Spectrochimica Acta Part A 61 (2005) 3013–3019

Simultaneous spectrophotometric determination of four metals by twokinds of partial least squares methods

Ling Gao, Shouxin Ren∗

Department of Chemistry, Inner Mongolia University, Huhehot 010021, Inner Mongolia, China

Received 17 August 2004; accepted 11 November 2004

Abstract

Simultaneous determination of Ni(II), Cd(II), Cu(II) and Zn(II) was studied by two methods, kernel partial least squares (KPLS) and waveletpacket transform partial least squares (WPTPLS), with xylenol orange and cetyltrimethyl ammonium bromide as reagents in the mediumpH = 9.22 borax-hydrochloric acid buffer solution. Two programs, PKPLS and PWPTPLS, were designed to perform the calculations. Datareduction was performed using kernel matrices and wavelet packet transform, respectively. In the KPLS method, the size of the kernel matrixis only dependent on the number of samples, thus the method was suitable for the data matrix with many wavelengths and fewer samples.Wavelet packet representations of signals provide a local time-frequency description, thus in the wavelet packet domain, the quality of then beches 12a .©

K

1

mobcsoacikTidtc

aper,firsth re-

ionasedveletue,WTinalof a

ry out.sionseir

ilityom-tionsingbin-oto-

1d

oise removal can be improved. In the WPTPLS by optimization, wavelet function and decomposition level were selected as Daund 5, respectively. Experimental results showed both methods to be successful even where there was severe overlap of spectra2004 Elsevier B.V. All rights reserved.

eywords:Partial least squares; Wavelet packet transform; Simultaneous multicomponent determination

. Introduction

The partial least squares (PLS) method is a generalizedethod used to build a predictive model between two blocksf variables: the C-block of predictor variables and the D-lock of response variables. PLS is a factor-based methodapable of using full spectra, which can include as muchpectral detail in the analysis as possible[1]. The advantagef PLS is the transformation of the numerous original vari-bles into a small number of latent vectors, which are a linearombination of the original variables[2,3]. New analyticalnstruments produce huge quantities of data that need someinds of reduction in order to be practicable to analyze[4].he analysis of large data arrays is emerging as a problem

n analytical chemistry. The best approach to this problem isata compression. Of course, the compression must be such

hat the loss of significant information is minimized. The in-reasing complexity of chemical data has recently stimulated

∗ Corresponding author. Tel.: +86 4714992125; fax: +86 47144951761.E-mail address:cersx@mail.imu.edu.cn (S. Ren).

the development of data compression methods. In this pwe try to apply two methods for data compression. Theapproach is based on some small kernel matrices, whicquire much less storage space than the original data[5,6].The method[6] provides a simple way of data compressfor speeding up the calculations. Another approach is bon wavelet transform and wavelet packet transform. Watransform (WT)[7–9] is a novel signal processing techniqwhich has dual localization in both frequency and time.can be used for the purpose of converting data from origdomain into wavelet domain, in which the representationsignal is sparse and makes signal denoising easier to carWavelet packet transform (WPT) is an important extenof wavelet transform[10,11]. Wavelet packet (WP) functioninherit the property of time-frequency localization from thcorresponding wavelet functions, but offer more flexibthan wavelet in representing different type of signals. Cparing to wavelet analysis, WPT has found a few applicato chemistry[12–14] and are still rare in the preprocessof PLS input data. In this paper, a WPTPLS method coming WPT and PLS was developed to perform spectroph

386-1425/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.saa.2004.11.020

3014 L. Gao, S. Ren / Spectrochimica Acta Part A 61 (2005) 3013–3019

metric multicomponent analysis. It not only has advantagesof WPT in extraction of characteristic information and de-noising but also advantages of PLS in regression capacityfor multicomponent systems. It is difficult to simultaneouslydetermine Ni(II), Cd(II), Cu(II) and Zn(II) with xylenol or-ange and cetyltrimethyl ammonium bromide (CTMAB) asreagents using traditional spectrophotometry because the ab-sorption spectra overlap. The determination of trace amountsof metals has recently received considerable attention owingto concern with the problems of environmental pollution. Theproposed two methods were successfully applied to simulta-neous determination of multicomponent overlapping spectra.

