Functional Cluster Analysis via Orthonormalized Gaussian Basis Expansions and Its Application

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.