Latent Functional PARAFAC for Modeling Multidimensional Longitudinal Data

In psychometric sciences, such as social or behavioral sciences, and, similarly, in medical sciences, it is increasingly common to deal with longitudinal data organized as high-dimensional multidimensional arrays, also known as tensors. Within this framework, the time-continuous property of longitudinal data often implies a smooth functional structure on one of the tensor modes. To help researchers investigate such data, we introduce a new tensor decomposition approach based on the PARAFAC decomposition. Our approach allows researchers to represent a high-dimensional functional tensor as a low-dimensional set of functions and feature matrices. Furthermore, to capture the underlying randomness of the statistical setting more efficiently, we introduce a probabilistic latent model in the decomposition. A covariance-based block-relaxation algorithm is derived to obtain estimates of model parameters. Thanks to the covariance formulation of the solving procedure and thanks to the probabilistic modeling, the method can be used in sparse and irregular sampling schemes, making it applicable in numerous settings. Our approach is applied in the psychometric setting to help characterize multiple neurocognitive scores observed over time in the Alzheimer's Disease Neuroimaging Initiative study. Finally, intensive simulations show a notable advantage of our method in reconstructing tensors.