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ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
Application of the Method of Rank Annihilation to Quantitative Analyses of Multicomponent Fluorescence Data from the Video Fluorometer C.-N. Ho, G. D. Christian, and E. R. Davidson" Department of Chemistry, BG- 10, Universlty of Washington, Seattle, 1Vashington 98 195
A new scheme for quantitative analysis of a multlcornponent fluorescent mlxture uslng the ExcHatlon-Emlsslon Matrix (EEM) acqulred by the vldeo fluorometer has been developed and tested. Thls method, rank annlhllation, Is capable of quantlfylng a partlcular component known to be present In the mlxture wlthout havlng also to know the identlty of the rest of the components. The theory of the method and the results of Its appllcatlon to a mlxture matrix are presented and comparison Is made with the method of least squares.
T h e need for reliable quantitative analysis of multicomponent systems has prompted many workers to develop new instrumentation capable of quickly acquiring data from which the identities and concentrations of the components can be readily extracted. Recently, a video fluorometer was described (1)for accomplishing this with fluorescing components. This instrument acquires an excitation-emission matrix (EEM) which has been shown by Warner et al. to be useful for qualitative (2) and quantitative (3) analyses of fluorescent mixtures. The method of least squares used by Warner et al. (3)and by Sternberg e t al. (4)is conceptually simple and easy to implement. For quantitative multicomponent analysis, however, the method of least squares yields predictably reliable results only if one has knowledge of all the major constituents present. Consequently other methods such as linear programming (31, non-negative least squares (5) and factor analysis (6) have been suggested as possible algorithms. In many analytical problems, one is confronted with an analyte which contains a few known fluorescing species of interest mixed with other fluorescing unknowns. In such cases, it would be very convenient if one could obtain quantitative information for the known compounds without having to worry about the other species present. The method of rank annihilation proposed here offers a promising approach to this problem when combined with the video fluorometer. T h e method of rank annihilation qualitatively can be described as follows. For a multicomponent solution emission-excitation matrix, M, the rank, ideally, should equal the number of components. If we know one of the components with E E M N present in the solution, and if we subtract the correct amount of N from M, the original rank of M should be reduced by one. In such an instance, we should observe the eigenvalue of M corresponding to N becoming zero. Because of errors in actual experimental data, we cannot expect the eigenvalue to vanish completely. However, it will attain a minimum. The amount of N subtracted to achieve a minimum in the corresponding eigenvalue will correspond to the relative concentration of the known component in the mixture. EXPERIMENTAL The video fluorometer previously described was used for acquiring the fluorescent data. The EEM was obtained by 0003-2700/78/0350-1108$01 .OO/O
summing 512 video frames of the fluorescence, followed by subtraction of the same number of frames of dark current. A scattered light EEM for the pure solvent was similarly acquired. This scattered light EEM was subtracted from each EEM prior to mathematical analysis. Data Analysis. The data produced by the video fluorometer were written onto floppy disks of the mini-computer system. The data were then transmitted to the CDC-6400 and mathematical reduction performed by a set of FORTRAN-IV programs. Reagents. The chemicals used in this study were zone refined perylene and anthracene (both from James Hinton, Valparaiso, Fla.), dissolved in spectral grade cyclohexane (MCB). THEORY OF RANK ANNIHILATION For a single-component the E E M is ideally of the form
where xi is proportional to the number of photons emitted at wavelength hi and y, is proportional to the number of photons absorbed at wavelength A,. If the xi and yl are normalized so that Z x: = I: y j = 1 and if the absorbance is low for all wavelengths, then the number a is independent of A, and X j and is proportional to the concentration. For a mixture of r components under ideal conditions
where Mck)is the matrix which would have resulted if component k were present alone. If Equations 1 and 2 are obeyed, then M can be written as
(3) where xk is the column vector containing the x i and Yk is a column vector containing the y, for component k . The superscript T denotes the operation of matrix transposition throughout this paper. An M of this form has rank r if the ( x & =and ~ ( Y k ) i = 1 are, respectively, sets of linearly independent vectors. Any matrix M of real numbers with dimension K X L and rank R can be written in the biorthogonal expansion
(4) Here the vectors uk and vk are defined by
or equivalently, by the matrix eigen-equations
in which the
appear as eigenvalues and the vk (or u k ) as
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 8, JULY 1978
eigenvectors. The (uk)and (vk)separately form orthonormal sets and the expansion
for S I R gives the best least squares approximation to M of rank S. Consequently, for a matrix M of the form of Equation 3 with r
