Anal. Chem. 1983, 55,567-570
Table 111. Condition K and the (Determinant)'" of K K T for Experimentally Obtained K Matrix wavelength corn hination
cond K
(det. KKT)"'
10, 25, 28, 34 1 , 3 , 5 , 7)...,55
65.6 20.3 12.1
2.21 x 10' 1.28 X 10' 3.22 X 10"'
1-36
the responses and the relative errors in the estimated calibration constants can be multiplied by 65.6 to yield the relative error in no,The worst accuracy and precision were found for the four sensor system. This is reflected both in cond ( K )and the relative errors and the RSDs listed in Table 11. The best accuracy and precision were found when all 36 sensors are used for analysis of the four-component system. This is also expressed in the cond ( K ) shown in Table I11 and the relative errors and the RSD's listed in Table 11. It appears that cond ( K ) can be used as an indication for the stability of the system to noise perturbations while at the same time it can be used for a measure of accuracy and precision. That is, a minimum cond (ac)means minimum relative errors and RSD for no. Kaiser has shown (28) that it is possible to define the total sensitivity of a multicomponent procedure as the absolute value of the determinant of the calibration matrix K. Maximum sensitivity will correspond to a K matrix with large diagonal elements andl low off-diagonal elements. These same circumstances will minimize cond ( K ) . Hence a minimum cond ( K ) should be reflected as a maximum determinant. "he method of Kaiser is limited to the case of p = r. Junker and Bergmann (19)have generalized Kaiser's definition of sensitivity for p 2 r as the square root of the determinant of the product of the K matrix and its transpose. Again, we would expect that for p > r, maximum sensitivity, greatest determinant would also be represented as a minimum in cond ( K ) . This was found to be true for the present study. Shown in Table I11 are the determinants of K p for the three sets of sensors. The sensor set containing all 36 wavelengths is seen to have the minimum cond ( K ) and the maximum sensitivity. Consequently, it appears that the sensitivity of the calibration matrix K compliments the information contained in cond ( K ) . Jochum et al. (5) have also used the cond ( K ) to determine the optimal sensors for multicomponent analysis. This is accomplished by obtaining the minimum cond ( K ) for various combinations of the available sensors. Similarly, Junker and Bergmann (19)have used the sensitivity number as a method of optimization of a multicomponent system. Other methods for the choice of optimal analytical sensors in multicomponent systems exist (8-10); these involve tedious mathematical calculations and necessitate the direct comparison of several numbers rather than the comparison of single numbers as in the methods of the GSAM and Junker and Bergmann. With the cond ( K ) or determinant of K p as criteria for optimal
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selection of sensors, all 36 sensors should be used for optimal performance. This agrees with the experimental findings presented in Table 11. These results suggest digitization at a higher resolution for optimal performance. Namely, measurement at 1.0 nm intervals should increase the precision and accuracy accordingly. Other studies have been performed to show the optimal number of standard additions that should be made (20, 21). This paper describes a new approach for using the GSAM to perform analytical multicomponent analysis and ai new interpretation of the condition number of matrix K. It was demonstrated th,at, whenever possible, the best results are obtained when p > r , as expressed by cond (K). This paper is in agreement with a previous study (22)where similar results were found for a series of computer-simulated experiments and real data. The real data were those obtained by Jochum et al. ( 5 ) ,but only consisted of selected wavelength positions in the absorption curves rather than the complete absorption spectra as done here.
