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Anal. Chem. 1985, 57,2880-2684
Selection of Analytical Wavelengths for Multicomponent Spectrophotometric Determinations S. D. Frans' and J. M. Harris* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The statlstlcai criterla for the selection of analytical wavelengths are derived in terms of the elements of the variance-covariance matrix. The uncertainty in the soivedhfor concentratlons Is evaluated as a function of wavelength choke. The variance factors identify analytbal wavelengths whlch consistently yield superlor concentration precision, especiaiky In the case of severe spectral overlap. The greatest Impact of Wavelength selection occurs when the total number of measurements Is small. Compared with the acquisition of a complete spectrum, replicated measurements at the smallest possible number of selected wavelengths improves concentratiQn precision but cannot detect lack-of-fit error which provldes information about spectral contamlnatlon.
1. Mixture spectra were generated by multiplying the standard spectrum of each component by a concentration value and adding the intensity of the Components in the mixture. Gaussian weighted noise of mean zero having a standard deviation of 0.01 relative to the maximum absorbance was added to the mixture absorption spectrum. The model of constant absorbance noise corresponds to transmission measurements limited by proportional noise in the light intensity. No noise was added to the design matrix since uncertainty present in the standard spectra could be mapped linearly onto the measured mixture spectrum.
THEORY The Beer-Lambert relationship between the absorbance of the sample and the component concentrations makes the application of linear least squares appropriate to the concentration determination. The absorbance, y,, of an n-component system a t wavelength i can be written as
+ XiZP2 +
+ X,,P, + ei
While the spectrophotometric determination of multicomponent samples has become a routine method in analytical practice, the quantitation of compounds in a mixture continues to be a difficult analytical problem. Much of the emphasis in the literature on this problem has been on mathematical techniques to effect the resolution of individual components of a mixture (1-1 7),the most common of which utilize linear regression (18-38). Crucial to the least-squares analysis of spectrophotometric data are the selection of the wavelengths at which measurements are made and the extent of wavelength overdetermination which is useful (39-54). Implicit in this selection is that some "analytical" wavelengths are better than others with respect to the accuracy and precision of the concentrations determined in the analysis. Statistical criteria for selection of analyticaI wavelengths in multicomponent analysis are considered in this work and compared with the use of an entire spectrum for analysis. The results indicate that there are potential improvements in precision for selecting a small number of optimal wavelengths but that a degree of caution is necessary in choosing this approach since lack-of-fit error cannot be detected.
(1) where X J is the molar absorptivity of component k a t wavelength i, & is the concentration of component k in the mixture, and e, represents the noise or error in measuring y z . A system of equations in the form of eq 1can be conveniently expressed as a matrix equation Y=Xj?+E (2) where Y and E are column vectors of length r corresponding to the number of wavelengths, X is an r by n matrix of the standard spectra of the n components, and B is column vector of concentrations of length n. If the model is accurate, that is, if the component standard spectra in X are those actually present in the mixture spectrum, Y, the residuals R resulting from the linear least squares fit, will be of the order of the measurement error or pure error, E R=(Y-X@)=E (3) where is the least squares best estimate of defined hy (55)
EXPERIMENTAL SECTION The synthetic spectra were generated and the subsequent analyses were performed on a DEC 20160. Solution to simultaneous equations and least-squaresanalysis of overdetermineddata sets were executed with subroutines from the IMSL, EISPACK, and LINPACK libraries of scientific subroutines. Typical execution times for the determinationof the 100 X 100 error surfaces were on the order of 10-30 s, depending on the error parameter being evaluated. Graphical presentation of the data utilized the PLOT79 graphics package. All programming was done in Fortran. Absorption spectra of individual componentswere modeled as Gaussian peaks over a range of 1-100 arbitrary wavelength units. The Gaussian mean of each component was adjusted to simulate various degrees of overlap, as indicated by a resolution factor, R, = AXm,/40, where AX,,, is the difference in the means and u is the standard deviation of the peaks, such that a resolution factor of unity, R , = 1.0, corresponds to base line resolution of the two peaks. The three cases studied are presented in Figure
The choice and number of wavelengths used in the measurement to construct eq 2 affect both thelaccuracy and precision of the concentrations determined, p. Criteria for the selection of these wavelengths are usually baaed on some measure of the linear independence or orthogonality of the equations represented in eq 2 (18). Sets of wavelengths are tested in combination, and for each set, the orthogonality criterion is evaluated. In the following paragraphs, the relative merits of several criteria for selecting analytical wavelengths are briefly discussed. Determinant. Kaiser has defined sensitivity, S,, to be a criterion for the selection of analytical wavelengths as the absolute value of the determinant of the design matrix X (18,
lPresent address: Spectra-Physics,3333 N. First St., San Jose, CA 95134. 0003-2700/85/0357-2680$01.50/0
Yt = X i l P l
e..
