Cluster Analysis and Applications

Description

Preface Contents 1 Introduction 2 Representatives 2.1 Representative of Data Sets with One Feature 2.1.1 Best LS-Representative 2.1.2 Best 1-Representative 2.1.3 Best Representative of Weighted Data 2.1.4 Bregman Divergences 2.2 Representative of Data Sets with Two Features 2.2.1 Fermat–Torricelli–Weber Problem 2.2.2 Centroid of a Set in the Plane 2.2.3 Median of a Set in the Plane 2.2.4 Geometric Median of a Set in the Plane 2.3 Representative of Data Sets with Several Features 2.3.1 Representative of Weighted Data 2.4 Representative of Periodic Data 2.4.1 Representative of Data on the Unit Circle 2.4.2 Burn Diagram 3 Data Clustering 3.1 Optimal k-Partition 3.1.1 Minimal Distance Principle and Voronoi Diagram 3.1.2 k-means Algorithm I 3.2 Clustering Data with One Feature 3.2.1 Application of the LS-Distance-like Function 3.2.2 The Dual Problem 3.2.3 Least Absolute Deviation Principle 3.2.4 Clustering Weighted Data 3.3 Clustering Data with Two or Several Features 3.3.1 Least Squares Principle 3.3.2 The Dual Problem 3.3.3 Least Absolute Deviation Principle 3.4 Objective Function F(c1,…,ck)=i=1m min1≤j≤kd(cj,ai) 4 Searching for an Optimal Partition 4.1 Solving the Global Optimization Problem Directly 4.2 k-means Algorithm II 4.2.1 Objective Function F using the Membership Matrix 4.2.2 Coordinate Descent Algorithms 4.2.3 Standard k-means Algorithm 4.2.4 k-means Algorithm with Multiple Activations 4.3 Incremental Algorithm 4.4 Hierarchical Algorithms 4.4.1 Introduction and Motivation 4.4.2 Applying the Least Squares Principle 4.5 DBSCAN Method 4.5.1 Parameters MinPts and ε 4.5.2 DBSCAN Algorithm Main DBSCAN Algorithm 4.5.3 Numerical Examples 5 Indexes 5.1 Choosing a Partition with the Most Appropriate Numberof Clusters 5.1.1 Calinski–Harabasz Index 5.1.2 Davies–Bouldin Index 5.1.3 Silhouette Width Criterion 5.1.4 Dunn Index 5.2 Comparing Two Partitions 5.2.1 Rand Index of Two Partitions 5.2.2 Application of the Hausdorff Distance 6 Mahalanobis Data Clustering 6.1 Total Least Squares Line in the Plane 6.2 Mahalanobis Distance-Like Function in the Plane 6.3 Mahalanobis Distance Induced by a Set in the Plane 6.3.1 Mahalanobis Distance Induced by a Set of Points in Rn 6.4 Methods to Search for Optimal Partition with Ellipsoidal Clusters 6.4.1 Mahalanobis k-Means Algorithm 6.4.2 Mahalanobis Incremental Algorithm 6.4.3 Expectation Maximization Algorithm for GaussianMixtures 6.4.4 Expectation Maximization Algorithm for Normalized Gaussian Mixtures and Mahalanobis k-Means Algorithm 6.5 Choosing Partition with the Most Appropriate Number of Ellipsoidal Clusters 7 Fuzzy Clustering Problem 7.1 Determining Membership Functions and Centers 7.1.1 Membership Functions 7.1.2 Centers 7.2 Searching for an Optimal Fuzzy Partition with Spherical Clusters 7.2.1 Fuzzy c-Means Algorithm 7.2.2 Fuzzy Incremental Clustering Algorithm (FInc) 7.2.3 Choosing the Most Appropriate Number of Clusters 7.3 Methods to Search for an Optimal Fuzzy Partition with Ellipsoidal Clusters 7.3.1 Gustafson–Kessel c-Means Algorithm 7.3.2 Mahalanobis Fuzzy Incremental Algorithm (MFInc) 7.3.3 Choosing the Most Appropriate Number of Clusters 7.4 Fuzzy Variant of the Rand Index 7.4.1 Applications 8 Applications 8.1 Multiple Geometric Objects Detection Problem and Applications 8.1.1 The Number of Geometric Objects Is Known in Advance 8.1.2 The Number of Geometric Objects Is Not Known in Advance 8.1.3 Searching for MAPart and Recognizing GeometricObjects 8.1.4 Multiple Circles Detection Problem Circle as the Representative of a Data Set Artificial Data Set Originating from a Single Circle The Best Representative Multiple Circles Detection Problem in the Plane The Number of Circles Is Known KCC Algorithm The Number of Circles Is Not Known Real-World Images 8.1.5 Multiple Ellipses Detection Problem A Single Ellipse as the Representative of a Data Set Artificial Data Set Originating from a Single Ellipse The Best Representative Multiple Ellipses Detection Problem The Number of Ellipses Is Known in Advance KCE Algorithm The Number of Ellipses Is Not Known in Advance Real-World Images 8.1.6 Multiple Generalized Circles Detection Problem Real-World Images 8.1.7 Multiple Lines Detection Problem A Line as Representative of a Data Set The Best TLS-Line in Hesse Normal Form The Best Representative Multiple Lines Detection Problem in the Plane The Number of Lines Is Known in Advance KCL Algorithm The Number of Lines Is Not Known in Advance Real-World Images 8.1.8 Solving MGOD-Problem by Using the RANSAC Method 8.2 Determining Seismic Zones in an Area 8.2.1 Searching for Seismic Zones 8.2.2 The Absolute Time of an Event 8.2.3 The Analysis of Earthquakes in One Zone 8.2.4 The Wider Area of the Iberian Peninsula 8.2.5 The Wider Area of the Republic of Croatia 8.3 Temperature Fluctuations 8.3.1 Identifying Temperature Seasons 8.4 Mathematics and Politics: How to Determine Optimal Constituencies? Defining the Problem 8.4.1 Mathematical Model and the Algorithm Integer Approach Linear Relaxation Approach 8.4.2 Defining Constituencies in the Republic of Croatia Applying the Linear Relaxation Approach to the Model with 10 Constituencies Applying the Integer Approach to the Model with 10 Constituencies 8.4.3 Optimizing the Number of Constituencies 8.5 Iris 8.6 Reproduction of Escherichia coli 9 Modules and the Data Sets 9.1 Functions 9.2 Algorithms 9.3 Data Generating 9.4 Test Examples 9.5 Data Sets Bibliography Index