2. Theory

2.1. KPLS method[6]

A kernel algorithm is based on eigenvectors to thekernel matrix (D DT C CT ). The KPLS regressionsolution can be written as:C=DBKPLS +F. The regressioncoefficients are expressed as:BKPLS =W(PTW)−1QT. Thesix steps of the algorithm were as follows:

(1) Calculate the covariance matrices (D DT ) and(C CT ). The kernel matrix (D DT C CT )

he

( r ofs:

(

(

( ted.( -

2

rm[ sef

W

w thel2 da

ferent scales, oscillations and localizations:

f (x) =∑

i

∑n

∑k

CjnkWjnk(x) (2)

whereCjnk is the wavelet packet coefficients. The coefficientsCjnk can be obtained from:

Cjnk =∫ ∞

−∞f (x)Wjnk(x) dx (3)

The whole set of orthonormal basis is called the waveletpacket library. The each element of the library is deter-mined by the functionWjnk(x). The discrete wavelet transform(DWT) can be implemented by means of Mallat’s pyramidalgorithm[8]. The theoretical background to DWT has beendescribed in details[15]. WPT can be considered as an ex-tension of DWT. A fast wavelet packet transform (FWPT) isexpressed as:

Wj+1,2n = HWjn (4)

Wj+1,2n+1 = GWjn (5)

whereW0,0 indicates the measured signalf,H={hl}l ∈ Z andG={gl}l ∈ Z are the low- and high-pass matrix filters. Thefirst and second indices ofW indicate the level of decompo-sition and its position at that level. The reconstruction can beimplemented by:

W

wd cri-

tg ume tri-b cal-c fewc oachz effi-c ly thec rede-fi ationi oho[ e es-t itherh thed ginald

2(

ovala ise int WPd hardS co-e T,

is then created as: (D DT C CT )a

=(D DT )

a(C CT )

a, where symbol a means t

rank index (a= 1, 2,. . ., n).2) The PLS score vectort is estimated as the eigenvecto

the kernel matrix (D DT C CT ), it is expressed ataλa = (D DT C CT )

ata.

3) The PLS score vectoru is calculated using the (C CT )covariance matrix:ua = (C CT )

ata.

4) Update the kernel matrices: (D DT C CT )a+1 =

(I − tatTa )(D DT C CT )

a(I − tat

Ta )

(D DT )a+1 = (I − tat

Ta )(D DT )

a(I − tat

Ta );

(C CT )a+1 = (I − tat

Ta )(C CT )

a(I − tat

Ta ). Here I

is thea×a identity matrix.5) Step 2–4 are repeated until all information is extrac6) The weight and loading matricesW, P andQ are calcu

lated andBkPLS is obtained.

.2. WPT denoising

WPT is an important extension of wavelet transfo10,11]. A wavelet packetWjnk is generated from the baunction:

jnk(x) = 2(−j/2)Wn(2−jx − k) (1)

here indicesj, n, k are the scale, the oscillation andocalization parameter, respectively,j, k∈Z, n= 0, 1, 2,. . .j − 1. In wavelet packet analysis a signalf(x) is representes sum of orthogonal wavelet packet functionWjnk(x) at dif-

jn = H∗Wj+1,2n + G∗Wj+1,2n+1 (6)

hereH* andG* represent the conjugate matrix ofH andG.The best basis is selected according to entropy-base

erion proposed by Coifman and Wickerhauser[10]. The al-orithm is based on finding the basis that gives the minimntropy or maximum information of the signal energy disution. In this case, Shannon entropy was applied andulated for finding a sparse representation where onlyoefficients have high amplitude and the remaining apprero. After finding the best basis, the obtained WP coients are selected according to their absolute value. Onoefficients whose absolute values are higher than a pned threshold value are retained. The thresholding opers implemented by the SURE method proposed by Don16] based on stein’s unbiased risk estimation, then thimated wavelet packet coefficients can be defined in eard or soft thresholding, and by utilizing inverse WPT,enoised coefficients can be converted back to the oriomain.

