA ACTA Analytica
Chimica Acta 348 (1997) 5 1-59
Multivariate calibration transfer applied to the routine polarographic determination of copper, lead, cadmium and zinc A. Herrero*, M.C. Ortiz Dpto. de Quimica Analitica, Facultad de C.EZA. y C. Quimicas, Vniversidad de Burgos, Pza. Misael Bafiuelos s/n, 09001 Burgos, Spain Accepted
10 December
1996
Abstract The application of calibration transfer methods has been successful in combination with near-infrared spectroscopy or fluorescence spectroscopy for prediction of chemical composition. One of the methods developed that provides accurate performance is the piecewise direct standardization method (PDS), which in this paper is applied to transfer from one day to another the partial least squares (PLS) models built in the polarographic determination of copper, lead, cadmium and zinc. This is an electrochemical example in which interferences have been found, making necessary the use of soft calibration models because of their ability to model this phenomenon, which implies a large number of standard samples for the calibration. Once the PLS models are built, the calibration transfer is carried out to overcome the instrumental change over time in this routine analysis, allowing one to reduce from 28 to 8 the number of calibration standards necessary for later determinations. Standard errors of prediction (SEP) are found similar to those given by the complete recalibration. Keywords: Multivariate calibration; Partial least squares; Calibration transfer; Polarography;
1. Introduction Electroanalytical techniques applied to the determination of elements at trace levels provide an important alternative to the already traditional spectroscopic methods. The use of a multivariate methodology, through suitable regression techniques such as principal component regression (PCR) or partial least squares regression (PLS), allows one to extract in an adequate way the information contained in electrochemical data, greatly increasing the possibilities of the electroanalytical techniques.
*Correspondingauthor. OOO3-2670/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII
SOOO3-2670(97)00154-2
Interferences;
Multicomponent
analysis
However, the applicability of these techniques has been limited by a problem that arises when a calibration model developed from a set of data in a particular situation cannot be applied to another, i.e. by a problem identified as ‘calibration transfer’. In this new situation some different responses are obtained, perhaps as the result of using other instruments, using the same instrument at a later time (drift, electronic fluctuations), changing the measurement conditions (temperature) or the physical constitution of the sample (particle size, surface texture), etc. Nevertheless, it is known that sometimes small variations in the analytical signals may be a sign of fundamental changes in the system under consideration. Therefore, small variations between the same
52
A. Herrero, M.C. OrtidAnalytica
measurements, carried out in different situations, could be sufficiently significant to prevent a calibration model being transferred, i.e. it could be applied to another situation providing different responses. One evident solution to the problems associated with a calibration transfer consists of performing a full recalibration in the new situation, which involves the repetition of all the calibration measurements with the aim of developing a similar calibration model. This procedure, traditionally used, requires considerable effort and cost, both in terms of experimental burden as in the time spent on the task, especially when the samples are numerous, chemically unstable, hazardous, etc. In any case, it would be desirable to calibrate one instrument in a new situation without feeling obliged to repeat the whole calibration procedure again. In spite of the importance of this question, there are not a lot of references in the bibliography on the matter, most of them in the field of NIR spectroscopy. Shenk and Westerhaus [l] proposed a univariate procedure to correct the experimental response, which algorithm has been modified and improved by Bouveresse et al. [2]. Wang et al. [3,4] compared five procedures, the Shenk procedure already cited and four multivariate standardization methods developed by themselves, in the determination of four analytes in gasoline samples by NIR spectroscopy, the piecewise direct standardization (PDS) method being that which gave the best standard errors of prediction (SEP) [5,6]. This method, based on PCR or PLS, has also been compared with a univariate slope correction of NIR spectra by Bouveresse et al. [7]. PDS method has been applied to the determination of two analytes in corn samples at different temperatures by NIR spectroscopy [8]. In the same way, a method has been developed to standardize second-order instruments together with an application to LC-UV data [9]. Forma et al. [lo], suggest a multivariate calibration transfer procedure that uses PLS regression twice: first to compute the relationship between the responses from two instruments and afterwards to compute the regression equation. Although most of the literature on multivariate calibration transfer comes from NIR spectroscopy, many of the strategies and methods described can also be applied to calibration transfer problems in other analytical fields [ 11,121. In this paper, the PDS
