Crisp and fuzzy k-means clustering algorithms for multivariate functional data

Functional data analysis, as proposed by Ramsay (Psychometrika 47:379–396, 1982), has recently attracted many researchers. The most popular approach taken in recent studies of functional data has been the extension of statistical methods for the analysis of usual data to that of functional data (e.g...

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Bibliographic Details
Published in:Computational Statistics Vol. 22; no. 1; pp. 1 - 16
Main Authors: Tokushige, Shuichi, Yadohisa, Hiroshi, Inada, Koichi
Format: Journal Article
Language:English
Japanese
Published: Heidelberg Springer Science and Business Media LLC 01.04.2007
Springer Nature B.V
Subjects:
Algorithms
Artificial neural networks
Calculus of variations
Cluster analysis
Clustering
Crisps
Data analysis
Data processing
Functionals
Fuzzy
Fuzzy logic
Mathematical analysis
Mathematical models
Multivariate analysis
Neural networks
Statistical methods
Statistics
Vector quantization
New York
United States > US
Japan
ISSN:0943-4062, 1613-9658
Online Access:Get full text
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Summary:Functional data analysis, as proposed by Ramsay (Psychometrika 47:379–396, 1982), has recently attracted many researchers. The most popular approach taken in recent studies of functional data has been the extension of statistical methods for the analysis of usual data to that of functional data (e.g., Ramsay and Silverman in Functional data Analysis Springer, Berlin Heidelberg New York, 1997, Applied functional data analysis: methods and case studies. Springer, Berlin Heidelberg New York, 2002; Mizuta in Proceedings of the tenth Japan and Korea Joint Conference of Statistics, pp 77–82, 2000; Shimokawa et al. in Japan J Appl Stat 29:27–39, 2000). In addition, several methods for clustering functional data have been proposed (Abraham et al. in Scand J Stat 30:581–595, 2003; Gareth and Catherine in J Am Stat Assoc 98:397–408, 2003; Tarpey and kinateder in J Classif 20:93–114, 2003; Rossi et al. in Proceedings of European Symposium on Artificial Neural Networks pp 305–312, 2004). Furthermore, Tokushige et al. (J Jpn Soc Comput Stat 15:319–326, 2002) defined several dissimilarities between functions for the case of functional data. In this paper, we extend existing crisp and fuzzy k-means clustering algorithms to the analysis of multivariate functional data. In particular, we consider the dissimilarity between functions as a function. Furthermore, cluster centers and memberships, which are defined as functions, are determined at the minimum of a certain target function by using a calculus-of-variations approach.
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ISSN:0943-4062
1613-9658
DOI:10.1007/s00180-006-0013-0