Second- and higher-order data generation and calibration: A tutorial

•Second- and third-order (multi-way) data and algorithms are reviewed.•Suitable examples of different complexity are provided.•The advantages of multi-way calibration are illustrated.•Multi-way analytical figures of merit are discussed.

An introduction to multi-way calibration based on second- and higher-order data generation and processing is provided, with emphasis on practical experimental aspects. After a discussion concerning a proper nomenclature scheme, a suitable classification of the obtainable data, and the general features of the available algorithms and their underlying models, a series of examples is discussed in detail, with the purpose of illustrating the great potentiality of the field for the analytical community. Emphasis is directed toward the most popular multi-way data, i.e., second-order or matrix data, which can be conveniently measured in a variety of instruments. Third-order data are being increasingly studied and are also discussed, along with the less explored field of fourth-order data. The estimation of figures of merit, which analysts need to report during method development, is now sufficiently mature to be provided for the general audience.

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