.3. The wavelet packet transform partial least squaresWPTPLS) method

In the method, WPT is used as a tool for noise remnd data compression. The main steps of signal deno

he case are: (1) original data are transformed into theomain by using FWPT; (2) best basis selection and theURE thresholding are employed to select suitable WPfficients in the WP domain; (3) by utilizing inverse FWP

L. Gao, S. Ren / Spectrochimica Acta Part A 61 (2005) 3013–3019 3015

the denoised coefficients can be converted back to the origi-nal domain. The reconstructed matrices from standard andunknown mixtures were obtained for further PLS opera-tion. The calibration with the use of the PLS approach isdone by decomposition of both the concentration and ab-sorbance matrices into latent variables,D = TPT + E andC = UQT + F . The aim of PLS is to find linear model that isgiven byC = DB, whereC is estimated concentrations. The

regression coefficients are expressed asB = W(PTW)−1

QT,whereW is a weight matrix. The PLS algorithm is built onthe properties of the nonlinear iterative partial least-squares(NIPALS) algorithm by calculating one latent vector at atime.

According to above-mentioned algorithms, two programscalled PKPLS and PWPTPLS were designed to perform datacompression, denoising and simultaneous determination.

3. Experimental

3.1. Apparatus and reagents

The Shimadzu UV-240 spectrophotometer furnished withOPI-2 function was used for all experiments; a LegendPentium IV microcomputer was used for all the calcula-t gitalp . Allr usedw tionsoa withw fromt d. A1b

3

iousr tan-d 2),5a ark.A ea-s m in-t atrixD hreer

4

SEPa erfor-m onenti

RSEP is given by Eq.(9) [17].

SEP=√∑m

j=1{Cij − Cij}2

m(7)

SEP=√∑n

i=1∑m

j=1{Cij − Cij}2

nm(8)

RSEP=

√√√√∑ni=1

∑mj=1{Cij − Cij}2∑n

i=1∑m

j=1C2ij

(9)

whereCij andCij are the actual and estimated concentrations,respectively, for theith component in thejth mixture,m thenumber of mixtures, andn the number of components.

4.1. Absorbance spectra of the system

Fig. 1 shows the absorption spectra of Cu(II), Cd(II),Zn(II), Ni(II) and their mixed solution with xylenol orangeand CTMAB as reagents. Their maximum absorptions were590, 594, 596, and 610 nm respectively. It can be seen fromFig. 1that the absorption spectra of these four metal ions ex-h ions,s zed.

F(1); 10.00× 10−6 mol l−1 Cd(II) (2); 16.00× 10−6 mol l−1 Zn (3);5.00× 10−6 mol l−1 Ni(II) (4); their mixture (5).

ions; pH measurements were made with a pH-3B diH-meter with a glass-saturated calomel dual electrodeeagents were of analytical reagent grade. The wateras doubly distilled and deionized. Stock standard soluf 0.01000 mol l−1 Cu(II) and Ni(II), 0.02500 mol l−1 Cd(II)nd Zn(II) were prepared from correspondent reagentsater as solvents. Standard solutions were then prepared

he stock standard solutions by serial dilution as require.000 g l−1 xylenol orange; 1.400 g l−1 CTMAB; pH = 9.22orax buffer solution were used.

.2. Procedures

A series of mixed standard solutions containing varatios of the four metal ions was prepared in 25 ml sard flasks, 10.00 ml of borax buffer solution (pH = 9.2.00 ml 1.400 g l−1 CTMAB, 1.00 ml 1.000 g l−1 xylenol or-nge were added and diluted with distilled water to mblank solution was prepared similarly. Spectra were m

ured in 1 cm cuvettes between 492 and 650 nm at 2 nervals with respect to a reagent blank. An absorption m

was built up. All the values measured were means of teplicates.