B
,i3 = (XTX)-1XT Y
(4)
XT signifying the transpose of X.
48)
S , = ldet XI
(5)
where X is a square design matrix, r = n. A large S, corresponds to large diagonal elements and small off-diagonal elements; therefore, by maximizing S , the molar absorptivity at wavelength i for component k , x , ~ becomes , large a t the 0 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985 2681
of Y is independent of wavelength. In many spectroscopic measurements, this assumption may not hold which would require one to minimize the product of the condition number and the relative error in Y. Since the wavelength dependence of the relative error in Y typically depends on the spectral intensity, which in turn depends on sample composition, relative error optimization based on condition number should be repeated for each sample. Variance. The variance associated with the least-squares solution vector, p, obtained by eq 4 can be written as (56,57)
V(& =
(XTX)-lu2
(8)
where V@)is the variance-covariance matrix, and uz (a scaler) is the variance of the least-squares fit. If the model, X, is accurate representation of the spectra in Y, then uz = u 2 p ~ , the pure or experimental error in measuring Y. For a twocomponent system, the variance-covariance matrix is
and u2b2being the variances of concentration estimate of components 1 and 2, respectively, and u2~lb2 the covariance between the components. The elements of the matrix (X%)-’ relate the errors in Y,u2, to the errors in the estimated concentrations @
Figure 1.
Synthetic data for two-component mixtures: (a) case A,
Xmx1,2 = 40, 80,R , = 0.5; (b) case B, h , = 45, 55, R , = 0.25; (c) case C,,,,x, = 48, 52,R , = 0.1. The ddth (standard deviation) of the Guassian peaks is 10 wavelength units.
expense of the off-diagonal elements 3cLJ (if lz). Modifications to this definition of S, have been presented by Junker and Bergman (18, 49-51) including one which allows for overdetermined measurements, r > n, where
S, = ldet XTXl
(6)
Although the methods of selection based on eq 5 or 6 reflect the degree of linear independence of the, set of equations represented in Y = X@,they provide no prediction as to the expected improvement in accuracy or precision of the concentrations determined using a particular set of wavelengths. As a result, one is not assured, using the determinant, that the selected waveIengths minimize the concentration error. Condition Number. Ebel and co-workers (53) have investigated the use of the condition number of the design matrix X as an inverse measure of orthogonality. As with the determinant approach, cond(X) values are determined for each set of wavelengths, and the set producing the minimum condition number is chosen as optimal. The condition number is a reliable measure of the degree of linear independence exhibited by a system of equations and is much more reliable than the determinant. The condition number is particularly attractive because it can be directly related to the error in by (55)
p,
IIAPII IlAYll -< cond(X) Itall IlYll
(7)
where llYll represents the Euclidian norm of the vector, Y (56), and likewise for the other vectors. IIA@IIand llAYll are norms of the errors in Y and 8, respectively, such that the ratios are simply the root mean squared relative errors in Y and 8. With the prior knowledge of the relative errors in Y, eq 7 can establish an upperbound for the relative error in 8. Ebel optimized a set of analytical wavelengths by minimizing only the condition number of X,which assumes the relative error
The variance and covariance factors, fll, fzz, f l z = fzl depend explicitly on the design matrix, X,which in turn depends on the choice of the wavelengths for the measurement. Because of their relationship to the variance of the estimated concentrations, 8, these factors are excellent criteria for both selecting analytical wavelengths and evaluating the error incurred for a particular set of wavelengths. For a two-component determination using N replicate measurements at a pair of analytical wavelengths, the variance factors can be shown using eq 10 to be proportional to the exactly determined, single measurement ( N = 1) of two wavelengths case by
where N is the number of measurements. The variance factors for the exactly determined system can therefore be used as the selection criteria for a pair of analytical wavelengths regardless of the number of replicate measurements or, more generally, for r = n analytical wavelengths for an n component mixture. Unlike the condition number wavelength selection, the variance factors are independent of spectral intensity, Y, and therefore need only be determined once for a given set of analyte spectra, X.
