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Spatial classification

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Date
2008
Dewey
Probabilités et mathématiques appliquées
Sujet
Pyramidal clustering ; Spatial classification ; Symbolic data analysis ; Conceptual Lattices ; Kohonen mapping
Journal issue
Discrete Applied Mathematics
Volume
156
Number
8
Publication date
2008
Article pages
1271-1294
Publisher
Elsevier
DOI
http://dx.doi.org/10.1016/j.dam.2007.04.031
URI
https://basepub.dauphine.fr/handle/123456789/397
Collections
  • CEREMADE : Publications
Metadata
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Author
Diday, Edwin
Type
Article accepté pour publication ou publié
Abstract (EN)
The aim of a spatial classification is to position the units on a spatial network and to give simultaneously a set of structured classes of these units “compatible” with the network. We introduce the basic needed definitions: compatibility between a classification structure and a tessellation, (m,k)-networks as a case of tessellation, convex, maximal and connected subsets in such networks, spatial pyramids and spatial hierarchies. As like Robinsonian dissimilarities induced by indexed pyramids generalize ultrametrics induced by indexed hierarchies we show that a new kind of dissimilarity called “Yadidean” induced by spatial pyramids generalize Robinsonian dissimilarities. We focus on spatial pyramids where each class is a convex for a grid, and we show that there are several one-to-one correspondences with different kinds of Yadidean dissimilarities. These new results produce also, as a special case, several one-to-one correspondences between spatial hierarchies (resp. standard indexed pyramids) and Yadidean ultrametrics (resp. Robinsonian) dissimilarities. Qualities of spatial pyramids and their supremum under a given dissimilarity are considered. We give a constructive algorithm for convex spatial pyramids illustrated by an example. We show finally by a simple example that spatial pyramids on symbolic data can produce a geometrical representation of conceptual lattices of “symbolic objects”

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