. Results and discussion

Absolute and relative standard errors of prediction (nd RSEP) were used as the criteria for comparing the pances of the test methods. The SEP for a single comp

s given by Eq.(7), that for all components by Eq.(8). The

ibited are seriously overlapped in their absorbing rego that for mixed solution only one peak can be recogni

ig. 1. The absorption spectra of 8.000× 10−6 mol l−1 Cu(II)

3016 L. Gao, S. Ren / Spectrochimica Acta Part A 61 (2005) 3013–3019

Table 1Composition of the training set

Sample Concentration (10−6 mol l−1) Sample Concentration (10−6 mol l−1)

Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II)

1 0.8000 0.8000 0.8000 0.8000 9 5.600 0.8000 4.000 9.6002 0.8000 6.000 4.000 5.200 10 5.600 6.000 7.200 14.003 0.8000 11.20 7.200 9.600 11 5.600 11.20 10.00 0.80004 0.8000 16.00 10.00 14.00 12 5.600 16.00 0.8000 5.2005 3.200 0.8000 4.000 9.600 13 8.000 0.8000 4.000 9.6006 3.200 6.000 7.200 14.00 14 8.000 6.000 7.200 14.007 3.200 11.20 10.00 0.8000 15 8.000 11.20 10.00 0.80008 3.200 16.00 0.8000 5.200 16 8.000 16.00 0.8000 5.200

4.2. Optimization of the experimental conditions

The effect of pH on this experimental system was stud-ied over the pH range 8.90–9.55. After trials, in this mediumthe Ni(II), Cd(II), Cu(II) and Zn(II) mixture had maximumabsorbance within the pH 9.14–9.30 range. Therefore, buffersolution (pH = 9.22) was used in the study. The influence ofxylenol orange was also investigated. A 1.00 ml xylenol or-ange solution was chosen because it ensures the absorbanceto be maximal and constant. The influence of the amountof buffer solution and CTMAB on this experimental systemand the selection of other experiment parameters were in-vestigated as well. Selected optimal experiment conditionswere the same as those described in experiment procedure.Under the selected experimental conditions, the absorbancespectrum of the quaternary mixture remains constant for atleast 1.5 h.

4.3. Estimating the number of factors

In the first step of multivariate statistical methods, thenumber of independent factors is determined by principalcomponent analysis (PCA). These techniques require therank of the absorbance matrix to be equal to the number of in-dependent factors. The essence of the techniques is the pseu-d Thep ce on om-p icalp tus oro ningp rma-t weret eriaw cipalf is ofp

4

ourk hog-o -

rizes the composition of the training set. Spectra measuredbetween 492 and 650 nm at 2 nm intervals were extractedfrom the training set asDmatrix. The spectra of the syntheticunknown samples were measured, then test matrixDu wasbuilt in the same way as the training matrix. Using PKPLSprogram the concentration of Ni(II), Cd(II), Cu(II) and Zn(II)were calculated. The actual concentrations, found concentra-tion and average recoveries are listed inTable 2.

It is obvious that theBKPLS can be calculated using the ker-nel matrices (D DT C CT ) and the associated matrices(D DT ) and (C CT ). Both the kernel matrix and the as-sociation matrices are of sizeN×N, whereN is the numberof solutions. This means that as long asN is small, no largematrices or vectors are used in the calculations. Thus, thisiterative process is much faster than that of the classical PLSalgorithm. After the first dimension has been determined, theprocedure of updating both (D DT ) and (C CT ) takeplace by subtracting a rank-one matrix. Both (D DT ) and(C CT ) can be updated by left and right multiplication us-ing the same matrixGa = I − tat

Ta , thereby, avoiding having

return to the original large matricesD andC.

4.5. Wavelet packet transform and wavelet packetdenoising

ls o-

orm.

orank determination of an experimental data matrix.seudorank means the mathematical rank in the absenoise[18,19]. However, experimental spectra are often clicated by noise, which may be due to physical and chemrocesses and imperfections in the experimental apparather reasons. With the definition, the problem of determiseudorank is one of separation between chemical info

ion and noise. In this case, two kinds of pre-treatmentsaken to eliminate the noise. After denoising, eight critere used to estimate the number of factors. Four prin

actors for both the methods were selected on the basrevious reported methods[20].

