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Alternating penalty quadrilinear decomposition algorithm for an analysis of four‐way data arrays

A novel algorithm, alternating penalty quadrilinear decomposition (APQLD), is developed as an extension of alternating penalty trilinear decomposition (APTLD) for decomposition of quadrilinear data and applied to third‐order calibration. The proposed method as well as four‐way parallel factor analysis (PARAFAC) not only retains the second‐order advantages possessed in second‐order calibration but also holds additional advantage, for example with trilinear data from one sample, the intrinsic profiles in each order can be determined uniquely for each species in the sample. From simulations, it is observed that another advantage is that the introduction of fourth mode can relieve the serious problem of collinearity. It can be defined the ‘third‐order advantage’. It was shown a much higher convergence rate compared with four‐way PARAFAC. Moreover, it is generally insensitive to the overestimates of the component number chosen. This offers the advantage that in third‐order calibration one need not pay much attention to determining a proper component number for the model, and it is difficult for four‐way PARAFAC to avoid it. By treating simulated and one real excitation‐emission‐pH data sets, the results indicated that both APQLD and PARAFAC work well, but the performance of APQLD is better than that of PARAFAC in the prediction of concentration even if the component number chosen is the same as the actual number of underlying factors in the real system. Copyright © 2007 John Wiley & Sons, Ltd.