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Graphs as Structural Models
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Table of Contents

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.

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