.4. KPLS method

A training set of 16 samples formed by the mixture of finds of metals was designed according to four-level ortnal array design with the L16(45) matrix. Table 1summa

fHere, we selected mean spectra of matrixD as origina

ignalf. WPT of the signalfwas carried out using FWPT alg

Fig. 2. The part of WP coefficients obtained by wavelet packet transf

L.Gao,S.Ren/Spectro

chimica

Acta

PartA

61(2005)3013–3019

3017

Table 2Actual concentration, found concentration and percentage recovery of the synthetic unknowns

Sample no. Actual concentration (10−6 mol l−1) Found concentration (10−6 mol l−1) Recovery (%)

WPTPLS KPLS WPTPLS KPLS

Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II)

1 4.000 10.00 4.800 6.400 4.520 9.829 4.533 6.898 4.552 10.25 4.070 6.750 113.0 98.3 94.4 107.8 113.8 102.5 84.8 105.52 4.000 11.00 5.600 8.000 4.109 11.20 5.202 7.718 3.872 11.50 4.986 7.686 102.7 101.8 92.9 96.5 96.8 104.6 89.0 96.13 4.000 12.00 6.400 9.600 3.622 11.38 7.030 9.605 3.756 11.20 7.141 9.678 90.5 94.8 109.8 100.0 93.9 93.3 111.6 100.84 5.000 10.00 5.600 9.600 4.860 10.56 5.415 9.546 4.884 10.54 5.405 9.600 97.2 105.6 96.7 99.4 97.7 105.4 96.5 100.05 5.000 11.00 6.400 6.400 4.728 11.09 6.137 6.255 4.968 10.68 6.484 6.332 94.6 100.8 95.9 97.7 99.4 97.1 101.3 98.96 5.000 12.00 4.800 8.000 4.658 11.47 5.202 8.411 4.388 11.62 5.174 8.429 93.1 95.6 108.4 105.1 87.8 96.8 107.8 105.47 6.000 10.00 5.600 9.600 5.924 10.59 5.700 9.629 6.058 10.35 5.883 9.664 98.7 105.9 101.8 100.3 101.0 103.5 105.1 100.78 6.000 11.00 6.400 6.400 5.994 11.19 6.235 6.109 6.274 10.81 6.530 6.124 99.9 101.7 97.4 95.5 104.6 98.2 102.0 96.79 6.000 12.00 4.800 8.000 6.586 11.70 4.945 7.830 6.246 12.06 4.727 7.737 109.8 97.5 103.0 97.9 104.1 100.5 98.5 96.7

Fig.3.

Originalraw

spectra(row

1),reconstructedspectra

(row2),and

theirdifference

obtainedby

means

ofWP

denoising(row

3)fordifferentamounts

ofnoiseadded

(indicatedatthe

headofthe

columns).

rithm.E

achlevelconsists

of2 jblocks.Thus,level0

consistsofa

singleblock

(2 0);level1consists

oftwo

blocks(2 1);level

2four

blocks(2 2).B

locksfor

subsequentlevelsare

derivedin

asim

ilarm

anner.T

hepart

ofW

Pcoefficients

obtainedby

FW

PT

isshow

ninF

ig.2.

Itis

obviousthat

thew(j,0)

onlyhave

apositive

partand

similar

tothe

approximations

Table3

Optim

izationofw

aveletfunctions

Waveletfunction

SE

P(10 −6

moll −

1)R

SE

P(%

)

Daubechies

40.354

4.53D

aubechies6

0.6087.77

Daubechies

80.812

10.4D

aubechies10

0.4876.23

Daubechies

120.337

4.31D

aubechies14

0.4836.18

Daubechies

160.569

7.27D

aubechies18

0.5016.40

Daubechies

200.550

7.0S

ymm

let50.955

12.2S

ymm

let60.914

11.7S

ymm

let70.937

12.0S

ymm

let80.940

12.0C

oiflet10.876

11.2C

oiflet20.738

9.44C

oiflet30.788

10.1C

oiflet40.993

12.7C

oiflet51.22

15.6

Table4

Optim

izationofw

aveletdecomposition

level

LS

EP

(10 −6

moll −

1)R

SE

P(%

)

