Clustering

The following sections are included:

• Dedication
• Preface
• Contents

Clustering is the process of grouping a set of objects into subsets referred to as clusters according to some dissimilarity measure between objects. The goal is to group together similar objects, and to ensure that objects placed in distinct groups are dissimilar…

The following sections are included:

• Introduction
• Sets and Set Operations
• Sequences
• Multisets
• Closure Systems
• Permutations
• Relations
• Partially Ordered Sets
• The Poset of Partitions of a Set
• Boolean Matrices
• Galois Connections and Formal Concepts
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Dissimilarity Spaces
• Similarities and Dissimilarities between Sets
• Norms and Metrics on ℝⁿ
• The Geometry of Ultrametric Spaces
• Matrices on Semirings
• Entropy
  • Partition Entropy
  • Generalized Entropy
  • Conditional Entropy and Entropy Gain
  • The Entropic Metric Space of Partitions of Finite Sets
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Convex Sets
• Operations on Convex Sets
• Extreme Points
• Convex Functions
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Vertices, Edges and Weights
• Trees
  • Heaps
  • kd-Trees
• Bipartite Graphs
• Co-cycles
• GF(2)-Linear Spaces and Graphs
• Graphs and Matrices
• Directed Graphs
• From Matrices to Graphs
• Minimum Spanning Trees
• Matrices and Dissimilarities
• Planar Graphs
• Voronoi Diagrams
• Graph Searching
• Flows in Graphs
• Hypergraphs
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Inertia of a Set of Vectors
• The k-Means Algorithm
• Clustering and Matrices
• The PAM Algorithm
• Kernel k-Means Clustering
• A Geometric Approach to k-Clustering
• Clustering and Singular Value Decomposition of Matrices
• Vector Quantization and Matching Pursuit
• Partitional Clustering in R
• Clustering in Python
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Sampling
• Likelihood Function
• Data Density Estimations
• Density Kernels for Unidimensional Data
• Density Kernels for Multidimensional Data
• Mean Shift Clustering
• Expectation Maximization and Clustering
• The EM Algorithm for Unidimensional Data
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Ultrametrics, Hierarchies, and Partition Chains
• Ultrametrics and Minimum Spanning Trees
• The Single-Link Algorithm
• Other Hierarchical Clustering Algorithms
• Hierarchical Clustering in R
• Hierarchical Clustering in Python
• Divisive Hierarchical Clustering
• The CURE Algorithm
• Complexity of Hierarchical Clustering
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Core, Border and Noise Objects in DBSCAN
• The OPTICS Algorithm
• Density-based Clustering in R
• Density-based Clustering in Python
• Subspace Clustering
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Market Basket Data and Clustering Sets
• The ROCK Algorithm
• The CACTUS Approach
• An Entropy-based Framework
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Data and Graphs
• The Ordinary Spectrum of a Graph
• The Laplacian Spectrum of a Graph
• Cuts, Separators, and Clusterings
• Spectral Clustering Algorithms
• Spectral Clustering in R
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Graphs and Correlation Clustering
• The NP-Completeness of Correlation Clustering
• Disagreements Minimization
• Consensus Clustering
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Internal Criteria
  • The Davies-Bouldin Criterion
  • The Dunn Quality Indices
  • The Silhouette Coefficient
• External Criteria
• Pairwise Measures
• Graph Clustering and Modularity
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Clustering Functions
• An Impossibility Result
• Antichains of Partitions
• Centroid-based Clustering and Consistency
• Partitioning Functions
• An Axiomatization of Clustering Quality Measures
• Exercises and Supplements

The following sections are included:

• Introduction
• Reorderable Matrices
• A Boolean Approach to Binary Data Sets Biclustering
• The Cheng and Church Algorithm
• Numerical Aspects of Biclustering of Binary Data Sets
• Exercises and Supplements

The following sections are included:

• Introduction
• A Semi-Supervised Variant of k-Means
• Constraint Propagation
• Semi-Supervised Hierarchical Clustering
• Learning Mahalanobis Metrics
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Appendix A Special Functions and Applications
  • Introduction
  • Euler’s Functions
  • Spheres and Cubes in ℝⁿ
• Appendix B Linear Algebra
  • Introduction
  • Matrices
  • Matrix Rank
  • Matrix Differentiation
  • Eigenvalues of Matrices
  • Optimization and Eigenvalues
  • Matrix Norms
  • Unitary Matrices and Orthogonal Projections
  • Positive Definite Matrices
  • Singular Values of Matrices
  • CS Decomposition
  • Geometry of Subspaces
  • Matrix Approximation via SVD
• Appendix C Linear Programming
  • Introduction
  • The Fourier-Motzkin Elimination
  • Primal and Dual LP Problems
  • The Bin Packing Problem
  • Bibliographical Comments
• Appendix D NP Completeness
  • Introduction
  • Words and Problems
  • Problems and Algorithms
  • Variables and Propositional Formulas
  • Turing Machines
  • NP Complete Problems
  • Parameterized Problems
  • Approximation Algorithms
  • Bibliographical Comments

The following sections are included:

• Bibliography
• Index