Abstract
We begin with pervasive ultrametricity due to high dimensionality and/or spatial sparsity. How extent or degree of ultrametricity can be quantified leads us to the discussion of varied practical cases when ultrametricity can be partially or locally present in data. We show how the ultrametricity can be assessed in text or document collections, and in time series signals. In our presentation we also discussed applications to chemical information retrieval and to astrophysics, in particular observational cosmology.
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References
BELLMAN, R. (1961): Adaptive Control Processes: A Guided Tour. Princeton University Press, Princeton.
BENZÉCRI, J.P. (1979): L’Analyse des Données, Tome I Taxinomie, Tome II Correspondances. 2nd ed., Dunod, Paris.
BUSTOS, D., NAVARRO, G. and CHÁVEZ, E. (2003): Pivot Selection Techniques for Proximity Searching in Metric Spaces. Pattern Recognition Letters, 24, 2357–2366.
CAILLIEZ, F. and PAGÈS, J.P. (1976): Introduction à l’Analyse de Données. SMASH (Société de Mathématiques Appliquées et de Sciences Humaines), Paris.
CAILLIEZ, F. (1983): The Analytical Solution of the Additive Constant Problem. Psychometrika, 48, 305–308.
CHÁVEZ, E. and NAVARRO, G. (2000): Measuring the Dimensionality of General Metric Spaces. Technical Report TR/DCC-00-1, Department of Computer Science, University of Chile.
CHÁVEZ, E., NAVARRO, G., BAEZA-YATES, R. and MARROQUÍN, J.L. (2001): Proximity Searching in Metric Spaces. ACM Computing Surveys, 33, 273–321.
CHÁVEZ, E. and NAVARRO, G. (2003): Probabilistic Proximity Search: Fighting the Curse of Dimensionality in Metric Spaces. Information Processing Letters, 85, 39–56.
DE SOETE, G. (1986): A Least Squares Algorithm for Fitting an Ultrametric Tree to a Dissimilarity Matrix. Pattern Recognition Letters, 2, 133–137.
FISHER, R.A. (1936): The Use of Multiple Measurements in Taxonomic Problems. The Annals of Eugenics, 7, 179–188.
HORNIK, K. (2005): A CLUE for CLUster Ensembles. Journal of Statistical Software, 14, 12.
LERMAN, I.C. (1981): Classification et Analyse Ordinale des Données. Dunod, Paris.
MURTAGH, F. (1985): Multidimensional Clustering Algorithms. Physica, Würzburg.
MURTAGH, F. (2004): On Ultrametricity, Data Coding, and Computation. Journal of Classification, 21, 167–184.
MURTAGH, F. (2005a): Identifying the Ultrametricity of Time Series. European Physical Journal B, 43, 573–579.
MURTAGH, F. (2005b): Correspondence Analysis and Data Coding with R and Java. Chapman & Hall/CRC, Florida.
MURTAGH, F. (2006a): A Note on Local Ultrametricity in Text, Literary and Linguistic Computing, submitted.
MURTAGH, F. (2006b): From Data to the Physics using Ultrametrics: New Results in High Dimensional Data Analysis. In: A.Yu. Khrennikov, Z. Rakić and I.V. Volovich (Eds.): p-Adic Mathematical Physics, American Institute of Physics Conf. Proc. Vol. 826, 151–161.
NEUWIRTH, E. and REISINGER, L. (1982): Dissimilarity and Distance Coefficients in Automation-Supported Thesauri. Information Systems, 7, 47–52.
RAMMAL, R., ANGLES D’AURIAC, J.C. and DOUCOT, B. (1985): On the Degree of Ultrametricity. Le Journal de Physique — Lettres, 46, L-945–L-952.
RAMMAL, R., TOULOUSE, G. and VIRASORO, M.A. (1986): Ultrametricity for Physicists. Reviews of Modern Physics, 58, 765–788.
TORGERSON, W.S. (1958): Theory and Methods of Scaling, Wiley, New York.
TREVES, A. (1997): On the Perceptual Structure of Face Space. BioSystems, 40, 189–196.
VAN RIJSBERGEN, C.J. (1979): Information Retrieval, 2nd ed., Butterworths.
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Murtagh, F. (2007). Identifying and Exploiting Ultrametricity. In: Decker, R., Lenz, H.J. (eds) Advances in Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70981-7_30
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DOI: https://doi.org/10.1007/978-3-540-70981-7_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70980-0
Online ISBN: 978-3-540-70981-7
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