RESULTS AND DISCUSSION Variance Selection of Analytical Wavelengths. The variance of concentration estimates was evaluated using (XTX)-lfor a two-wavelength measurement in order to select analytical wavelengths. A contour plot of the sum of variance factors for components 1 and 2, (til+ f& for case B (A, = 45,55;R, = 0.25) is shown in Figure 2a. The contour plot presented here is a result of taking constant variance sections, or slices, through the three-dimensional error surface. The surface has minima (fIl + fzz) = 3.9 for the wavelength pairs [X1,X2]= [40,60] and [60,40] as indicated by the crosses; notice that these minima do not correspond to the Amax’s. The maximum for the surface occurs at [20,80] where (Ill+ fzz) = 3 x lo7;however for clarity the surface has been truncated at (fll + f z z ) = 20. A one-dimensional slice taken through
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985 w
1
a
20
20
30
40
60
30
*avelerlgth,A~
70
80
I
I
0' 20
I
30
I
I
,
I
40 50 60 70 Wavelength, A p
I
I
-
1.8 20
I
80
Flgure 2. (a) Summed variance contour plot for case B. Mimina are 3.90; the indicated by crosses, at [40,601 where ( f , , + f,,), increment between contour lines is 1.73. (b) Horizontal slice through (a)at wavelength, Ai = 40. Both (a)and (b) are truncated at (fli f,*), = 20.
+
30
40
50
60
Wavelength, A 4
70
80
Figure 4. (a) Summed variance contour plot for surface of Figure 3, with wavelengths 40 and 60 preselected. Minima, indicated by crosses at [40, 601 for ( f , , f 2 2 ) f = 1.95; the increment between contour lines is 0.87. (b) Two-dimensional slice through (a)at wavelength, A3 = 40.
+
4.0,
,
,
,
,
,
,
n
-
2.8 20
30
40
60
60
Wavelength, hg
70
80
Figure 5. Summed variance for the selection of the third wavelength, where [A,, A,] = [40,BO] are preselected.
Flgure 3. Three-dimensional variance factor surface, in which [A,, A,] = [40, 601 are preselected and A3 and A, are varied.
