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This paper describes the estimation of reaction rate constants and pure species UV‐vis spectra of the consecutive reaction of 3‐chlorophenylhydrazonopropane dinitrile with 2‐mercaptoethanol. The reaction rate constants were estimated from the UV‐vis measurements of the reacting system using the generalized rank annihilation method (GRAM) and the Levenberg–Marquardt/PARAFAC (LM‐PAR) algorithm. Both algorithms can be applied in cases where the contribution of different species in the mixture spectra is of exponentially decaying character. From a single two‐way array, two two‐way datasets are formed by means of splitting such that there is a constant time lag between the two two‐way datasets. By stacking these two two‐way datasets, the reaction rate constants can be estimated very easily from the third dimension. GRAM, which is fast and non‐iterative, decomposes the trilinear structure using a generalized eigenvalue problem (GEP). The iterative algorithm LM‐PAR consists of a combination of the Levenberg–Marquardt algorithm and alternating least squares steps of the PARAFAC model using GRAM results as a set of initial starting values. Pure spectra of the absorbing species were estimated and compared with their measured pure spectra. LM‐PAR performed the best, giving the lowest relative fit error. However, the relative fit error obtained with GRAM was acceptable. Since a lot of measurements are based on exponentially decaying functions, GRAM and LM‐PAR can have many applications in chemistry. Copyright © 1999 John Wiley & Sons, Ltd.

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Two cross-validation methods are presented for multiway component models. They are used for choosing the numbers of components to use in Tucker3 models describing three-way data. The approach is general and can easily be adapted to other three-way and multiway models. A model is estimated after leaving out a small part of the multiway data array. The predictive residual error sum of squares (PRESS) is calculated for the eliminated part of the data by comparing the model values with the actual data. PRESS of the entire data set can be calculated like this sequentially. The methods are the leave-bar-out cross-validation method, which leaves out data slices in all modes, and the EM cross-validation method, which handles eliminated data as missing values. A method to calculate the statistical significance of the PRESS reduction for an additional component, the so called W-statistic, is provided for Tucker3 models. A strategy is proposed to search along an efficient path, to reduce computation time, since the number of feasible models as a function of the total number of components summed over the modes increases fast. monitoring batch processes, for process optimization and to improve process understanding. Other applications areas are image analysis 6,7 examining complex biochemical systems, 8 multiway calibration in three-dimensional quantitative structure-activity relationships (3D-QSAR) to predict the activity of potential drugs, 9 and modeling a five-factor factorial design of enzymatic activities with a five-way PARAFAC model. 10 Two specialized international meetings on three-way methods have taken place already (in 1993 and 1997), showing that the interest in multiway methods and their application areas is growing.The tools for handling multiway data are mainly generalizations of principal component analysis (PCA), singular value decomposition (SVD) or factor analysis (FA) of two-way data. Roughly, these generalizations can be divided into parallel factor analysis (PARAFAC) models and Tucker models, where the latter can be subdivided into Tucker1, Tucker2, Tucker3 (three-mode PCA) and, more recently, constrained Tucker models. Kroonenberg 11 explains the Tucker1, Tucker2 and Tucker3 models extensively in an excellent book on this subject. Constrained Tucker models are explained by Kiers, Smilde and co-workers. [12][13][14][15] The PARAFAC and Tucker models generalize certain aspects of PCA and FA. The Tucker models are very versatile. Nomikos and MacGregor 3,4 use a Tucker1 model for MSPC, de Ligny et al 16 use a Tucker3 model, and when certain constraints are known beforehand, a Tucker2 model or a constrained Tucker3 model can be used. Kiers 12 and Smilde et al 13 show the use of PARAFAC and Tucker3 models in second-order calibration, and how to translate specific (chemical) knowledge that is available in constraints on the Tucker3 model. Only specific interactions between the three modes are allowed, simultaneously with non-negativity constraints for specific modes.One of the major problems of using these m...