2091
Anal. Chem. 1991, 63,2091-2094
Vibrational Absorption Intensities in Chemical Analysis. 5. Constrained Factor Analysis Richard 5.Emmence and Derek Steele* Department of Chemistry, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW2 OEX, England
The use of factor analysis Is explored for determlnlng the intendty distributlon arlsing from speclfic functional groups. A nonorthogonal trandormatlon to produce positive absorbances and concentrations ylelds recognizable CH, and CH, spectra when applied to n-alkanes. Thls is a partlculariy favorable case, however, since the group contributions are known to be additlve to wHhln experimental error. Even so, dgnificant errors occur due to the absence of speclflc frequencles at which only one grouping absorbs. The effect Is most marked on the elgenvectors (ioadlngs), whlch fall to show proportlonailty to the concentratlons of the groupings in the compounds. Constraining the eigenvectors (in a leastsquares sense) to the form appropriate to the relevant concentrations leads to excellent separations. The potentlal use of the technlqtm to detmine the lntenslty dlstributlons arlsing from other structural groups Is discussed brlefly.
INTRODUCTION In previous studies by our group (2-3) the object has been to establish the intensity arising from various structural groups of atoms throughout the fundamental region of the infrared spectrum. This has been attained by least-squares analysis fitting of the spectra to the number present of each structural group. Such a procedure involves the assumption that the spectral contributions are strictly additive. For normal hydrocarbons such additivity of CH2 and CH, intensities was first recognized by Rose (overtone region) ( 4 ) and Fox and Martin (fundamental region) (5). In branched hydrocarbons interaction between the CH2 and CH3 units is substantial, though the integrated intensity over the CH stretching region is reasonably proportional to the number of CH bonds. In the HCH bending (scissoring) region, splitting of the umbrella vibration near 1370 cm-' is well-known and characteristic for the isopropyl and tertiary butyl units. In addition to these characteristic modes, intensity appears also in the C-C stretching and HCC bending regions as a consequence of the removal of phase effects down the CH2 chain. In a similar manner, further chain-branching results in increasing complexity of the observed spectrum. Some success has been achieved in reducing the spectral intensities of the branched hydrocarbons to a sum of additive group components (5-7). It is not clear, however, that the optimum model has been employed. For instance we have distinguished CH3 and CH2 units according to whether they are adjacent to primary, secondary, or tertiary carbons. In another model entities such as -CH2-CH(CH3)-CH2 have been considered as functional groups (7). Factor theory (8) can be used to identify the number of significant additive contributions to a series of data sets. As such, it has a significant role to offer in this problem. In its basic form the results are not readily interpretable in terms of functional group contributions. We shall begin by dis-
cussing briefly the factor method and a procedure by which spectra of "additive units" may be extracted. Application to the n-alkanes will then be described. Various problems will be identified, which leads us to introduce a constrained procedure to separate mixed contributions from recognized contributing groups.
THEORY Consider a data set D consisting of n column vectors of dimension nwave (or n spectra a t nwave wavenumbers, presented as columns). The covariance matrix Z (n X n) is defined by
Z = DtD (1) where t denotes the matrix transpose. Let Q (n X number of eigenvectors, m) and A (m X m) be the eigenvector and eigenvalue matrices of Z, respectively. ZQ = AQ (2) Then an eigenspectrum matrix S (nwave X m) can be defined bY
S=DQ
* To whom correspondence should be addressed. 0003-2700/91/0363-2091$02.50/0
(3)