LITERATURE CITED (1) Saxberg, B. E.; Kowalski, B. R. Anal. Chem. 1979, 57, 1031--1038. (2) Kaiivas, J.; Kowalski, B. R. Anal. Chem. 1981, 53, 2207-221 2. (3) Gerlach, R. W.; Kowalski, B. R. Anal. Chem. Acta 1982, 734, 119. (4) Moran, M.; Kowalski, B. R., Laboratory for Chemometrlcs, Department of Chemistry, Unlversity of washington, unpublished work, June 1981. (5) Jochum, C.; Jochum, P.; Kowalski, B. R. Anal. Chem. 1961, 53, 85-92. (6) Kalivas, J.; Kowaiski, B. R. Anal. Chem. 1982, 54,560-565. (7) Kalivas, J.; Jochum, C.; Kowalski, B. R. Presented as paper NO. 440 at the Pittsburgh Conference, Atlantic Clty, NJ, March 1982. (8) Sustek, J. Anal. Chem. 1974, 46, 1676-1679. (9) Zscheile, F. P.; Murray, H. C.; Baker, G. A.; Peddicord, R. 0. Anal. Chem. 1962, 3 4 , 1776-1780. (10) Sustek, J.; Llvar, M.; Schiessl, 0. Chem. Listy 1972, 66, 168. (11) Certontaln, H.; Duin, H. 0.J.; Vollbracht, L. Anal. Chem. 196:3, 35, 1005-1007. (12) Milano, M. J.; Kim, K. Anal. Chem. 1977, 49,555-559. (13) Ratzlaff, K. L. Anal. Chem. 1980, 52, 1415-1420. (14) Hirschfeld, T. A@. Spectrosc. 1976, 30, 67-69 (15) Neter, J.; Wassorman, W. "Applied Linear Statistical Models"; Rlchard D. Irwln, Inc.: Homewood, IL, 1974;Chapter 6. (16) Naylor, T. H.; Baiintfy, J. L.; Burdlck, D. S.; Chu, K. "Computer Simulatlon Techniques"; Wiley: New York, 1966;Chapter 4. (17) Dahlquist, G.;Bjorck, A.; Anderson, N. "Numerical Methods", Pretntica Hall: Englewood Cliffs, NJ, 1974;Chapter 5. (18) Kaiser, H. Pure Appl. Chem. 1973, 34, 35-61. (19) Junker, A.; Berpmann, G. 2.Anal. Chem. 1974, 272, 267. (20) Franke, J. P.; de Zeeuw, R. A. Anal. Chem. 1978, 50, 1374-1380. (21) Ratziaff, K. L. Anal. Chem. 1979, 57, 232-235. (22) Kalivas, J.; Kowaiski, B. R., Laboratory for Chemometrics, Department of Chemistry, University of Washington, unpublished work, June 1982.
'
Present address: Laboratory for Chemometrics, Department of Chemistry, BElO, University of Washington, Seattle, WA 98195.
J. H.Kalivas' Department of Chemistry Montana State University Bozeman, Montana 59717 RECEIVED for review August 19, 1982. Accepted November 12, 1982. This work was supported by the Department of Chemistry, Montana State University, Bozeman, MT.
Temperature! Dependence of the Modified Pararosaniline Method for the Determination of Formaldehyde in Air Sir: There is currently considerable interest in methods for the measurement of formaldehyde in nonindustrial indoor environments, in particular, buildings insulated with urea formaldehyde foam insulation. The N.I.O.S.H. recommended method ( I , 2 ) employri chromotropic acid and sulfuric acid
for the analysis of the sampled air. The chromotropic ,acid method is relativelly insensitive and is potentially subject to interferences by both organic and inorganic compounds (3). Miksch et al. have recently published ( 3 ) a modified pararosaniline method. which has superior sensitivity for the
0003-2700/83/0355-0567$01.50/0 0 1983 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 3, MARCH 1983
Table I. Slopes and Intercepts from Calibration Lines Obtained from Pararosanaline Analysis of Standard Formaldehyde Solutionsa at 570 nm date 7120182 7/21/82 7/21/82 7/25/82 7/27/82 a
slope,b interAU.mL. cept,b fig-' AU 0.690 0.672 0.745 0.680 0.670
blank AU
0.110 0.104 0.149 0.167 0.145 0.096
0.135 0.160 0.142 0.104
R
lab temp, "C
0.9994 0.9998 0.9988 0.9995 0.9987
21.@ 25.5 28.3 25.5 21.0
Method of analysis as described in the text.
Line
of best fit determined by method of least squares.
analysis of
