Wavelet-based functional mixed models

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2006-04, Vol.68 (2), p.179-199
• Main Authors: Morris, Jeffrey S., Carroll, Raymond J.
• Format: Article
• Language: English
• Subjects: Bayesian analysis; Bayesian method; Bayesian methods; Coefficients; Corn oil; Covariance; Covariance matrices; Crypts; Data analysis; Diet; Exact sciences and technology; Fourier analysis; Functional data analysis; Inference; Linear inference, regression; Mathematical independent variables; Mathematical methods; Mathematical models; Mathematics; Methodology; Mixed models; Model averaging; Modeling; Nonparametric inference; Nonparametric regression; Numerical analysis; Numerical analysis. Scientific computation; Probability and statistics; Proteomics; Regression analysis; Sciences and techniques of general use; Statistical methods; Statistical models; Statistical variance; Statistics; Studies; Variance; Wavelets
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1467-9868.2006.00539.x

Increasingly, scientific studies yield functional data, in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid. We present new methodology that generalizes the linear mixed model to the functional mixed model framework, with model fitting done by using a Bayesian wavelet-based approach. This method is flexible, allowing functions of arbitrary form and the full range of fixed effects structures and between-curve covariance structures that are available in the mixed model framework. It yields nonparametric estimates of the fixed and random-effects functions as well as the various between-curve and within-curve covariance matrices. The functional fixed effects are adaptively regularized as a result of the non-linear shrinkage prior that is imposed on the fixed effects' wavelet coefficients, and the random-effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient. Because we have posterior samples for all model quantities, we can perform pointwise or joint Bayesian inference or prediction on the quantities of the model. The adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks.