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 -- 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) = Pm i=1 min 1≤j≤k d(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 97 4.5.2 DBSCAN algorithm -- 4.5.3 Numerical examples -- 5 Indexes -- 5.1 Choosing a partition with the most appropriate number of 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 R n -- 6.4 Methods to search for optimal partition with ellipsoidal clusters -- 6.4.1 Mahalanobis k-means algorithm 139 CONTENTS v -- 6.4.2 Mahalanobis incremental algorithm -- 6.4.3 Expectation Maximization algorithm for Gaussian mixtures -- 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 Multiple circles detection problem -- 8.1.2 Multiple ellipses detection problem -- 8.1.3 Multiple generalized circles detection problem -- 8.1.4 Multiple lines detection problem -- 8.1.5 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? -- 8.4.1 Mathematical model and the algorithm -- 8.4.2 Defining constituencies in the Republic of Croatia -- 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.