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Recently, there has been growing interest in the decomposition of multilinear component models in the field of multi-way calibration. In this work, a novel quadrilinear decomposition method, i.e. four-way algorithm combination method (FACM), is proposed for four-way calibration. The FACM skillfully integrates the alternating quadrilinear decomposition (AQLD) algorithm with the four-way parallel factor analysis (FPARAFAC) algorithm and gives full play to their advantages. The performance of the FACM and the existing five algorithms are compared by using two simulated data sets. Moreover, a published four-way data array and a new four-way data array, which record a series of fluorescence spectra information of biomarkers in biological samples, are investigated by the proposed method respectively. The results demonstrate that the novel method can accurately and effectively extract the qualitative and quantitative information of analytes of interest even in the presence of unknown interferents and varying background. Besides, the FACM has attractive properties, such as very fast convergence, insensitive to initial values and excess number of components, and suitable for high noise level as well as severely collinear data. In addition, the proposed method can also be extended to combine other iterative algorithms, providing a feasible idea for the exploration and development of new algorithms.

A three-way data array in the trilinear component model can be described into sliced matrices and then decomposed into three underlying matrices by iterative procedure. In this paper, we make an in-depth study of the quadrilinear component model, generalize the “slice” to the four-way scenario, and develop a novel quadrilinear decomposition algorithm for third-order calibration, i.e., slicing alternating quadrilinear decomposition (SAQLD). The presently developed algorithm can be considered as a generalization of ATLD to four-way case. In the algorithm, updates of four underlying matrices are alternately iterated until convergence is reached. Operation of extracting diagonal elements is adopted, which makes SAQLD focus on extracting the quadrilinear part in data, leading to a significant decrease in the loss function and finally a high-performance computing strategy for SAQLD, i.e., fast convergence. Owing to its specific optimization approach, the proposed SAQLD algorithm recovers parameter matrices faster when compared with the existing quadrilinear decomposition algorithms. Both numerical simulations and experimental measurements demonstrate that third-order calibration based on the SAQLD algorithm allows one to obtain quantitative information regarding known constituents present in samples without worrying about other interferents. Moreover, quantitative results supplied by the SAQLD algorithm are still satisfying when the number of components used in calculation is excessive. Such a feature is very useful in quantitative chemical analysis since it is not easy to accurately determine the appropriate number of components due to the complex of chemical substrates.

Mycotoxins are a class of highly carcinogenic substances often naturally occurring in the moldy foods especially cereals. In the present work, a smart chemometrics-assisted analytical strategy that combines liquid chromatography-full scan-mass spectrometry (LC-MS) detection with second-order calibration method based on alternating trilinear decomposition (ATLD) algorithm is developed for direct, fast and interference-free determination of multi-class regulated mycotoxins in complex cereal samples with one-step ultrasound-assisted extraction. Ten mycotoxins with different property are fast eluted out (9.0 min) under a simple gradient LC-separation and detected by full scanning MS with a segmented fragment program. With the aid of prominent "second-order advantage" of the algorithm, the problems of co-eluted peaks, unknown interferences and baseline drifts occurring in the LC-MS profiles are mathematically resolved, making "pure signals" of targeted analytes can be extracted out from heavily interferential information. The new strategy avoids intricate physical/chemical clean-up steps to significantly improve the accuracy of the trace analysis of mycotoxins, average recoveries of ten analytes in both complex cereal samples (maize and rice) range from 93.8 to 109% with standard deviations (SD) lower than 9.8%, and the limits of detection (LOD) range from 0.01 to 1.17 μg kg −1 . In order to further confirm the reliability of the method, the same batch of samples is analyzed using LC-MS 2 method with complicated immunoaffinity column (IAC) purification step, the elliptical joint confidence region (EJCR) tests of quantitative results of ten mycotoxins show that higher precision is obtained using the proposed method. Therefore, this new strategy can be as an attractive alternative for simple, fast and accurate determination of multi-class mycotoxins in complex cereal samples.