wavelength, A, = 40, is presented in Figure 2b, indicating that the minimum is very sharp and distinct at Az = 60. On the basis of eq 11, these "analytical" wavelengths represent the best possible choice with respect to the absolute errors in concentrationfor replicated measurementsat two wavelengths. While the error surface in Figure 2 identifies the pair of wavelengths which minimize the variance sum, one can also use the method to check the next best pair of Wavelengths and the sensitivity of the variance to their choice. Figure 3 shows the plot of the summed variance factors for the design matrix [A,, A,] = [40, 601 has been preselected, and the effect of a third and fourth wavelength on (fil +- fz2)1 is plotted. The contour plot, Figure 4a shows not surprisingly that [A,, A,] = [40, 601 is still the wavelength pair of lowest summed variance. The value of variance factor sum at the minimum is half the value of the two-wavelength measurement, which
is consistent with eq 11. The variance range for this surface, however, is much smaller than the two-wavelength case indicating less sensitivity to wavelength choice. The wavelength pairs near [A,, A,] [40,40] and [60,60] show local minima, see also Figure 5.3c, although not as distinct as the global minima at [A3, A,] = [40, 601 and [60, 401. This behavior is similar to the intermediate case where a third wavelength measurement is being selected as shown in Figure 5. In this figure, the summed variance is plotted vs. the third wavelength, AB, with [A,, A,] having been preselected as [40, 601 as before. The choice of the third wavelength indicates two degenerate minima at A, = 41 or 61, where the value of the variance factor sum is slightly larger than the 2.6 value predicted by interpolation of the two- and four-wavelength results. The cause for the minima not to be located at [40,60] is due to an asymmetry from the odd number of wavelengths being measured which also decreases the orthogonality of the design matrix. On selection of a pair of analytical wavelengths, the variance factor sum is made up of equivalent contributions from each component. The wavelengths [40,60] were consistently selected as the fifth and sixth, and subsequently (up to the 99th
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
2683
Table I. Estimated Variance and Mean for Concentrations of Multiwavelength and Replicated Analytical Wavelength Measurements case C, R, = 0.1
case B, R, = 0.25
case A, R, = 0.5 lOOX
2Xb
lOOh
2XC
lOOX
2Xd
0.49997’ 5.89 X lo4 0.49989 4.12 X 10” 6.52 X -2.40 X l0”f
0.50015a 2.1 x 10” 0.50001 2.10 x 10” 2.10 x 104 -0.46 X 10”
0.49994 12.15 X 10” 0.49989 9.26 X lo4 14.3 X 10” -11.2 x 10”
0.50020 4.20 X 10” 0.50001 4.04 X lo4 3.90 X 10“ -2.53 X 10“
0.49993 58.9 X 10“ 0.49989 48.3 X 10” 73.4 x 10“ -70.5 X 10“
0.50039 21.3 X 10“ 0.50001 19.2 X 10” 18.1 X 10“ -16.8 X 10”
“Result of 100 trials of the overdetermined case, true concentrations are 0.5000. *Wavelengths(39, 61) selected by f11 + f2z criterion. Wavelengths (40, 60) selected by fll + fzz criterion. Wavelengths (40, 60) selected by fll + fZz criterion. e uZpl= uZp2,therefore only u2b1is presented. u2p1 = f l l u 2 p ~ where , u2pE = (0.01)2(relative). f u e g 1 ~ = 2 f&p~. and 100th) analytical wavelengths as predicted by eq il. At the higher order pairs, however, the surfaces for the summed variance become more shallow, continuing the trend shown between Figures 2 and 4. While the optimum wavelengths for the measurement remain the same, the selection of wavelength affects the variance less as the number of measurement increases. To study this trend, the behavior of replicated measurements of this one wavelength pair was compared to that of a multiwavelength determination made with an equal number of measurements. Comparison of the Method of Replicate Analytical Wavelengths with Multiwavelength Determination. The use of a selected pair of wavelengths in a N = 50 replicate experiment was compared to an equivalent number of measurements over the entire absorption spectrum, and the results are summarized in Table I. The variance, ,:u derived from the variance-covariance matrix agree well with the sample variances, s2, determined from 100 numerical trials of the multiwavelength and replicated data sets. The improvement in precision obtained using analytical wavelength measurements increases as the component spectra become more overlapped. Table I also shows that most of the variance, especially for cases of severe spectral overlap, arises from the covariance contribution. While replicate measurements at the optimum pair of analytical wavelengths minimize the variance of the concentrations determined, such a strategy provides no information about lack of fit or variance in excess of pure error. Although knowledge of the constituent components is required in both the analytical wavelength and the multiwavelength cases, the residuals resulting from the least-squares fit of a multiwavelength measurement of Y are sensitive to a lack of fit from the presence of an unknown component, while the residuals in the analytical wavelength case are not. Figure 6a shows case B (R,= 0.25) with one added component whose spectrum (1- = 50) contaminates both components 1and 2. Since the analytical wavelength criteria are derived solely from the spectra of the target components, X, the same analytical wavelengths are selected regardless of excess error in Y from contamination. Both replicated Analytical wavelength and multiwavelength measurements give comparably large determinant errors in the solved-for concentrations due to the particular distribution contaminant intensity. Assuming a random probability of encountering contamination across the wavelength range, the analytical wavelength approach would, on the average, produce better accuracy than the multiwavelength method as shown by ita performance with random error. For the analysis of a mixture spectrum made up of equal concentrations from component 1,2, and the contaminant, the residuals in the multiwavelength case, Figure 6c, show clearly the presence of a component not included in the model. The replicated analytical wavelength method, Figure 6b, does not indicate any systematic error since the contaminant in-
0.05
b
a
ci 0.00 0
Ba?