Chimica Acta 348 (1997) 51-59
method has been applied to electrochemical data provided by a single instrument on different days. To carry out the work, two calibrations were developed with the aim of determining simultaneously copper, lead, cadmium and zinc by differential pulse polarography. Frequently, when several metals are simultaneously on a mercury electrode intermetallic compounds are formed between them [13]. Numerous intermetallic compounds have been reported (Au-Zn, Cu-Cd, CuZn, Cu-Sn, Co-Zn, Ni-Zn,...), but most of the works on intermetallics have focused on the Cu-Zn compound because of the presence of these metals in several analytical samples. The formation of intermetallic compounds on the electrode may be reflected in the electrochemical response recorded, causing its severe depression or shift, which generates large errors in the determinations made by stripping analysis [14,15]. Several methods have been suggested for minimizing or eliminating errors caused by intermetallic compounds formation, such as the addition of a third element [16], the use of hyphenated methods [17], or usually, if the concentration of the metals in the sample is above 1 ppm, the analysis can be carried out by differential pulse polarography [ 181. But even in this last case, interferences are found in the analysis, suggesting the suitability of using multivariate calibration techniques capable of modelling phenomena of interferences, which allows one to analyse the problem in a different way [19]. In this paper, the simultaneous polarographic determination of copper, lead, cadmium and zinc in aqueous solution has been carried out, using a technique of multivariate regression (PLS) that has been successfully applied to the resolution of electrochemical problems in which interferences exist [20,21]. The determination of these four metals is usually made in water and food routine analysis [22,23] and it is common to find in the bibliography references to the interferences found in their electrochemical analysis [24,25].
2. Materials and equipment Analytical-reagent grade chemicals were used without further purification. All the solutions were pre-
53
A. Herrero, M.C. Ortiz/Analytica Chimica Acta 348 (1997) 51-59
pared with deionised water obtained with a Barnstead NAN0 Pure II system. Nitrogen (99.99%) was used to remove dissolved oxygen. The buffer solution (pH = 4.335) was acetic acid 2 M and ammonium hydroxide 1 M. Polarographic measurements were carried out using a Metrohm 646 VA processor with a 647 VA stand in conjunction with a Metrohm multimode electrode (MME) used in the static mercury drop electrode (SMDE) mode. The three-electrode system was completed by means of a platinum auxiliary electrode and an Ag-AgCl-KCl(3 M) reference electrode. Analysis of the data was done with PARVUS [26] STATGRAPHICS [27] and MATLAB [28].
This method uses a moving window so each predictor variable from situation A is related to the variables in this window from situation B. The window Zi used to calculate the correction of the variable xi is given according to
3. Experimental
F = diag (br, bl, . . . . b:, . . . . b;)
procedure
Polarographic measurements were carried out by the following procedure. The solution was placed in a polarographic cell and purged with nitrogen for 10 min. Once the solution had been deoxygenated a polarogram was recorded from 0.095 V to - 1.117 V, using the differential-pulse mode with a pulse amplitude of -50 mV. The drop time was 0.6 s, the drop area was 0.40 mm2 and the scan rate was - 10 mV s-i. After each addition the solution was stirred and purged for 15 s.
4. Results
and discussion
4.1. Piecewise
direct standardization
(PDS) method
Piecewise direct standardization (PDS), a method developed by Wang et al. [3], is based on establishing a relationship between each experimental point from a situation, A, with a small window of predictor variables close to itself in a different situation, B, by means of a transfer matrix F, itA = XnF
(1)
where XA and iin are the transfer subset (small series of measurements, usually a subset of the training set) of situations A and B respectively, and F is a square matrix dimensioned p x p, p being the number of predictor variables measured.
