Combinatorial Data Analysis: Optimization by Dynamic by Lawrence Hubert, Phipps Arabie, Jacqueline Meulman
By Lawrence Hubert, Phipps Arabie, Jacqueline Meulman
I wished to benefit approximately combinatorial optimization for a specific software in cluster research, and this booklet hit the mark. it is a in actual fact written evaluate of the applying of basic dynamic programming to cluster research, item sequencing and seriation, and different information research difficulties "in which the association of a set of items is admittedly central." [from the Preface]
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Example text
Conversely, any partition hierarchy of the form PI , . . , PT can be identified with the equivalence class of all ultrametric matrices that induce it. 2. HIERARCHICAL CLUSTERING 37 to the characterization of a full partition hierarchy PI , . . , Pn used throughout the previous section. Given some fixed partition hierarchy, PI, ... ,Pr, there are an infinite number of ultrametric matrices that induce it, but all can be generated by (restricted) monotonic functions of what might be called the basic ultrametric matrix U° = {u°j}, whose entries are defined by u°j = min[/c — 1 | objects Oi and Oj appear within the same class in the partition Pk].
QK-, increasing the size of the problems that might be effectively approached. Several of these admissibility issues will be considered in the following sections. ,SM, such that for some measure of heterogeneity H(-} that attaches a value to each possible subset of 17 18 CHAPTERS. CLUSTER ANALYSIS S, either the sum or, alternatively, is minimized. This stipulation assumes that heterogeneity has a cost interpretation and that smaller values of the heterogeneity indices represent the 'better' subsets (or clusters).
Moreover, because u°j for i ^ j can be only one of the integer values from 1 to T — 1, each ultrametric in the equivalence class that generates the fixed hierarchy may be defined by one of T — 1 distinct values. ,PT, and implicitly to all object pairs that appear together for the first time within a subset in Pk. To provide an alternative interpretation, the basic ultrametric matrix can also be characterized as defining a collection of linear equality and inequality constraints that any ultrametric in a specific equivalence class must satisfy.