It can be shown (9) that selection of the m highest eigenvalues and their corresponding eigenvectors from the matrix Z of order n, m < n, is equivalent to least-squares fitting of the data set, D, to m vectors. Thus for m = 1 the optimum fit is the mean, and the first eigenspectrum s1 = Dq, is just that. If the original data set is perfectly additive in j components, then the original data set will be reproduced by the first j eigenvectors. The contributions of each eigenspectrum to D is given by the corresponding elements of Q . In the case of real spectra, noise and nonadditivity corrections (due, for example, to collisional perturbations) result in additional non-zero terms to A. Recognition of the number of significant contributing factors (or effective rank) can be achieved in a number of ways, the most reliable of which is probably the determination of the number of vectors of Q necessary to reproduce the data to acceptable accuracy-e.g. 3 times the spectral noise limit. If the object is to extract the spectra of the constituent components of the mixture, then the eigenspectra must be transformed. sl, as we have seen, is the mean of the original spectra, d, (the spectral vectors from which D is constructed). s2 then is that spectral vector which yields alone the best correction to the mean. As such it consists of positive and negative contributions. Real absorption spectra however are everywhere positive. Visual projection procedures are commonly employed to extract real components in available software. Muller and Steele (IO) have discussed the problem and developed an algorithm for extracting real components under computer control. The matrix S is postmultiplied by a matrix T of the form
0 1991 American Chemical Society
2092
ANALYTICAL CHEMISTRY, VOL. 63, NO. 19, OCTOBER 1, 1991
where x i are parameters to be varied to generate minimum negative spectral elements sii (in a least-squares sense). Additional constraint is also imposed by requiring that the transformed factor concentrations, T-'Qt, are also positive. This is accomplished by generating a vector of length equal to the sum of the total number of spectral elements and of the product of the number of factors and the number of spectra. The latter product is the number of terms in T-'Qt. All elements of the vector are initially zeroed, and where a negative absorbance or concentration element is encountered, the corresponding vector element is set to this value. It is required that the parameters be varied to minimize the length of this vector. It has been shown (IO, 11)that the procedure is effective in separating the IR spectra of mixtures. In the work reported here a routine E04FDF from the NAG library was used. This routine has since been found to be unstable in some cases when used for the present type of minimization. This appears to arise from the cutoff in the error vector terms as they move from negative to positive, and thereby generating discontinuities in the derivatives. We now employ a simplex routine, as the simplex procedure does not depend on smoothly varying derivatives of the function to be minimized with respect to the parameters. Clearly, factor theory is a potential tool for exploring the feasibility of generating the spectra of a group of related compounds by the use of a smaller subset of spectra. Transformation in the sense described above to generate positive absorbances and concentrations may hopefully lead to spectral components that can be associated with recognizable groupings of atoms. This paper reporb an exploration of this idea using a real, but near-ideal, case-the homologous series of n-alkanes. BACKGROUND T O THE SPECTROSCOPY I t has long been recognized that in the homologous series of n-alkanes significant spectral absorption occurs near to only eight spectral centers, four associated with the CHz groupings and four with the CH3 (see ref 12, Chapter 1). Rose ( 4 ) , Francis ( 6 ) ,and Hastings and colleagues (13)demonstrated that the concentration of CH2 and CH3 units could be estimated to within a few percent from the group absorption intensities. In an earlier work in this series it was argued that, to extend such investigations to a wider group of molecular structures, it was advisable to integrate absorbances over spectral regions larger than the bandwidths and than the frequency shifts generally experienced through intermolecular interactions. This also has the merit of reducing the data vectors to manageable sizes without the unsatisfactory procedure of discarding the vast majority of data points. The chosen range for integration was 25 cm-'. The spectral absorbances of CH2 and CH3 units in a variety of X(CHz),CH3 situations (X = -CH3, -CHO, -(C:O)CH3) were reported by using least-squares fitting procedures. In all