Combining Functional Data Registration and Factor Analysis

ABSTRACT

We extend the definition of functional data registration to encompass a larger class of registration models. In contrast to traditional registration models, we allow for registered functions that have more than one primary direction of variation. The proposed Bayesian hierarchical model simultaneously registers the observed functions and estimates the two primary factors that characterize variation in the registered functions. Each registered function is assumed to be predominantly composed of a linear combination of these two primary factors, and the function-specific weights for each observation are estimated within the registration model. We show how these estimated weights can easily be used to classify functions after registration using both simulated data and a juggling dataset. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian modeling
• Factor analysis
• Functional data
• Registration
• Variational Bayes

Supplementary Materials

R Code: Code to register functions and perform factor analysis using MCMC methods can be found in Reg_and_FA_MCMC.R. Code for the variational Bayes approximation to the MCMC analysis can be found in Reg_and_FA_VB.R. Appendices: Appendix A specifies the complete registration and factor analysis model and provides all of the details of the MCMC sampling scheme used for inference. Appendix B provides the details of the variational Bayes approximation to the MCMC sampling scheme. Appendix C includes further analysis of the complete juggling data set. In Section 4, analysis was performed on a random subset of 25 of the 113 available observations. Here we perform the same analysis on each of 4 subsets that together form a partition of the remaining 88 functions. This analysis is provided to illustrate the consistency in registration and factor estimation amongst the 5 subsets of the original data when inference is performed with the proposed model.

Acknowledgments

This research was partially supported by NSF grants DEB-0813743, CMG-0934735, and DMS-1053252.