LYNNE BILLARD, PHD, is University Professor in the Department of Statistics at the University of Georgia, USA. She has over two hundred and twenty-five publications mostly in leading journals, and co-edited six books. Professor Billard is a former president of ASA, IBS, and ENAR. EDWIN DIDAY, PHD, is the Professor of Computer Science at Centre De Recherche en Mathematiques de la Decision, CEREMADE, Université Paris-Dauphine, Université PSL, Paris, France. He has published fifty-eight papers and authored or edited fourteen books. Professor Diday is also the founder of the Symbolic Data Analysis field.

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• 1 Introduction 12 Symbolic Data: Basics 72.1 Individuals, Classes, Observations, and Descriptions 82.2 Types of Symbolic Data 92.2.1 Multi-valued or Lists of Categorical Data 92.2.2 Modal Multi-valued Data 102.2.3 Interval Data 122.2.4 Histogram Data 132.2.5 Other Types of Symbolic Data 142.3 How do Symbolic Data Arise? 172.4 Descriptive Statistics 242.4.1 Sample Means 252.4.2 Sample Variances 262.4.3 Sample Covariance and Correlation 282.4.4 Histograms 312.5 Other Issues 38Exercises 39Appendix 413 Dissimilarity, Similarity, and Distance Measures 473.1 Some General Basic Definitions 473.2 Distance Measures: List or Multi-valued Data 553.2.1 Join and Meet Operators for Multi-valued List Data 553.2.2 A Simple Multi-valued Distance 563.2.3 Gowda–Diday Dissimilarity 583.2.4 Ichino–Yaguchi Distance 603.3 Distance Measures: Interval Data 623.3.1 Join and Meet Operators for Interval Data 623.3.2 Hausdorff Distance 633.3.3 Gowda–Diday Dissimilarity 683.3.4 Ichino–Yaguchi Distance 733.3.5 de Carvalho Extensisons of Ichino–Yaguchi Distances 763.4 Other Measures 79Exercises 79Appendix 824 Dissimilarity, Similarity, and Distance Measures: Modal Data 834.1 Dissimilarity/Distance Measures: Modal Multi-valued List Data 834.1.1 Union and Intersection Operators for Modal Multi-valued List Data 844.1.2 A Simple Modal Multi-valued List Distance 854.1.3 Extended Multi-valued List Gowda–Diday Dissimilarity 874.1.4 Extended Multi-valued List Ichino–Yaguchi Dissimilarity 904.2 Dissimilarity/Distance Measures: Histogram Data 934.2.1 Transformation of Histograms 944.2.2 Union and Intersection Operators for Histograms 984.2.3 Descriptive Statistics for Unions and Intersections 1014.2.4 Extended Gowda–Diday Dissimilarity 1044.2.5 Extended Ichino–Yaguchi Distance 1084.2.6 Extended de Carvalho Distances 1124.2.7 Cumulative Density Function Dissimilarities 1154.2.8 Mallows’ Distance 117Exercises 1185 General Clustering Techniques 1195.1 Brief Overview of Clustering 1195.2 Partitioning 1205.3 Hierarchies 1255.4 Illustration 1315.5 Other Issues 1466 Partitioning Techniques 1496.1 Basic Partitioning Concepts 1506.2 Multi-valued List Observations 1536.3 Interval-valued Data 1596.4 Histogram Observations 1696.5 Mixed-valued Observations 1776.6 Mixture Distribution Methods 1796.7 Cluster Representation 1866.8 Other Issues 189Exercises 191Appendix 1937 Divisive Hierarchical Clustering 1977.1 Some Basics 1977.1.1 Partitioning Criteria 1977.1.2 Association Measures 2007.2 Monothetic Methods 2037.2.1 Modal Multi-valued Observations 2057.2.2 Non-modal Multi-valued Observations 2147.2.3 Interval-valued Observations 2167.2.4 Histogram-valued Observations 2257.3 Polythethic Methods 2367.4 Stopping Rule R 2507.5 Other Issues 257Exercises 2588 Agglomerative Hierarchical Clustering 2618.1 Agglomerative Hierarchical Clustering 2618.1.1 Some Basic Definitions 2618.1.2 Multi-valued List Observations 2668.1.3 Interval-valued Observations 2698.1.4 Histogram-valued Observations 2788.1.5 Mixed-valued Observations 2818.1.6 Interval Observations with Rules 2828.2 Pyramidal Clustering 2898.2.1 Generality Degree 2898.2.2 Pyramid Construction Based on Generality Degree 2978.2.3 Pyramids from Dissimilarity Matrix 3098.2.4 Other Issues 312Exercises 313Appendix 315References 317Index 331

LYNNE BILLARD, PHD, is University Professor in the Department of Statistics at the University of Georgia, USA. She has over two hundred and twenty-five publications mostly in leading journals, and co-edited six books. Professor Billard is a former president of ASA, IBS, and ENAR. EDWIN DIDAY, PHD, is the Professor of Computer Science at Centre De Recherche en Mathematiques de la Decision, CEREMADE, Université Paris-Dauphine, Université PSL, Paris, France. He has published fifty-eight papers and authored or edited fourteen books. Professor Diday is also the founder of the Symbolic Data Analysis field.

