Symmetry Studies: An Introduction to the Analysis of Structured Data in Applications (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 26)

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. Half-title......Page 3 Title......Page 5 Copyright......Page 6 Contents......Page 7 Preface......Page 11 1.1 Introduction......Page 13 1.2 Symmetry and Classification......Page 14 1.3 Data Indexed by Symmetries......Page 15 1.4 Symmetry and Data Reduction......Page 17 1.5 Statistical Aspects......Page 20 1.6 Algebraic Aspects......Page 21 1.7 Structured Data......Page 23 1.8 Partitions......Page 27 A view from mechanics......Page 28 Canonical projections......Page 33 Further Reading......Page 35 Exercises......Page 37 Appendix A......Page 41 2.2 Permutations......Page 42 Parity of a permutation......Page 43 Conjugacy classes......Page 44 Integer partitions and Young frames......Page 45 2.3 Groups and Homomorphisms......Page 46 Group homomorphisms......Page 47 Cyclic groups......Page 48 Cosets......Page 49 Semidirect and direct products of groups......Page 52 Dihedral groups in the plane......Page 54 Matrix groups......Page 55 Isometry groups......Page 56 The group of the quaternions......Page 57 Orbits, stabilizers, and transitive actions......Page 59 Group actions and permutations......Page 60 Binary sequences in length of 2......Page 61 Cyclic orbits for binary sequences in length of 4......Page 62 Dihedral orbits for binary sequences in length of 4......Page 63 Ternary sequences in length of 4......Page 64 2.6 Genotypic Classification......Page 65 Data indexed by short branches or transitions......Page 67 2.8 Counting Orbits in Linkage Analysis......Page 68 Binary sequences in length of 4......Page 70 Exercises......Page 71 Permutation representations......Page 75 Equivalent representations......Page 76 The regular representation......Page 77 3.3 Unitary Representations......Page 78 3.4 Regular Representations and Group Algebras......Page 80 3.5 Tensor Representations......Page 81 3.6 Matrices with Group Symmetry......Page 82 Matrices with dihedral structure......Page 83 Matrices with complex structure......Page 84 Matrices with quaternionic structure......Page 85 Stable subspaces......Page 86 Irreducible representations......Page 87 3.8 Schur’s Lemma......Page 91 3.9 Constructing the Irreducible Representations of Sn......Page 94 An irreducible representation of S4......Page 98 Further Reading......Page 99 Exercises......Page 100 4.2 Characters of a Linear Representation......Page 103 4.3 Orthogonality Relations for Characters......Page 104 Irreducible characters......Page 105 The characters of the regular representation......Page 107 Class functions......Page 110 Reducing the conjugacy action on S3......Page 112 The canonical projections for the Sloan Fonts study......Page 113 The canonical projections for the binary sequences study......Page 115 The canonical projections for the regular representation of S3......Page 116 The canonical projections for the regular representation of D4......Page 119 Invariant plots......Page 123 4.5 The Standard Decomposition......Page 124 Sampling considerations......Page 126 Matrices with the symmetry of Sn......Page 127 Linear representations of order statistics and ranks......Page 128 4.6 Harmonic Analysis......Page 129 A decomposition for x ∈ F (G)......Page 131 A decomposition for x ∈ F (S3)......Page 132 A Poisson summation formula......Page 133 Exercises......Page 134 5.2 Analysis of Variance......Page 140 Analysis of variance for a simple triangular array......Page 141 One-way analysis of variance......Page 142 Two-way analysis of variance......Page 143 Latin squares......Page 144 5.5 Cyclic Reduction of Binary Sequences......Page 146 5.6 Dihedral Reduction of Binary Sequences......Page 148 Dihedral stratifications for voting preferences......Page 149 5.7 Projections in the Space of Scalar-Valued Functions......Page 150 5.8 Decompositions in the Dual Space......Page 151 Planar rotations......Page 152 Coinvariants of C4......Page 154 The Standard Decomposition of Entropy......Page 160 Invariant plots in the H1 × H2 space......Page 161 The standard decomposition of the entropy of the Sloan fonts......Page 162 Geological compositions......Page 163 The regular decomposition of entropy......Page 164 5.10 A Two-Way Cyclic Decomposition......Page 165 Data structures induced by C3v......Page 166 5.12 Data Indexed by Homogeneous Polynomials......Page 168 5.13 Likelihood Decompositions......Page 170 Further Reading......Page 172 Exercises......Page 173 6.2 Symmetry Studies of Four Sequences in Length of 3......Page 177 6.3 Reductions by Position Symmetry......Page 178 Determining the classes of transitivity......Page 179 Canonical decompositions in the partition λ = 210(2)......Page 180 Canonical decompositions in the partition λ = 1(3)0......Page 181 6.4 Reductions by Symbol Symmetry......Page 183 6.5 Dihedral Studies......Page 185 Exercises......Page 187 Appendix A......Page 189 Appendix B......Page 190 7.1 Introduction......Page 192 7.3 Astigmatic and Stigmatic Constraints......Page 193 7.4 Ranking Permutations......Page 194 Two-color topography......Page 195 Three-level gray scale......Page 196 7.5 Classification of Astigmatic Mappings......Page 197 The distribution of the y'P y components......Page 199 The likelihood of an astigmatic mapping......Page 200 7.7 Dihedral Fourier Analysis......Page 202 The dihedral Fourier coefficients......Page 203 Dihedral Fourier analysis of refractive profiles......Page 204 Dihedral Fourier analysis-related applications......Page 205 Further Reading......Page 206 Appendix A......Page 210 Appendix B......Page 211 Appendix C......Page 212 8.1 Introduction......Page 213 8.2 Characterizing Rotations and Reversals......Page 215 8.3 Canonical Classification of Handedness in Elementary Images......Page 217 D4 symmetry......Page 218 Line (y = x) symmetry......Page 219 C4 symmetry......Page 220 Analytically generated images......Page 221 Sampling the mapping space......Page 223 Exercises......Page 225 Appendix A......Page 229 Appendix B......Page 230 Appendix C......Page 232 Appendix A: Computing Algorithms......Page 233 Appendix B: Glossary of Selected Symbols, Notations, and Terms......Page 237 Bibliography......Page 239 Index......Page 245

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.

"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 visual field and corneal topography arrays, or in experimental refractive optics. The symmetry studies introduced in this book describe the interplay among symmetry transformations that are characteristic of these sets of labels, the resulting algebraic decomposition of the data that are indexed by them, and the research questions that are induced by those transformations. 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-analytic applications of symmetry."--Jacket