Clustering for Functional Data via Nonlinear Mixed Effects Models

Abstract

We consider the problem of clustering functional data using nonlinear mixed effects models along with the technique of basis expansions. With the help of fixed and random effects functions, the nonlinear mixed effects model makes it easy to handle unbalanced or sparse data which are highly occurred in the longitudinal study. We assume different numbers of basis functions for fixed and random effects functions. Unknown parameters included in the model are estimated by the maximum likelihood method along with the EM algorithm, and then the numbers of basis functions included in the model are selected by model selection criteria.
We then apply hierarchical and non-hierarchical clustering methods to the predicted coefficients of the random effect terms of functional data in order to highlight the features of each subject. The hierarchical clustering such as the Ward's method proceeds in successive steps from smaller to larger clusters, which can be directly observed visually. In contrast, the non-hierarchical clustering such as the self-organizing maps consists of progressively refining the data partitions to obtain a given number of clusters. In functional cluster analysis, we can remove the measurement errors of observed data and therefore we can capture the functional structure behind the data. We report the results of application of the proposed method to some real data sets such as environmental data and weather data.