A new method, alternating penalty trilinear decomposition (APTLD), is developed for the decomposition of three-way data arrays. By utilizing the alternating least squares principle and alternating penalty constraints to minimize three different alternating penalty errors simultaneously, the intrinsic profiles are found. The APTLD algorithm can avoid the two-factor degeneracy problem and relieve the slow convergence problem, which is difficult to handle for the traditional parallel factor analysis (PARAFAC) algorithm. It retains the second-order advantage of quantification for analytes of interest even in the presence of potentially unknown interferents. In additions, it is insensitive to the estimated component number, thus avoiding the difficulty of determining a correct component number for the model, which is intrinsic in the PARAFAC algorithm. The results of treating one simulated and one real excitation-emission spectral data set showed that the proposed algorithm performs well as long as the model dimensionality chosen is not less than the actual number of components. Furthermore, the performance of the APTLD algorithm sometimes surpasses that of the PARAFAC algorithm in the prediction of concentration profiles even if the component number chosen is the same as the actual number of underlying factors in real samples.

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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.

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