An introduction to multi-way analysis

Multi-way analysis is the analysis of multi-way data. If a single measurement on a chemical system generates a single number and this measurement is repeated, then a sequence of numbers is the result. For example, a titration is repeated four times, resulting in the volumes x1, x2, x3 and x4. This is a sequence of numbers and this sequence has to be analyzed with one-way tools, such as calculating the mean and the standard deviation of these four numbers.

Going up one level to instruments that generate sequences of numbers, e.g. a UV-Vis spectrum consisting of absorptions at J wavelengths. Upon taking I of those measurements, a matrix X of dimensions I × J results. This is a two-way array and such a matrix can be analyzed appropriately with two-way analysis tools, such as principal component analysis. If a single instrument generates a table of numbers for each sample, e.g., a fluorescence emission/excitation landscape, this results in a matrix X (J × K), where J excitation wavelengths are used for measuring emission at K wavelengths. Taking I of these measurements, for example at I different occasions, generates a three-way array of size I × J × K. Such arrays can be analyzed with three-way analysis methods. It is even possible to generate four-way, five-way, in general multi-way data.

This free chapter from Multi-way Analysis: Applications in the Chemical Sciences discusses multi-way analysis, including:

• What is multi-way analysis?
• Conceptual aspects of multi-way data analysis.
• Hierarchy of multivariate data structures in chemistry.
• Principal component analysis and PARAFAC.

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This chapter is taken from Multi-way Analysis: Applications in the Chemical Sciences

This book is an introduction to the field of multi-way analysis for chemists and chemometricians. Its emphasis is on the ideas behind the method and its pratical applications. Sufficient mathematical background is given to provide a solid understanding of the ideas behind the method. There are currently no other books on the market which deal with this method from the viewpoint of its applications in chemistry.

• Applicable in many areas of chemistry.
• No comparable volume currently available.
• The field is becoming increasingly important.