Graphs as Structural Models: The Application of Graphs and Multigraphs in Cluster Analysis

ISBN-13: 9783528063122 / Angielski / Miękka / 1988 / 214 str.

Erhard Godehardt

Opis

The advent of the high-speed computer with its enormous storage capabilities enabled statisticians as well as researchers from the different topics of life sciences to apply mul tivariate statistical procedures to large data sets to explore their structures. More and more, methods of graphical representation and data analysis are used for investigations. These methods belong to a topic of growing popUlarity, known as "exploratory data analysis" or EDA. In many applications, there is reason to believe that a set of objects can be clus tered into subgroups that differ in meaningful ways. Extensive data sets, for example, are stored in clinical cancer registers. In large data sets like these, nobody would ex pect the objects to be homogeneous. The most commonly used terms for the class of procedures that seek to separate the component data into groups are "cluster analysis" or "numerical taxonomy." The origins of cluster analysis can be found in biology and anthropology at the beginning of the century. The first systematic investigations in cluster analysis are those of K. Pearson in 1894. The search for classifications or ty pologies of objects or persons, however, is indigenous not only to biology but to a wide variety of disciplines. Thus, in recent years, a growing interest in classification and related areas has taken place. Today, we see applications of cluster analysis not only to. biology but also to such diverse areas as psychology, regional analysis, marketing research, chemistry, archaeology and medicine."

Spis treści

0 Mathematical Symbols and Notation.- 1 Introduction, Basic Concepts.- 1.1 Modelling in Medicine and Biology.- 1.2 Graphs as Tools in Mathematical Modelling.- 1.3 The Scope of Exploratory Data Analysis.- 1.4 The Basic Concepts of Cluster Analysis.- 2 Current Methods of Cluster Analysis: An Overview.- 2.1 The Aim of Cluster Analysis.- 2.2 The Different Steps of a Cluster Analysis.- 2.2.1 Data Sampling and Preparation.- 2.2.2 Measures of Similarity or Distance.- 2.2.3 Types of Classification.- 2.2.4 Procedures of Classification.- 2.2.4.1 Optimization Methods.- 2.2.4.2 Recursive Construction of Groups.- 2.2.4.3 Analysis of the Point Density.- 2.2.4.4 Linkage Methods.- 2.3 A Short Review of Classification Methods.- 2.4 Preparation and Presentation of Results.- 3 Graph-theoretic Methods of Cluster Analysis.- 3.1 Classification by Graphs.- 3.1.1 The Classification at Level d.- 3.1.2 Single-Linkage Clusters as Components of a Graph.- 3.1.3 Modifications of the Cluster Definition.- 3.2 Classifications by Multigraphs.- 3.2.1 Undirected, Completely Labelled Mulitgraphs.- 3.2.2 Application to Classification Models: The ( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGRbGaaiilaiqadsgagaWca8aadaahaaWcbeqaa8qacaWGubaa % aOGaai4oaiaadohaaaa!3B95! $$k,{\vec d^T};s$$ )-Cluster.- 3.2.3 Discussion of the New Cluster Definition.- 3.3 An Algorithm for the Construction of ( % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGRbGaaiilaiqadsgagaWca8aadaahaaWcbeqaa8qacaWGubaa % aOGaai4oaiaadohaaaa!3B95! $$k,{\vec d^T};s$$ )-Clusters.- 3.4 The Construction of Dendrograms of (k; s)-Clusters.- 4 Probability Models of Classification.- 4.1. Current Probability Models in Cluster Analysis.- 4.2. Graph-Theoretic Models of Classification.- 4.2.1. The Model of R.F. Ling.- 4.2.2. A Probability Model Based on Random Multigraphs.- 4.3. Discussion of the Graph-Theoretic Probability Models.- 5 Probability Theory of Completely Labelled Random Multigraphs.- 5.1 Definitions and Notation.- 5.2 A Probability Model of Random Multigraphs.- 5.2.1 Definition of the Probability Space.- 5.2.2 Definition of the Random Variables.- 5.2.3 Relations to Current Probability Models.- 5.3 Some Results for Random Graphs ?nN and Gnp.- 5.4 Limit Theorems for Random Multigraphs.- 5.5 Discussion of the Results.- 5.6 Hints for the Numerical Computation of the Expectations and Distributions.- 6 Classifications by Multigraphs: Three Examples from Medicine.- 6.1 Pharmacokinetics of Urapidil in Patients with Normal and Impaired Renal Function.- 6.1.1 Material and Methods.- 6.1.2 Biometrics: Basic Pharmacokinetics of Urapidil.- 6.1.3 Cluster Analysis of the Urapidil Data.- 6.2 Pharmacokinetics of Lidocaine in Patients with Kidney or Liver Impairments.- 6.2.1 Material and Methods.- 6.2.2 Biometrics: Basic Pharmacikinetics of Lidocaine.- 6.2.3 Cluster Analysis of the Lidocaine Data.- 6.3 Pregnancy-Induced Hypertension.