& = [xB,i-j> XS,i-j+l>
“‘> XE,i+k-13
XB,i+k]
(2)
where a window from index i-j to i+k is used (not necessarily symmetric around i). Then it is possible to establish a local multivariate regression given by xA,i = Zibi
(3)
where each regression vector, bi, can be calculated by means of PCR or PLS regression. These regression vectors bi are arranged along the main diagonal of the transformation matrix F while the rest of the elements are zero, which results in a banded diagonal matrix, (4)
Using the transfer matrix F, the predictor variables of a new sample measured in situation B (row vector XL) can be transformed to the A format: i;
= x;F
(5)
Each predictor variable Xi from situation A is obtained by multiplying the predictor variables from situation B in a window Zi by a transformation matrix F, and then predictions can be made using the original model. The use of a moving window to establish the relationship between the experimental variables from two different situations, instead of using all the predictor variables, avoids coming up against an illconditioned problem when calculating the transfer matrix because the number of variables is much larger than the number of standard samples, which is very common in electrochemical data. 4.2. Calibration
standards
Electroanalytical techniques are frequently used to determine several analytes in the same sample, the number of experimental samples necessary to carry out a multicomponent calibration being variable. If the number of standard measurements required for a successful calibration is quite high, their influence on time and cost of the experience (preparation of samples, cleaning of polarographic cell and material,
54
A. Herrero, M.C. Ortiz/Analytica
Chimica Acta 348 (1997) 51-59
Table 1 Experimental design for calibrates A and B. Each unit in the table represents an addition of 100 pl of standard solutions, 5.0 x 10e4 M copper, 5.0 x 10m4 M lead, 7.9 x 10e4 M zinc and 5.0 x 10m4 M cadmium respectively. First column lists the index of the samples and second column indicates that the additions were carried out on the same initial solution, using a different symbol for each initial solution used Run
1 2 6 3 9 10 7 4 12 13 15 14 16 11 8 5
Experimental
X
; X
A ;, 0 w
0 _ A X
design
Run
cu
Pb
Cd
Zn
2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4
2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4
2 2 4 4 2 2 4 4 2 2 4 4 2 2 4 4
2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4
deoxygenation and scan time, etc.) is negative. This fact indicates the advisability of a tool that allows the optimization of the number of experimental runs in order to obtain the most useful information from the analytical signal recorded with reasonable experimental effort. Therefore, an experimental design has been used, in which calibration samples are distributed approximately like a central composite design [29,30] but making equal volume additions of each analyte, as is shown in Table 1, so each factor (metal) takes five different levels of concentration. Following this design two complete calibrations have been carried out with the aim of analysing the variability due to the day to day effect between them, and afterwards developing a calibration transfer. It is necessary to model all pertinent variations expected in the samples to be analysed. So there are two sets of data, calibration A (carried out first) and calibration B (carried out several days later). Fig. 1 shows the polarograms of the test set samples from calibration B. The peaks appearing in the polarograms are well defined and characteristic in the
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Experimental
v v v v v 0 0 0 0 0 0 0 0 0 0 0 V V V V
100 loo
design
CU
Pb
Cd
Zn
1 2 3 4 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 1 2 3 4 5 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 1 2 3 4 5 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2 3 4 5
-140
380
-620
-860
-1100
Potential / mV Fig. 1. Polarograms concentration levels 4.83 pM and 7.61 cadmium, 4.83 pM
of the test set samples in calibration B, whose are between 4.83 pM and 7.61 pM for copper, pM for lead, 6.OOpM and 11.92 pM for and 7.61 pM for zinc.
A. Herrero, M.C. Ortiz/Analytica
polarographic analysis of these metals, but in spite of the clear similarities found between the polarograms from the two calibrations they are not exactly equal for the same samples. After additions of the different analytes it was evident that there exists severe interference between metals, as was expected, which will probably render difficult a univariate analysis. 4.3. Development
and evaluation
of PLS models
First, the prediction ability of univariate methods has been evaluated by means of a classical method of univariate standard addition procedure using the peak currents of calibration B, which will be used in subsequent analyses to evaluate the multivariate models. A statistically evaluated univariate model has been built for each metal with samples whose levels go from 2 to 5 in this metal (3 being the level in the rest of metals), see Table 1, and the concentration of the metal in the first of these samples is then calculated (samples l&23,28 and 33, for copper, lead, cadmium and zinc, respectively). The concentration of these samples will also be calculated in multivariate analysis to allow comparisons. In the analyses carried out the results yielded by lead and cadmium can be considered acceptable (~2% of relative error), but those of copper and zinc (~40%) are unacceptably high, showing the