Abstract
Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.
Highlights
With the technological advances seen in recent years, functional datasets are increasingly multivariate
Another important contribution is that our approach directly models the multivariate covariance structure of all random effects included in the model using multivariate functional principal components (FPCs) and implicitly models the covariances between the dimensions
Note that the estimated univariate error variances have been proposed as weights for two separate and independent modelling decisions: as weights in the scalar product of the multivariate functional principal component analysis (MFPCA) and as regression weights under heteroscedasticity across dimensions
Summary
With the technological advances seen in recent years, functional datasets are increasingly multivariate. The main advantages of our multivariate regression model, compared to Goldsmith and Kitago (2016) and Zhu et al (2017), are that it is readily available for sparse and irregular functional data and that it allows to include multiple nested or crossed random processes, both of which are required in our data examples Another important contribution is that our approach directly models the multivariate covariance structure of all random effects included in the model using multivariate functional principal components (FPCs) and implicitly models the covariances between the dimensions. The two measured modes (acoustic and articulatory, see Figure 3) are expected to be closely related but joint analyses have not yet incorporated the functional nature of the data These two examples motivate the development of a regression model for sparse and irregularly sampled multivariate functional data that can incorporate crossed or nested functional random effects as required by the study design in addition to flexible covariate effects.
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