such cases it is implicitly assumed the absorbances could be assigned to individual CH2, CH3, or X units. Work on branched alkanes have, not surprisingly, shown the need for larger units, and measured spectra have been interpreted by using a number of models (7). It is not clear, however, on the basis of the procedures adopted that such models are optimized. EXPERIMENTAL PROCEDURE The data processed here are the same as in the earlier work and are restricted to spectral regions 3075-2675 and 1500-1325 cm-I. The only significance of these range selections is that the absorbances outside are almost zero (within noise level),or in the case of the wagging region, 750-700 cm-', the absorbances were masked by the CC14absorption. Spectra were measured in tetrachloromethane at 0.1- or 0.2-mm path lengths by using concentrations between 3 and 20% on a Perkin-Elmer 983 instrument. All spectra were checked at a minimum of two concentrations and
Table I. Initial and Final Values of the T Matrix Variables
initial set
final
X1
x2
1 2 3
10.0 1.0 0.1
10.0 2.0
4
-0.059
5
-1.0
-3.093 -2.0
0.2
311
312
-0.059 -0,059 -0.059 -0.059 -0.029
-4.330
-3.554 -0.923 -3.093 -2.289
normalized to molar-integrated intensities. Spectra resolutions were about 4 cm-' (mode 5). Five n-alkanes, CH3(CH2)$H3 (n = 4,7,8,10, and 14)were available. All computer programs were written in Fortran 77 and developed by the authors from the program of Muller and Steele (IO). RESULTS The number of factors was unambiguously 2. The IND function of Malinowski (14) shows a clear minimum. A generally more useful measure of the significance of additional factors is whether or not the inclusion of an additional factor leads t o a significant improvement in the data reproduction-significant, that is, with respect to the noise level associated with the data. It is a measure of the quality of the data that two factors led to a data fit as defined by
D = SaQt of better than 0.3% at any frequency interval. This is significantly better than the authors would have claimed for the reproducibility of their data. S2represents the eigenspectrum matrix of the major two spectral vectors. It has been shown (IO)that a unique matrix T-and hence a unique set of transformed factor spectra-requires that there exist one frequency for each component (or all but one) that it alone absorbs at. The extent to which this condition is not met determines the range of nonunique solutions or mixing of the true component spectra. It follows then that the converged transformed eigenspectra, S,T are dependent on the route to convergence, that is on the initial guesses to the transformation matrix terms. With this in mind the set of five spectra were transformed by using a wide range of initial elements. As seen in Table I, x 2 in particular varied significantly depending on the initial guesses. Nevertheless, the transformed eigenspectra (Tables I1 and IV) are remarkably constant for all five solutions and generally match well to the spectra given by Steele (2) for the CH2 and CH3 groupings (second columns of Tables I1 and IV). The principal discrepancies are a t 2975 cm-' for the CHz-matched eigenspectrum, and in the case of the CH,-like spectrum, solution 5 is generally poor. In the former case the contribution at 2975 cm-' is clearly due to a mixing in of some CH3 component, intensity at this wavenumber dominating the CH, spectrum. The CH2spectrum is very intense in the 2950- and 2925 cm-' blocks, and solution 5 has clearly left a great deal of CH2mixed in the CH,. Set 4 deserves comment. This is the set resulting from application of additional constraints (see below). That it is a perfectly stable solution in the absence of the Constraints is readily apparent. In light of the fact that we are dealing with an almost ideal case, such failures of separation are of considerable concern. An examination of the eigenvectors (or loadings in factor theory language) emphasizes the problem. If the first and second transformed eigenspectra are indeed the CH2and CH3 spectra, respectively, then the eigenvectors should represent the relative concentrations of these groups in the molecules. For the "CH2" eigenvector the results are good for solutions 1-4 in that the eigenvector terms divided by the number of CHis are reasonably constant (*3%) (Table 111) whereas for starting set 5 the results are far less satisfactory (Table V).