inviability of the univariate analysis. So it is necessary to apply another kind of mathematical technique able to consider the existent interferences. In this sense, multivariate data analysis allows one to find underlying or latent information in electrochemical data, where the partial least squares [31,32] regression is increasingly used [21,33]. This multivariate regression method has been applied to both calibrations, A and B, developing one different regression model for each metal, so that eight PLS models have been built. Every one of these models has been developed taking the concentration of one of the metals as response variable and the currents recorded at 203 equally spaced potentials from 0.095 V to - 1.117 V as predictor variables. The data sets have been divided in training set (28 objects) and test set (samples 1820, 23, 25, 28, 30,33 and 35) to build and evaluate the regression models. For calibration A 5,4,4 and 5 latent variables were required for copper, lead, cadmium and zinc models
Chimica Acta 348 (1997) 51-59
55
respectively, the explained variance for each model being 99.66,99.83,99.79 and 99.14%, while the cross-validated variance was 99.55,99.77, 99.73%, and 99.11% respectively. However, for calibration B 8, 8, 5 and 6 latent variables were necessary for copper, lead, cadmium and zinc models respectively, obtaining in this case explained variance values of 99.87, 99.90,99.55 and 99.42%, and cross-validated variance values for each model of 99.45,99.54,99.51 and 99.19% respectively. Although there are similarities between both data sets, their internal structures do not coincide so much, as the different latent structures of the models demonstrate. Every model developed gives satisfactory explained variance values, while the proximity of the corresponding cross-validated variance values to them is a sign of the stability of the models built and of their prediction ability. Moreover, the calculated concentration values for test set samples provide another way to evaluate the real prediction ability of these models. The standard deviation (between brackets) and the mean of the absolute values of the relative prediction errors obtained for each metal with calibrations A and B are respectively: 1.34% (1.66) and 2.11% (2.64) for copper, 3.09% (5.44) and 2.50% (4.00) for lead, 1.57% (1.54) and 2.60% (3.04) for cadmium, and 3.49% (5.40) and 2.41% (2.94) for zinc. These errors are much smaller than those obtained in the univariate analysis (around 40% for copper and cadmium); in fact, 27 out of 32 error values are lower than 3% in calibration A, indicating the suitability of the multivariate models built. 4.4. Calibration
transfer
The calibration transfer has been carried out from situation A (standard situation) to B, which implies a calibration transfer from one day to another. The purpose of this procedure is to achieve a reduction of the experimental effort that will be necessary in subsequent calibrations, by means of reducing the number of samples required to calibrate. A m-function, pdsgen, implemented in the PLS Toolbox by Wise [34] to spectroscopic instrument standardization has been used to obtain the calibration transfer matrix. The inputs were as follows:
56
A. Herrero, M.C. Ortiz/Analytica
Table 2 Subsets of samples from training set used to calculate matrix Transfer subject (no)
Samples
Transfer subset (no)
Samples
1 2 3 4 5
all 1 to 5 1 to 8 1to11 1to14
7 8 9 10 11
6
1 to 15
12
1 to 16 6to 11 6 7 8 12 9 to 14 17 19 21 31 32 34 17 21 22
the transfer
13 14 22 24 26 27 29 36 26 27 31 32 36
A is the matrix of predictor variables referring to situation A, dimensioned 36 x 203. B is the matrix of predictor variables referring to situation B, dimensioned 36 x 203. Samples are in the same order in A and B. w stands for the number of predictor variables to consider in the calculation of the transformation, i.e. moving window size. The following values have been used: 3, 5, 7, 9, 11 and 13. no is the vector indicating which samples will be used in the calibration transfer subset, dimensioned 1 x d, where d depends on the number of transfer subset samples. Table 2 shows the different transfer subsets that have been used in the calibration transfer procedures. Each one of the raw data, A and B, is considered as divided into three groups. One of them, the test set, is formed in every case by the same eight test samples that were in the PLS models. The second is formed in each case by the transfer subset samples, and the third contains the rest of the samples. As result of applying the pdsgen function two outputs were generated, the matrix which transforms the later day polarograms to the first, i.e. transfer matrix F, and a vector with the index of the samples used in its calculation, sub. The transfer matrix is a banded diagonal matrix, dimensioned 203 x 203, which changes when the parameters w or ~to vary. Different values of these two parameters have been used to evaluate their effect on the calibration transfer procedure. Once the transfer matrix has been calculated, it is possible to transform the polarograms from situation
Chimica Acta 348 (1997) 51-59
