Hierarchy of multivariate data structures in chemistry

In order to set the stage for introducing different types of methods and models for analyzing multi-way data it is convenient to have an idea of what type of multivariate data analysis problems can be encountered in chemical practice. A categorization will be developed that uses the types of arrangements of the data set related to the chemical problem. 

Multi-way Analysis With Applications in the Chemical Sciences 

The simplest multivariate data set is a data set consisting of measurements (or calculated properties) of J variables on I objects. Such a data set can be arranged in an / x. / matrix X. This matrix X contains variation which is supposed to be relevant for the (chemical) problem at hand. Several types of methods are available to investigate this variation depending on the purpose of the research and the problem definition. 

If the purpose of analyzing the data set is to find patterns, relations, differences and agreements between objects and/or variables, then decomposition methods can be used to summarize the data conveniently and explore the data set using plots and figures. Typically, 

While PCA is a linear projection method, there also exist nonlinear projection methods, e.g. multidimensional scaling [Mardia et al. 1979] and nonlinear PCA [Dong McAvoy 1996], A good overview of nonlinear multivariate analysis tools is given by [Gift 1990], 

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