Second-Order Calibration Based on Alternating Trilinear Decomposition: A Comparison with the Traditional PARAFAC Algorithm.

Abstract

Quantitative analysis of p-chlorotoluene in the presence of o-chlorotoluene as an interferent was attempted by using HPLC with diode array detection and second-order calibration methods. An emphasis was placed on a further comparison between two methods, i.e., the traditional parallel factor analysis (PARAFAC) and the alternating trilinear decomposition (ATLD) methods. The following conclusions have been confirmed for the calibration examples tested. PARAFAC does not always converge to chemical meaningful solutions and its convergence rate is slow for the . This seems to be owing to the deficiency of a real trilinear sense in PARAFAC. The convergence rate of the ATLD algorithm is extremely fast. The ATLD-based calibration not only retains the advantage of second-order calibration but can give satisfactory concentration predictions.Keywords Second-order calibration, trilinear model, PARAFAC, alternating trilinear decomposition, singular value decomposition, Moore-Penrose inverse Second-order calibration has an advantage that quantitative analysis can be performed even in the presence of uncalibrated interfering species.'-9 Second-order calibration is usually performed by decomposition of a three-way data array and prediction of unknown concentrations based on regression of the relative concentration contributions of each component of interest in sample space against its standard concentrations. The three-way data array is composed of the responses from both the calibration samples and the unknown mixture samples. The typical methods are the direct trilinear decomposition (DTLD) 7, 8 and the PARAFAC algorithm.'0-17 However, theoretically, DTLD seems to be useful only when the cumulative percent variance of the first two principal components is close to 100% and otherwise other errors would be introduced because the number of factors in sample space is artificially restricted to two.The PARAFAC/CANDECOMP algorithm is only based on a series of the frontal slice matrices and includes both the three procedures of computing inverse matrices and the four addition procedures of a series of matrices containing plus or minus elements in each iterative cycle. Hence, the traditional PARAFAC algorithm is guaranteed to reduce the residual sum of squares at each iteration, but it does not always converge to chemical meaningful solutions. 14, In addition, as reported by most authors the convergence rate of the traditional PARAFAC algorithm is rather slow.16, 17 We recently developed an alternating trilinear decomposition method (ATLD) as an alternative method for decomposition of threeway data array. 18, 19 It has been found that the ATLD algorithm proposed by us has a capability to converge the iterative process more rapidly than the traditional PARAFAC algorithm does. The ATLD-based second-order calibration retains

Simultaneous Fluorometric Resolution of Pyridoxal, Pyridoxamine and 4-Pyridoxic Acid Using Chemometric Methods

“…Chemometric methods, especially three-way resolution and calibration methods, [10][11][12][13][14][15] can solve the problem of closely overlapping fluorescence spectra easily. These methods [10][11][12][13][14][15] utilize a "mathematical separation procedure" to substitute the traditional chemical separation procedure, which are different from those.…”

“…These methods [10][11][12][13][14][15] utilize a "mathematical separation procedure" to substitute the traditional chemical separation procedure, which are different from those. [1][2][3] In addition, these approaches [10][11][12][13][14][15] can not only determine the concentrations, but can also provide spectral profiles of the components in the mixtures, which are different from those. [6][7][8][9] In this work, the present authors simultaneously resolved the mixtures of pyridoxal, pyridoxamine and 4-pyridoxic acid using the newly proposed alternatively minimizing covariant matrix error (AMCME) algorithm 16 with the purposes of solving the spectral overlapping problem and of further extending the applications of chemometric methods.…”

Spectrofluorometric Resolution of Closely Overlapping Drug Mixtures Using Chemometrics Methods

“…Chemometric methods, such as factor analysis, [6][7][8][9][10] have become even more popular in solving problems that are difficult to handle using conventional techniques, without having to resort to expensive or time-consuming procedures. Especially, three-way resolution and calibration methods [11][12][13] play important roles in solving the problem of closely overlapping fluorescence spectra.…”