Title: Fr\'echet estimation of dynamic covariance matrices, with application to regional myelination in the developing brain
Authors: Alexander Petersen - Brigham Young University (United States) [presenting]
Hans-Georg Mueller - University of California Davis (United States)
Sean Deoni - University of Colorado Denver (United States)
Abstract: Assessing brain development for small infants is important for determining how the human brain grows during the early period of life when the growth rate is at its peak. MRI techniques enable the quantification of brain development, with a key quantity being 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 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, so that one has cross-sectional data available, while aiming at the time-varying dynamics. 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 that 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.