ANALYTICAL CHEMISTRY, VOL. 63,NO. 19, OCTOBER 1, 1991
Table 11. Eigenspectra for CH2 (Original Program) wavenumber/ cm-l
intens values (ref 2)
3075 3050 3025 3000 2975 2950 2925 2900 2875 2850 2825 2800 2775 2750 2725 2700 1500 1475 1450 1425 1400 1375 1350
9 14 22 39 161 1140 950 276 599 232 42 20 14 14 12 15 11 136 49 10 12 37 16
Table V. Eigenvectors for CH, (Original Program)
intens values for different starting points set 1 set 2 set 3 set 4 set 5 6 10 15 13 1 1140 952 251 581 235 40 19 12 12 13 17 7 118 45 8 3 26 16
8 13 19 29 99 1140 948 266 591 233 41 20 13 13 13 17 10 129 47 9 8 27 16
13 21 35 90 474 1140 930 322 626 222 47 23 15 17 15 17 19 171 55 14 29 29 17
10 16 27 57 271 1140 940 292 607 228 44 21 14 15 14 17 14 148 51 11 18 28 17
9 14 22 39 160 1140 945 275 596 231 42 20 13 14 14 17 11 136 49 10 12 27 16
Table 111. Eigenvectors for CHz (Original Program) starting point 4 7 8 10 14
eigenvector/N eigenvector/N eigenvector/N eigenvector/N eieenvectorlN
1
2
3
4
5
0.038 0.038 0.040 0.038 0.039
0.039 0.039 0.041 0.040 0.040
0.045 0.045 0.048 0.046 0.046
0.042 0.040 0.042 0.041 0.041
0.004 0.022 0.026 0.030 0.035
Table IV. Eigenspectra for CH8 (Original Program) wave-
intens
number/ cm-I
values (ref 2)
3075 3050 3025 3000 2975 2950 2925 2900 2875 2850 2825 2800 2775 2750 2725 2700 1500 1475 1450 1425 1400 1375 1350
39 53 99 390 2390 260 105 415 360 0 42 28 16 29 18 12 60 293 60 33 134 13 10
intens values for different starting points 1
2
3
4
5
39 54 103 389 2390 307 144 427 383 0 44 29 19 29 13 3 60 296 61 33 134 19 11
39 54 103 389 2390 307 144 427 383 0 44 29 19 29 13 3 60 296 61 33 134 19 11
39 54 103 389 2390 307 144 427 383 0 44 29 19 29 13 3 60 296 61 33 134 19 11
39 54 103 389 2390 307 144 427 383 0 44 29 19 29 13 3 60 296 61 33 134 19
48 70 130 409 2390 2110 1650 824 1303 372 107 59 38 48 34 30 71 483 133 45 138 61 36
11
2093
For the “CH3”eigenvectors the terms ought to be all equal, since all the malkanes contain two CH3groups. In this case set 3 is very poor (Table V) with the number of methyls decreasing monotonically by an overall factor of 3 from hexane to hexadecane. Quite remarkably however the “CH,”eigen-
no. of
starting point
CH2, N
1
2
3
4
5
4 7 8 10 14
1.362 1.502 1.613 1.622 1.783
1.265 1.332 1.408 1.376 1.435
0.875 0.645 0.580 0.383 0.311
1.204 1.225 1.279 1.221 1.216
0.744 0.699 0.708 0.636 0.551
Table VI. Eigenspectra for CHI and CHSwith Eigenvectors Constrained as in Eq 4 with a and b Varying wavenumber / cm-I 3075 3050 3025 3000 2975 2950 2925 2900 2875 2850 2825 2800 2775 2750 2725 2700 1500 1475 1450 1425 1400 1375 1350
CHZ __ __
CHS
ref 2
calc
ref 2
calc
9 14 22 39 161 1140 950 276 599 232 42 20 14 14 12 15 11 136 49 10 12 37 16
9 14 22 39 162 1140 945 275 596 231 42 20 13 14 14 17 11 136 49 10 12 27 16
39 53 99 390 2390 260 105 415 360 0 42 28 16 29 18 12 60 293 60 33 134 13 10
39 53 103 389 2390 307 144 427 383 0 44 29 19 29 13 13 60 296 61 33 134 19 11
spectrum is virtually unaltered throughout starting seta 1-4, even though the contribution of the methyls is barely visible in the higher n-alkanes. This substantial insensitivity of transformed spectra to the eigenvalues has also been noted in separating spectra of mixtures (IO, 11). These observations prompt the question “If a given eigenspectrum can be recognized as pertaining to a given structural group, will a constraint of the eigenvector improve the overall transformation?” In Table VI the results are shown of expanding the condition for optimum transformation to a minimum of the resultant of (i) the sum of the squares of the negative terms of the transformed eigenspectrum, (ii) the sum of the squares of the negative concentrations (loadings), and (iii) (new condition) the sum of squares of the deviations of the eigenvector terms from the form
(4)
where n&, = number of methylene units in the j t h alkane; a and b were taken as the mean of the eigenvector elements from previous unconstrained calculations for that vector identified as corresponding to the CH2and CH, groups, respectively. Only a marginal improvement is seen and s2 in particular is disappointing with a range of 2 in the eigenvector components. T o accommodate condition iii the length of the vector was increased by 2n.