1
a
-0.05
20
’40 60 60 Measurement Number
0
100
C
a
i f 0.0 --0.2
-
0
20
40 60 Wavelength
80
100
Figure 6. Contaminant study: (a)three-component system consisting of the spectra in case B and an addltional component at,,A = 50; (b) the residuals in Y for the replicated analytical wavelength determination (wavelength determination (wavelengths [40, 601 repeated 50 times);(c) the residuals in Y for the multlwaveiength determination. (Wavelengths 1 through 100 were measured once.)
tensity does not vary with measurement number. Furthermore, there are no degrees of freedom in the replicated, two-wavelength design matrix to test for error due to lack of fit. The standard deviation of the residuals in Figure 6b is equal to the Gaussian weighted noise added to the mixture spectrum Y showing no indication of the inaccuracy of the results, while the size of the residuals in Figure 6c are 50 times larger than the purely experimental error. In addition, the nonrandomness of the multiwavelength residuals would be easily detected by a partial autocorrelation criterion (58),for example. In conclusion, the variance factors from the variance-covariance matrix identify analytical wavelengths which consistently yield superior concentration precision, especially in cases of severe spectral overlap. The greatest impact of wavelength selection on the concentration variance occurs
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
when the total number of measurements is small. This improvement in precision is tempered, however, by the forfeiture of knowledge about the accuracy of the model describing the chemical system which can be obtained by a multiwavelength measurement over an entire spectrum. Even in carefully characterized chemical systems, the risk of contamination not revealed by lack-of-fit errors using a replicated r = n wavelength strategy might outweigh the improvement in the concentration variance. A proposed compromise between these two approaches would be to gather absorbance data over an entire spectrum weighing the time spent or number of measurements at a given wavelength according to its relative effect on concentration error. Further research will be required to assess the merits of this proposal.
ACKNOWLEDGMENT The authors are grateful to S. N. Deming, whose lecture notes on optimization and experimental design motivated this study. LITERATURE CITED (1) Hirshfeld, T. Anal. Chem. 1976, 48, 721-723. (2) Saguy, I.;Mizrahl, S.;Kopelman, I.J. J . Food Scl. 1978, 43, 121-134. (3) Giilette, P. C.; Lando, J. B.; Koenlg, J. L. Anal. Chem. 1983, 55, 630-633. (4)Honigs, D. E.; Hleftje, G. M.; Hlrschfeld, T. Appl. Spectrosc. 1984, 38, 317-322. (5) Arunachaiem, J.; Gongadhoran, S. Anal. Chlm. Acta 1984, 757, 245-260. (6) Barker, B. E.; Fox, M. F.; Hayon, E.; Ross, E. W. Anal. Chem. 1974,
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RECEIVED for review May 14,1985. Accepted July 12,1985. This research was supported in part by the donors of the Petroleum Research Fund, administered by the American Chemical Society, and by the National Science Foundation under Grant CHE 85-06667.
