Fr\'echet Estimation of Time-Varying Covariance Matrices From Sparse Data, With Application to the Regional Co-Evolution of Myelination in the Developing Brain

Abstract

Assessing brain development for small infants is important for determining how the human brain grows during the early period of life when the rate of brain growth is at its peak. The development of MRI techniques has enabled the quantification of brain development. A key quantity that can be extracted from MRI measurements is the level of myelination, where myelin acts as an insulator around nerve fibers and its deployment makes nerve pulse propagation more efficient. The co-variation of myelin deployment across different brain regions provides insights into the co-development of brain regions and can be assessed as a correlation matrix that varies with age. Typically, available data for each child are very sparse, due to the cost and logistic difficulties of arranging MRI brain scans for infants. We showcase here a method where data per subject are limited to measurements taken at only one random age, while aiming at the time-varying dynamics. This situation is encountered more generally in cross-sectional studies where one observes $p$-dimensional vectors at one random time point per subject and is interested in the $p \times p$ correlation matrix function over the time domain. The challenge is that at each observation time one observes only a $p$-vector of measurements but not a covariance or correlation matrix. For such very sparse data, we develop a Fr\'echet estimation method. Given a metric on the space of covariance matrices, the proposed method generates a matrix function where at each time the matrix is a non-negative definite covariance matrix, for which we demonstrate consistency properties. We discuss how this approach can be applied to myelin data in the developing brain and what insights can be gained.