2094
ANALYTICAL CHEMISTRY, VOL. 83, NO. 19, OCTOBER 1, 1991
Table VII. Eigenvectors for CHI and CH3 with Eigenvectors Constrained As Defined in Table VI
no. of CH,, N eigenvector 1, CH2 eigenvector 1/N eigenvector 2, CH,
4
7
8
10
14
0.160 0.040 1.202
0.282 0.040 1.222
0.339 0.042 1.275
0.408 0.041 1.217
0.576 0.041 1.210
Table VIII. Eigenvectors Arising from Conditions 1-3O with Varying Weighting Given to (3)
no. of CH,. N weighting 1.0 0.1
10.0
a
4
eigenvector (CHJ/N eigenvector (CH,) eigenvector (CHd/N eigenvector (CH,) eigenvector (CHz)/N eigenvector (CH,)
7
8
10
14
0.040 0.040 0.042 0.041 0.041 1.204 1.225 1.279 1.221 1.216 0.040 0.040 0.042 0.041 0.041 1.203 1.222 1.275 1.217 1.210 0.038 0.038 0.040 0.038 0.039 1.363 1.504 1.615 1.625 1.787
See text.
The factors a and b were then allowed to vary as part of the optimization routine. The results are shown in Tables VI and VII. Now the terms of s2 are constant to h370,and the accord between the results of ref 2 and the new transformed eigen component intensities is excellent. We recognize that the relative weightings of the data fit and of the desired eigenvector fits may need to be varied in order to produce the optimum results. Table VI11 shows the variations in the eigenvector values caused by first decreasing the weighting of the new condition by a factor of 10 and then increasing the weighting by a factor of 10. Decreasing the weighting had no significant effect on the eigenvectors or eigenspectra. However the increase of the factor caused a noticeable change in the eigenvectors and produced a CH3 eigenspectrum in which agreement with results from ref 2 was slightly worse. In
general, for equal weighting of all terms, a weighting factor equal to the ratio of the average spectral element to the average loading will be required. In conclusion, a method is proposed for determining additive intensity components on the basis of chemically recognizable entities. This involves (i) the evaluation of the number of significant factors, (ii) transformation of the eigenspectra using a constraint of minimum negative absorbances and concentrations, and (iii) assuming that an eigenspectrum arising fom a specific atomic grouping is recognized, through perhaps its spectral distribution and/or its integrated intensity in certain regions and/or the form of its eigenvector, constraining the relative values of the eigenvector elements. This method will be demonstrated in a later publication applied to branched alkanes.
ACKNOWLEDGMENT The program used was modified from one produced by Anna Muller, whose comments and cooperation are gratefully acknowledged.
LITERATURE CITED Steele, D.; Hamitton, K.; Stevens, J. Croat. Chem. Acta 1988. 67, 551. Steele. D. Spectrochim. Acta 1988, 44A, 1255. Muller, A.; Steele, D. J . phvs. Chem., in press. Rose, F. w. J . Res. &ti. B U ~ .stand. 1938, 2 0 , 129. Fox, J. J.; Martin, A. E. Roc. R . SOC.1940, A775, 208; IbM. 1940, A775, 234. Francis, S. A. J . Chem. phys. 1950, 78, 861. Parker, M.; Steele, D.unpublished work. See for example: Martens, H.; Naes. T. Multivariate Cai/bratlon; John Wlley and Sons Ltd.: Chichester. U.K., 1989. Muller, A. Fh.D. Thesls, University of London, 1991. Mulier, A.; Steele, D. Spectroch/m. Acta 1990, 46A. 817. Muller, A.; Steele, D. Spectrochim. Acta 1990, 46A, 1177. Bellamy, L. J.; The Infrared Spectra of Complex M k u l e s , 3rd ed.; Chapman and Hall: London, 1975; Vol. 1, Chapter 1. Hastings, S. H.; Watson, A. T.; Williams, R. 8.; Anderson, J. A,, Jr. Anal. Chem. 1952, 2 4 , 611. Maiinowski, E. R. Anal. Chem. 1977, 4 9 , 612.
RECEIVEDfor review April 17, 1991. Accepted July 3, 1991. This work has been supported by Perkin-Elmer LM., to whom we are very grateful.
