Marlos A. G. Viana

This book presents an introduction to the foundations, interpretations, and data-analytic applications of symmetry studies with an emphasis on applications in optical sciences. Symmetry studies connect group theoretic and statistical methods for data summary and inference. Readers should have an understanding of calculus and linear algebra as well as introductory statistics. The book reviews finite group theory in the introductory chapters. Computational tools used in the text are available for download in the form of Mathmaticaâ notebooks or R scripts. This book: Demonstrates the usefulness of a unified view of algebra and symmetry studies to address data-analytic questions in optics and vision science Offers a brief review of finite group theory and elements of multivariate analysis Includes various examples from diverse areas of optical science

Dihedral Fourier Analysis introduces the theory and applications necessary to study experimental data indexed by, or associated with, the points in a dihedral symmetry orbit. This book looks at experimental data and analytical models indexed by certain dihedral rotations and reversals realized as vector fields. Its particular relevance as a research tool in areas such as optical and molecular biology statistics appears when formulated within the context of symmetry studies, which formally connects algebraic and statistical reasoning together in one methodology for data summary and inference. Chapter 1 presents an overview of the theory and methods of dihedral analysis. It introduces data sets and examples defining and connecting the algebraic notions of symmetry with those of statistical summaries and inference. Chapter 2 includes the required algebraic aspects and data-analytic results. Chapters 3-6 offer applications of the methods presented in the text. This book is intended for data analysts of both theoretical and applied interests.

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 interplay between symmetry-related research questions and their statistical analysis.

Algebraic methods and arguments in statistics and probability are well known, from Gauss' least squares principle through Fisher's method of variance decomposition. The relevance of group-theoretic arguments, for example, became evident in the 1980s. Such techniques continue to be of interest today, along with other developments, such as the use of graph theory in modelling complex stochastic systems. This volume is based on lectures presented at the AMS Special Session on Algebraic Methods and Statistics held at the University of Notre Dame (Indiana) and on contributed articles solicited for this volume. The articles are intended to foster communication between representatives of the diverse scientific areas in which these functions are utilized and to further the trend of utilizing algebraic methods in the areas of statistics and probability. This is one of few volumes devoted to the subject of algebraic methods in statistics and probability. The wide range of topics covered in this volume demonstrates the vigorous level of research and opportunities ongoing in these areas.