10.374

4.792

0.4065.19

30.402

5.144

0.6328.08

50.354

4.536

0.87411.2

3018 L. Gao, S. Ren / Spectrochimica Acta Part A 61 (2005) 3013–3019

Table 5SEP (10−6 mol l−1) and RSEP (%) values for Ni(II), Cu(II), Cd(II), Zn(II) system by two methods

Methods SEP (10−6 mol l−1) Total elements RSEP (%) Total elements

Ni(II) Cd(II) Cu(II) Zn(II) Ni(II) Cd(II) Cu(II) Zn(II)

WPTPLS 0.331 0.412 0.325 0.266 0.337 6.53 3.73 5.76 3.28 4.31KPLS 0.318 0.429 0.441 0.251 0.368 6.28 3.89 7.81 3.09 4.71

part of wavelet transform. The others are composed of bothpositive and negative parts. Each block of the coefficientsdescribes the components of the signalfrelated to a certainfrequency band. This time-frequency resolution enables theWP more flexible to represent different types of signals. Inthe spectrophotometric measurements, the analytical signalsusually center in low frequency part, whereas the noise inhigh frequency part. The aim of WP denoising is to extractthe desired information from a complex instrument output.Random Gaussian noise was added to the mean spectra for as-sessing the WP denoising ability. Original and reconstructedspectra as well as their difference with different added noiseare displayed inFig. 3. It was found that WPT provides anappropriate approach for denoising even in case where 2%noise is added. Thus, the method is safe for preprocessingtwo-dimensional raw data matrix in the following operation.

4.6. The wavelet packet transform partial least squares(WPTPLS) method

Each of the wavelet functions has different characteristics.The wavelet function, which is optimal for a given signal, isnot necessarily the best for another type of signal. Thus, thechoice of the wavelet functions is empirical. It is possibleto use the predictive parameters SEP and RSEP to find theoptimum choice of functions. In this case, different waveletf SEPa wn inT ms too . Theib onglyds

pre-p tb pro-g II)w tra-tT sat-i (II),C eds

4

red.U n of

Ni(II), Cd(II), Cu(II) and Zn(II) and their average recoverieswere calculated. The SEP and RSEP for the two methodsused for the four component systems are given in theTable 5.In this case, the RSEP values for all components with thetwo methods are 4.31 and 4.71, respectively. FromTable 5it can be seen that both the PLS methods are successful andWPTPLS method give a little better prediction results thanKPLS method.

5. Conclusion

The two PLS methods, KPLS and WPTPLS, have beenshown to be successful approaches to simultaneous spec-trophotometric determination of Ni(II), Cd(II), Cu(II) andZn(II). The difficulty imposed by overlap of the absorptionspectra was overcome by both the methods. The two meth-ods can be used to analyze the whole spectra rather than justpicking out a few characteristic values. Properly designedcomputer programs according to chemometric algorithmscan provide successful tools for simultaneous determination.

Acknowledgements

nceF nnerM

R

.art,

996)

89..92)

ork,

[ 38

[[ 997)

unctions are tested under SURE hard thresholding. Thend RSEP of total components were computed and shoable 3. It can be seen that the Daubechies 12 (Db12) seeutperform the others, so Db12 was selected in the study

nfluence of decomposition levelL is listed inTable 4. It cane seen that the SEP and RSEP of all components strepend on the decomposition level. In this case,L= 5 waselected.

The same standard solutions described as 4.4 wereared. The training matrixD and the test matrixDuwere gey experiment procedures as above. Using PWPTPLSram, the concentrations of Ni(II), Cd(II), Cu(II) and Zn(ere calculated with four factors. Their actual concen

ions, found concentration and recoveries are listed inTable 2.he experimental results showed that this method gave

sfactory results for simultaneous determination of Nid(II), Cu(II) and Zn(II) mixture with severely overlapppectra.

.7. A comparison of KPLS and WPTPLS

A set of eight synthetic unknown samples was prepasing PWPTPLS and PKPLS, the found concentratio

The authors would like to thank National Natural Scieoundation of China and Natural Science Foundation of Iongolia for financial support of this project.

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