B to the format expected for them in situation A simply applying Eq. (1) having as a result a new matrix C, dimensioned 36 x 203, with the corrected signals. Next, PLS models built for calibration A can be used to predict concentration values from polarograms recorded in situation B by using the regression coefficients, calculated in the PLSC programm [26] by the procedure proposed by Marengo and Todeschini [35], taking into account the optimum number of latent variables indicated above for each model. Then, it is only necessary to multiply the corrected signals, C, by the coefficients of this closed form, saved in a matrix D (dimensioned 203 x 4, 4 being the number of analytes), to obtain the concentration values calculated for the later situation, Ycalc. Following this procedure, illustrated as a whole in Fig. 2, and taking the different values of w and no specified above, 72 different sets of concentration values have been calculated for each analyte from the corrected signals of calibration B, i.e. as many sets as transfer matrices have been computed. These results have been evaluated by means of the standard error of prediction (SEP), that is given for each metal by the squared root of
(6) where yj and jsj are the true concentrations and those computed by means of the PLS model built for situation B respectively, and J is the number of objects in the test set. To evaluate the effect that both transfer subset and window size have on the calibration transfers, an analysis for the SEP values has been made through the multiple range tests of Tukey and Newman-Keuls [29]. The analysis suggests that the window size used in the calibration transfer does not lead to statistically significant different SEP values, except for cadmium, which shows groups between certain levels. However, there is a major advantage of using the same window for all the metals in the calibration transfer procedure, since in this way it should be necessary to calculate only one transfer matrix, and the execution of the whole calibration transfer procedure should be carried out for the four metals at the same time. This fact indicates the selection of only one window size, 5,
A. Herrero, M.C. OrWAnalytica
5-l
Chimica Acta 348 (1997) 51-59
36x
203
3
pdwn 36
x
203
36 x 203
Ixd
IXd
I
J
36x
203 203x4
Fig. 2. Schematic
representation
of the calibration
which has been chosen on the basis of the results of the tests. With regard to the transfer subset, as was expected, both range tests found statistically significant differences of SEP between levels, pointing out which of them reach the best values. The lowest SEP values are related with transfer subsets 1, 11 and 12, the first of these subsets being the whole training set, while subset 11 is formed by 12 samples and subset 12 by only 8, as is shown in Table 2. As the aim of this calibration transfer procedure was to minimize the experimental effort necessary to carry out the determination, subset number 12 has been chosen as the best because of having two advantages: low number of samples that need to be measured in the new situation, and good prediction results in the calibration transfer procedure. The last affirmation is supported by the relative prediction errors obtained for the test set samples, which standard deviations (between brackets) and means of the absolute values are: 2.58% (3.71) for copper, 1.02% (1.42) for lead, 2.61% (3.15) for cadmium and 4.28% (3.91) for zinc. These results have been compared with those obtained in the full recalibration by means of a least squares regression between the concentration values calculated without and with calibration transfer,
transfer procedure
carried out.
Fig. 3. The parameters corresponding to this regression have been evaluated using a join confidence interval test [26] which accepts the hypothesis of zero intercept and unity slope for a level of significance o =5% (Fcalc < F, “-2, bil = 2.042), i.e. the concentra-
..
.,. . .?
4
P
’
*. 4
I 5
6 Conccnmtion
7 cahlated
10 8 9 with full redibmtion I pM
11
12
Fig. 3. Concentration values of the test set calculated with the standardization procedure versus full recalibration. Least squares regression y = b+ + bo, where bl = 1.0013 (sb, = 0.02), b,, = -0.0268 (sb = 0.12), p = 0.9906 and the standard deviation of regression is syx = 0.18.
58
A. Hewem, M.C. Ortiz/Analytica
tions calculated with the standardization procedure are statistically equal to those corresponding to the whole recalibration. With the calibration transfer procedure the number of samples necessary to effect the determination has been reduced from 28 to 8, which results after only two additions of each metal. This would be the minimum number of experiments necessary for a univariate analysis, that in this case would not be adequate because of the high errors reached. Whereas with the same experimental effort, the calibration transfer procedure applied achieves accurate results, assured by a multivariate methodology that has considered the internal relationships between metals.
5. Conclusions Partial least squares regression has been applied successfully to the resolution of the interference problem in the polarographic determination of copper, lead, cadmium and zinc. By means of a calibration transfer procedure it has been possible to use the experimental effort required to build a PLS model with a whole calibration, and also the information about the variability modelled (both experimental and internal), in later calibrations with less experimental effort (lower number of samples) and a certain guarantee in the results.
Acknowledgements
This work has been partially supported by Consejeria de Education y Cultura de la Junta de Castilla y Leon under project BU22/96.
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Electroanal., 2 (1990) 309. [34] B.M. Wise, PLS Toolbox for use with MATLAB, Ver. 1.3, available from the author. [35] E. Marengo and R. Todeschini, Chemom. Intell. Lab. Syst., 12 (1991) 117.