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• 1 Introduction 12 Symbolic Data: Basics 72.1 Individuals, Classes, Observations, and Descriptions 82.2 Types of Symbolic Data 92.2.1 Multi-valued or Lists of Categorical Data 92.2.2 Modal Multi-valued Data 102.2.3 Interval Data 122.2.4 Histogram Data 132.2.5 Other Types of Symbolic Data 142.3 How do Symbolic Data Arise? 172.4 Descriptive Statistics 242.4.1 Sample Means 252.4.2 Sample Variances 262.4.3 Sample Covariance and Correlation 282.4.4 Histograms 312.5 Other Issues 38Exercises 39Appendix 413 Dissimilarity, Similarity, and Distance Measures 473.1 Some General Basic Definitions 473.2 Distance Measures: List or Multi-valued Data 553.2.1 Join and Meet Operators for Multi-valued List Data 553.2.2 A Simple Multi-valued Distance 563.2.3 Gowda–Diday Dissimilarity 583.2.4 Ichino–Yaguchi Distance 603.3 Distance Measures: Interval Data 623.3.1 Join and Meet Operators for Interval Data 623.3.2 Hausdorff Distance 633.3.3 Gowda–Diday Dissimilarity 683.3.4 Ichino–Yaguchi Distance 733.3.5 de Carvalho Extensisons of Ichino–Yaguchi Distances 763.4 Other Measures 79Exercises 79Appendix 824 Dissimilarity, Similarity, and Distance Measures: Modal Data 834.1 Dissimilarity/Distance Measures: Modal Multi-valued List Data 834.1.1 Union and Intersection Operators for Modal Multi-valued List Data 844.1.2 A Simple Modal Multi-valued List Distance 854.1.3 Extended Multi-valued List Gowda–Diday Dissimilarity 874.1.4 Extended Multi-valued List Ichino–Yaguchi Dissimilarity 904.2 Dissimilarity/Distance Measures: Histogram Data 934.2.1 Transformation of Histograms 944.2.2 Union and Intersection Operators for Histograms 984.2.3 Descriptive Statistics for Unions and Intersections 1014.2.4 Extended Gowda–Diday Dissimilarity 1044.2.5 Extended Ichino–Yaguchi Distance 1084.2.6 Extended de Carvalho Distances 1124.2.7 Cumulative Density Function Dissimilarities 1154.2.8 Mallows’ Distance 117Exercises 1185 General Clustering Techniques 1195.1 Brief Overview of Clustering 1195.2 Partitioning 1205.3 Hierarchies 1255.4 Illustration 1315.5 Other Issues 1466 Partitioning Techniques 1496.1 Basic Partitioning Concepts 1506.2 Multi-valued List Observations 1536.3 Interval-valued Data 1596.4 Histogram Observations 1696.5 Mixed-valued Observations 1776.6 Mixture Distribution Methods 1796.7 Cluster Representation 1866.8 Other Issues 189Exercises 191Appendix 1937 Divisive Hierarchical Clustering 1977.1 Some Basics 1977.1.1 Partitioning Criteria 1977.1.2 Association Measures 2007.2 Monothetic Methods 2037.2.1 Modal Multi-valued Observations 2057.2.2 Non-modal Multi-valued Observations 2147.2.3 Interval-valued Observations 2167.2.4 Histogram-valued Observations 2257.3 Polythethic Methods 2367.4 Stopping Rule R 2507.5 Other Issues 257Exercises 2588 Agglomerative Hierarchical Clustering 2618.1 Agglomerative Hierarchical Clustering 2618.1.1 Some Basic Definitions 2618.1.2 Multi-valued List Observations 2668.1.3 Interval-valued Observations 2698.1.4 Histogram-valued Observations 2788.1.5 Mixed-valued Observations 2818.1.6 Interval Observations with Rules 2828.2 Pyramidal Clustering 2898.2.1 Generality Degree 2898.2.2 Pyramid Construction Based on Generality Degree 2978.2.3 Pyramids from Dissimilarity Matrix 3098.2.4 Other Issues 312Exercises 313Appendix 315References 317Index 331