Conventional spectrophotometric analysis generally measures the signal as dependent variable vs. wavelength or energy as the independent variable. Addition of a second independent variable in the detection system, such as time, for example, greatly increases the information available for each component in a mixture. This is called multichannel or multivariate detection. The larger amount of information present in the data facilitates the mathematical resolution of the signals of the individual components.
An example of a method featuring multivariate detection is time-resolved fluorescence , In the usual algebraic approach used in this book, the model for time resolved fluorescence of a mixture of n components can be expressed as
I(t,A) = measured fluorescence intensity as a function of A = wavelength (A) and time (i),
A(X)k = proportionality constant between intensity and concentration for the &th component in the mixture, C0,k = concentration of the k\h component in the mixture, Tk = decay lifetime of the fcth component in the mixture.
For a completely unknown system, we would need to estimate the number of components n, and the set of A(X)kC,xk, and rk. Knorr and Harris  have applied nonlinear regression to the resolution of overlapped peaks in time-resolved fluorescence spectra by using a matrix approach, which is well suited to dealing directly with the multivariate data sets. The approach of these authors is summarized next.
The data consisted of fluorescence spectra of a mixture of n compounds collected at nanosecond time intervals following excitation by a pulse of light. The data were placed in a w X t matrix [£>]. with one row for each wavelength observed and the t columns corresponding to the number of time intervals used. A linear response of the detector with changing concentration of each of the n independently emitting compounds was assumed. The elements of this data matrix at wavelength i and time interval j can be written
where Ai k is the molar emission intensity of the klh compound at wavelength i, and Ckj is the concentration of the &th excited state compound at time interval j. The data matrix was expressed as
[y4] = w X n matrix containing the fluorescence spectra of the n compounds,
[C] = n X t matrix containing the decay of excited state concentrations with time.
Resolution of the spectral data into its individual components requires decomposition of the data matrix into [A] and [C], Assuming first-order decay kinetics for all excited states, a trial matrix [C'] was defined by
where /, = measured time response function of the instrument and * represents the convolution of the two functions (see eqs. (14.1) and (14.2)).
Equation (14.6) contains one parameter for each compound, tk. Initial guesses of these Tk provide an intial [C']. The best spectral matrix [A'] for these parameters is given by
The [A'] found from eq. (14.7) for a given [C'] minimized the error sum S expressed in matrix form (Section 2.B.3). This error sum is equivalent to the usual sum of squares of deviations between the experimental data and the model. The model can be expressed as
The regression analysis employed the n lifetimes as parameters and the minimum in the error sum S was found by using the modified simplex algorithm. Because negative elements in [A'\ are not physically reasonable, S was multiplied by a function that penalized the occurrence of any such negative elements during the analysis.
The method was evaluated by using noisy simulated data representing a two-component mixture. Real samples containing two fluorescing compounds were also analyzed. Better precision was found for lifetimes and pre-exponential factors when compared to analysis of decay data at a single wavelength . The number of compounds present in the mixture was found by comparing goodness of fit criteria for a series of models with successively increasing integer n values. The n value providing the smallest minimum error sum was taken as the best model (cf. Chapter 6). This type of analysis would also benefit from employing the extra sum of squares F test (Section 3.C.1).
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