Functional Cluster Analysis via Orthonormalized Gaussian Basis Expansions and Its Application

• Published in: Journal of classification Vol. 27; no. 2; pp. 211 - 230
• Main Authors: Kayano, Mitsunori, Dozono, Koji, Konishi, Sadanori
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.09.2010; Springer; Springer Nature B.V
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Bioinformatics; Cluster analysis; Clustering; Computer science; control theory; systems; Exact sciences and technology; Functional analysis; Marketing; Mathematical analysis; Mathematics; Mathematics and Statistics; Methods; Methods of scientific computing (including symbolic computation, algebraic computation); Multivariate analysis; Normal distribution; Numerical analysis. Scientific computation; Pattern Recognition; Probability and statistics; Proteins; Psychometrics; Sciences and techniques of general use; Signal,Image and Speech Processing; Statistical methods; Statistical Theory and Methods; Statistics; Theoretical computing; Cholesky decomposition; means; Radial basis functions; Self-Organizing Maps; Functional data; Gram-Schmidt orthonormalization; Protein structure; Cluster analysis; Three-dimensional calculations; Cluster analysis (statistics); Decomposition method; Decomposition; Multivariate analysis; Multidimensional analysis; Cholesky method; k-means; Integral; Data structure; Mathematical expansion; Expansion; Measurement error; Discriminant analysis; Data analysis; Functional analysis; Functional; Observation data
• ISSN: 0176-4268; 1432-1343
• DOI: 10.1007/s00357-010-9054-8

We propose functional cluster analysis (FCA) for multidimensional functional data sets, utilizing orthonormalized Gaussian basis functions. An essential point in FCA is the use of orthonormal bases that yield the identity matrix for the integral of the product of any two bases. We construct orthonormalized Gaussian basis functions using Cholesky decomposition and derive a property of Cholesky decomposition with respect to Gram-Schmidt orthonormalization. The advantages of the functional clustering are that it can be applied to the data observed at different time points for each subject, and the functional structure behind the data can be captured by removing the measurement errors. Numerical experiments are conducted to investigate the effectiveness of the proposed method, as compared to conventional discrete cluster analysis. The proposed method is applied to three-dimensional (3D) protein structural data that determine the 3D arrangement of amino acids in individual protein.