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Anal. Chem. 1990, 62, 2220-2224
Nonlinear LeasbSquares Fitting of Multivariate Absorption Data Marcel Maeder* Department of Chemistry, University of Newcastle, Newcastle, New South Wales 2308, Australia
Andreas D. Zuberbuhler Institute of Inorganic Chemistry, University of Basel, 4056 Basel, Switzerland
The Introduction of fast scannlng and dlode-array spectrophotometers facllltates the acqukltlon of large series of absorption spectra as a functlon of reaction thne (klneUcs), elutkn tlme (chromatognrphy), or added reagenl (equlybrkm investlgatlons). I t Is Important to develop appropriate programs that are able to handle the wealth of data and to extract all informatlon. In this contrbutlon a new appllcaUon of factor anIn a nonlinear least-squares fitting algorkh Is presented. By prkr factor analydr of the raw mutlkhannel data, essentlal savings In memory aUocatlon and computing t h e are achleved. Separation of the llnear and the nodhear parameters Is accomplished and, specWcaWy, the computatkn of the Jacoblan Is dramatlcally enhanced by specifkally exploltlng the orthononnallty of the eigenvectors. The performance of the new algorithm Is compared wlth establlshed programs for examples of complex equmbrlum studles.
INTRODUCTION Multiwavelength or multichannel absorption detection is gaining increasing importance in very diverse fields of chemistry such as chromatography (1-5) or investigations on kinetic (6-9) and equilibrium systems (10-14). Two developments have enabled the widespread and advantageous application of these methods. On the hardware side, this is the commercial introduction of fast-scanning and diode-array spectrophotometers equipped with multipurpose interfaces for on-line data transfer and on the software side it is the availability of the appropriate programs to handle the large amount of data and to extract as much information as possible from the measurement within a reasonable time. These developments on the software side are results of chemometrics, a relatively new field of chemistry (15-17). For all types of multichannel spectrophotometric investigation, the raw data set consists of M spectra measured at W wavelengths. The individual absorbance readings thus can be arranged in a M X W matrix Y; the M spectra form the rows of Y. The columns of Y are the W response curves gathered at the different wavelengths. According to Beer’s law, for a system with N absorbing components, Y can be decomposed into the product of a concentration matrix C (M X N ) and a matrix of the molar absorptivities A ( N x W). However, because of the inherent noise in any measured data, the decomposition does not represent Y exactly. The matrix R of residuals is given by the difference between C A and Y
R=CA-Y
(1) In a fitting procedure, those matrices C and A are determined which best represent the original matrix Y. Generally the least-squares criterion is used to define the optimum. In most cases the chemist wishes to explain the measurements by using a chemical model and to determine the actual values of the parameters, i.e. the rate or equilibrium constants 0003-2700/9010362-2220$02.50/0
etc. The model is a function that describes the matrix C of the concentrations of all components in the system. In kinetics, integration of the differential equations of the kinetic model yields the function (18),in equilibrium studies it is the law of mass action (19), and in chromatography it is the function used to describe the concentration profiles (20-22). The parameters of the function, the rate or equilibrium constants in the cases of kinetic or equilibrium investigations or peak position, width and skewness in chromatography, are usually nonlinear parameters. Initial guesses for these parameters have to be iteratively refined until the best fit is achieved as indicated by the least-squares criterion. The elements of A, the molar absorptivities, are linear parameters, which can be dealt with differently (see below). The simplex algorithm and gradient method of the Newton-Gauss type are the most commonly used methods: the simplex algorithm being very simple, easy to understand and to program as well as relatively insensitive to initial guesses of the parameters (16);the Newton-Gauss method being more sophisticated but much faster and yielding not only optimal parameters but also statistical information such as the standard deviations of the parameters and the cross-correlation between them (23-26). There are many advantages in gathering whole spectra as a function of the independent variable and using multivariate methods for the data treatment relative to measuring at one single wavelength and reducing Y and A to column vectors. The most impostant points are as follows: (a) Factor analysis can be applied to get the number of components in the system under investigation, thus defining the complexity of the model needed to explain the data (27-31). Reasonable models or hypotheses can further be tested by target factor analysis (32-34). An additional feature of factor analysis is the possibility of data reduction which is important from the point of view of memory allocation and speeds up the calculations considerably (25, 26, 35). Even a completely model-free analysis of the original measurement is possible under certain conditions (1-4,11,36,37). (b) With measurements at many different wavelengths, the correlation between the nonlinear parameters defining the matrix C is greatly reduced and, as a consequence, the least-squares fit is much more robust and thus runs troublefree even in tricky cases ( 6 , 8 , 2 5 , 2 6 ) . (c) Intermediates, especially minor ones, are detected more reliably and the identification of unknown intermediates is supported by the availability of their absorption spectra. The drawbacks of multichannel detection are also evident: the large number (MX W) of data points to be handled, and the large number of parameters to be fitted which includes the nonlinear ones and the matrix A of linear parameters. To account for these problems, the combination of factor analysis and a sophisticated nonlinear least-square fitting algorithm is needed. Factor analysis is f i t l y used to reduce the number of data to be fitted, the original matrix Y of dimension M X W is replaced by a reduced matrix Y’ of dimension M X N , where N is the number of species in the chemical system (25). In addition, we propose an improved Newton-Gauss algorithm to specifically take advantage of the orthogonality of the 0 1990 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 62, NO. 20, OCTOBER 15, 1990
reduced matrix Y’. This has further advantages with respect to computing time, memory requirements, and program length.
THEORY OF NONLINEAR LEAST-SQUARES FITTING (a) General Considerations. The task of the fitting procedure is to optimise the matrix R of residuals (eq 1) according to the least-squares criterion. In eq 2, sq is the sum of the squares of all elements of R. I t is the task of the nonlinear least-squares fit to find the set of parameters that results in a minimal sq (23, 38). M W
sq =
C CR(iJ2= Min
i=’ j=1
(2)
(b) Data Reduction. By singular value decomposition
(SVD)( I , 32,39) the matrix Y of measurements is decomposed into the product of three matrices U, S, and V (eq 3). U,
Y = usv
(3)
S, and V have the following properties: U (M X N) is formed by the significant eigenvectors of Y Yt and V (N X W) by those of YtY. The columns of U as well as the rows of V are orthonormal, U t U = VVt = I (identity matrix). S ( N x N) is a diagonal matrix with the singular values in descending order as entries. The singular values are the square roots of the eigenvalues of the second moment matrices Y Yt or YtY. Generally N
