Symmetry studies: an introduction to the analysis of structured data in applications

Table of contents:

1 - Symmetry, Classification, and the Analysis of Structured Data
2 - Sorting the Labels: Group Actions and Orbits
3 - Connecting Symmetries and Data: Linear Representations
4 - Data Reduction and Inference: The Canonical Projections and Their Invariants
5 - Examples and Techniques
6 - Symmetry Studies of Short Symbolic Sequences
7 - Symmetry Studies of Curvature Data
8 - Symmetry Studies of Planar Chirality

Experimental data can often be associated with or indexed by certain symmetrically interesting structures or sets of labels that appear, for example, in the study of short symbolic sequences in molecular biology, in preference or voting data, in (corneal) curvature data, and in studies of the handedness and entropy of symbolic sequences and elementary images. The symmetry studies introduced in this book describe the interplay among symmetry transformations that are characteristic of these sets of labels, their resulting classification, the algebraic decomposition of the data indexed by them, and the statistical analysis of the invariants induced by those decompositions. The overall purpose is to facilitate and guide the statistical study of the structured data from both a descriptive and inferential perspective. The text combines notions of algebra and statistics and develops a systematic methodology to better explore the many different data-analysis applications of symmetry.