Symmetry studies an introduction to the analysis of structured data in applications 1st Edition by Marlos Viana ISBN 978-0511447907 1281717290

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ISBN 10:  1281717290

ISBN 13: 978-0511447907

Author: Marlos A. G. Viana

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.

Table of contents: 

1. Symmetry, Classification, and the Analysis of Structured Data

1.1 Introduction

1.2 Symmetry and Classification

1.3 Data Indexed by Symmetries

1.4 Symmetry and Data Reduction

1.5 Statistical Aspects

1.6 Algebraic Aspects

1.7 Structured Data

1.8 Partitions

1.9 Summary

Further Reading

Exercises

Appendix A

2. Sorting the Labels: Group Actions and Orbits

2.1 Introduction

2.2 Permutations

2.3 Groups and Homomorphisms

2.4 Group Actions

2.5 Actions on Mappings

2.6 Genotypic Classification

2.7 Actions on Binary Trees and Branches

2.8 Counting Orbits in Linkage Analysis

2.9 Burnside’s Lemma

Further Reading

Exercises

3. Connecting Symmetries and Data: Linear Representations

3.1 Introduction

3.2 Representations

3.3 Unitary Representations

3.4 Regular Representations and Group Algebras

3.5 Tensor Representations

3.6 Matrices with Group Symmetry

3.7 Reducibility

3.8 Schur’s Lemma

3.9 Constructing the Irreducible Representations of S Further Reading

Exercises

4. Data Reduction and Inference: The Canonical Projections and Their Invariants

4.1 Introduction

4.2 Characters of a Linear Representation

4.3 Orthogonality Relations for Characters

4.4 The Canonical Projections

4.5 The Standard Decomposition

4.6 Harmonic Analysis

Exercises

5. Examples and Techniques

5.1 Introduction

5.2 Analysis of Variance

5.3 Classical Structures and Their Analysis of Variance

5.4 The Analysis of Variance for Dihedral Data

5.5 Cyclic Reduction of Binary Sequences

5.6 Dihedral Reduction of Binary Sequences

5.7 Projections in the Space of Scalar-Valued Functions

5.8 Decompositions in the Dual Space

5.9 Canonical Decompositions of Entropy Data

5.10 A Two-Way Cyclic Decomposition

5.11 Data Structures Induced by Point Groups

5.12 Data Indexed by Homogeneous Polynomials

5.13 Likelihood Decompositions

Further Reading

Exercises

6. Symmetry Studies of Short Symbolic Sequences

6.1 Introduction

6.2 Symmetry Studies of Four Sequences in Length of 3

6.3 Reductions by Position Symmetry

6.4 Reductions by Symbol Symmetry

6.5 Dihedral Studies

Further Reading

Exercises

Appendix A

Appendix B

7. Symmetry Studies of Curvature Data

7.1 Introduction

7.2 Keratometry

7.3 Astigmatic and Stigmatic Constraints

7.4 Ranking Permutations

7.5 Classification of Astigmatic Mappings

7.6 Astigmatic Covariance Structures

7.7 Dihedral Fourier Analysis

Further Reading

Exercises

Appendix A

Appendix B

Appendix C

8. Symmetry Studies of Planar Chirality

8.1 Introduction

8.2 Characterizing Rotations and Reversals

8.3 Canonical Classification of Handedness in Elementary Images

8.4 Handedness of Short Symbolic Sequences

Further Reading

Exercises

Appendix A

Appendix B

Appendix C

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Tags: Marlos Viana, Symmetry studies, an introduction, data in applications