Lerman I.C., Leredde H. Seriation in Combinatorial and Statistical Data Analysis

Cham: Springer, 2022. — 287 p.This monograph offers an original broad and very diverse exploration of the seriation domain in data analysis, together with building a specific relation to clustering.Relative to a data table crossing a set of objects and a set of descriptive attributes, the search for orders which correspond respectively to these two sets is formalized mathematically and statistically.State-of-the-art methods are created and compared with classical methods and a thorough understanding of the mutual relationships between these methods is clearly expressed. The authors distinguish two families of methods:Geometric representation methods.
Algorithmic and Combinatorial methods.Original and accurate methods are provided in the framework for both families. Their basis and comparison are made on both theoretical and experimental levels. The experimental analysis is very varied and very comprehensive. Seriation in Combinatorial and Statistical Data Analysis has a unique character in the literature falling within the fields of Data Analysis, Data Mining, and Knowledge Discovery. It will be a valuable resource for students and researchers in the latter fields.General Introduction: Methods and History.Seriation from Proximity Variance Analysis.Definition of a Seriation; It's Unicity.
Definition of a σ Form.
Unicity.
Association Coefficient Between Columns of a σ Seriation Form.
Preamble.
The Association Coefficient.
Equation of the Association Coefficient in the Case of a σ Form.
Representation on a Directed Unit Line Segment.
Simultaneous Mean and Variance Analysis of Pairwise Proximities.
Preamble.
Optimal Properties of an Extreme Point of Rectilinear Cloud.
Discovering from an Optimal Property, a σ Form.
Block Seriation from Simultaneous Mean and Variance Analysis of Pairwise Proximities; The ``Attraction Poles'' Method.
Proximity Variance Analysis Equation.
Geometrical Representation by the Attraction Poles Method.Main Approaches in Seriation: The Attraction Pole Case.Visual and Combinatorial Methods.
Graphical Methods of J Bertin.
Block Seriation Method of F Marchotorchino.
Methodological Developments.
Elisséeff Heuristic.
The Deutsch and Martin Algorithm.
The Niermann Algorithm.
Seriation Defined as a Combinatorial Optimization Problem.
Preamble.
General Facets.
A Criterion for Evaluating the Seriation.
Usual Criteria for Evaluating a Seriation.
Spectral Approaches.
Some Methodological Points and Extensions.
Methods Using a Planar Geometrical Representation.
Introduction: The Most Classical Methods.
The Horse-Shoe Method of Kendall.
Combinatorial and Algorithmic Approaches of the Pole Attraction Method.Comparing Geometrical and Ordinal Seriation Methods in Formal and Real Cases.Methods.
Similarities and Distances.
Multidimensional Data Analysis.
Techniques and Algorithms.
Results in Processing Data.
Preamble.
Simulated Data According to σ Forms Models.
Real Data.
Some Concluding Remarks.A New Family of Combinatorial Algorithms in Seriation.
Introduction: The Fundamental Principles.
Row Data Table Ranking Associated with a Column Ordering in Seriation.The Combinatorial Algorithm.
The Statistical Algorithm.
Seriation Algorithms.
The Seriation Search Space Is Associated with the First Element.
The Seriation Search Space Is Associated with Sized Sequences of Elements.
Applying on σ Forms and Real Data.
Preamble.
Simulated σ Forms.
Real Data.Clustering Methods from Proximity Variance Analysis.
Introduction and General Presentation.
The Method Family of Attraction Poles by Successive Reallocations (APMSR).
Data, Similarities, and Distances.
Determination of a System of Attraction Poles.
Cluster Formation Around the Attraction Poles.
Assignment Criteria.
Criteria and Algorithmic of the Attraction Pole Methods by Successive Aggregations (APMSA).Criteria for APMSA.
On the Number of Poles to Extract and on the Quality of the Clusterings Obtained.
The Fitting Criterion.
Inertia or Ward Criterion.
Interesting Partitions from APMSR and APMSA.
Developments.
Applying Clustering Attraction Pole Methods in Real Data.Ruspini Data.
Fisher Data.
Merovingian Belt Buckles-Plates Data.
Some Words to Conclude.Conclusion and Developments.The Algorithms and Their Respective Logics.
Geometric Representations of the Descriptive Attributes and the Objects Described.
Row Seriation from Column Seriation.
Algorithms by Successive Chaining.
Attraction Pole Clustering Algorithms.
Structures and Algorithms in Asymmetrical Hierarchical Clustering.
Directed Ascendant Hierarchical Clustering.
Oriented Hierarchical Clustering.
Hierarchical Clustering of Successive Intervals.
Some Additional Interactions Between Seriation and Hierarchical Clustering.Appendix A Index.