This text is an introduction to data-analytic applications of symmetry principles and arguments or symmetry studies. Its motivation comes from a variety of disciplines in which these principles continue to play a significant role in describing natural phenomena, and from the goal of methodologically applying them to classification, description, and analysis of data. The product of the methodology presented here is a broader class of data-analytic tools derived from well-established and theoretically related areas in algebra and statistics, such as group representations and analysis of variance.

The principles discussed in the text reflect many defining aspects of symmetry, a Greek conception dating from the Hellenic Era and part of a class of terms and forms of expression that designated harmony, rhythm, balance, stability, good proportions, and evenness of structure. Early Greek art and architecture often capture the outstanding dualism intrinsic to the original notion of symmetry – that of retaining the static uniqueness of one's being and at the same time promoting its dynamical multipresent realizations. This dualism is only apparently hidden in the 12th-century Athenian detail shown in the front cover. I invite you to recognize the presence of these pleasant concepts in the methodology to be introduced in the coming pages.

The text is divided as follows. Chapter 1 gives a complete overview of the methodology, including an introduction to the concepts of data indexed by symmetries, finite groups, group actions, orbits and classification, and representations in the data space. At the same time it outlines the step-by-step connection between the algebra and statistical inference, in the context of analysis of variance.