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2024-11-22
Yi Zhao, Ph.D., Department of Biostatistics and Health Data Science, Indiana University School of Medicine
"Object-On-Object Regression"
In this presentation, two approaches of object-on-object regression will be introduced. The first is to perform a Covariance-on-Covariance Regression (CoCR), which transforms covariance matrix objects to Euclidean space for regression; and the second is to perform a Density-on-Density Regression (DoDR), which directly defines regression in the correspondingly constructed Riemannian manifold. The CoCR model assumes that there exists (at least) a pair of linear projections on outcome covariance matrices and predictor covariance matrices such that a log-linear model links the variances in the projection spaces, as well as additional covariates of interest. An ordinary least square type of estimator, which relaxes the distributional assumption, is proposed to simultaneously identify the projections and estimate model coefficients. Under regularity conditions, the proposed estimator is asymptotically consistent. This type of model is motivated by utilizing functional connectivity within the resting-state network to predict brain connectivity within a corresponding task-state network. Applying to data collected in the Human Connectome Project Aging study, three networks, corresponding to a global signal network, a task-related network, and a task-unrelated network, are identified.
The DoDR model is introduced to elucidate the association between densities via a warping function. The proposed model has the advantage of a being straightforward demonstration of how one density transforms into another. Using the Riemannian representation of density functions, which is the square-root function (or half density), the model is defined in the correspondingly constructed Riemannian manifold. To estimate the warping function, it is proposed to minimize the average Hellinger distance, which is equivalent to minimizing the average Fisher-Rao distance between densities. An optimization algorithm is introduced by estimating the smooth monotone transformation of the warping function. Asymptotic properties of the proposed estimator are discussed. Simulation studies demonstrate the superior performance of the proposed approach over competing approaches in predicting outcome density functions. Applying to a proteomic-imaging study from the Alzheimer’s Disease Neuroimaging Initiative, the proposed approach illustrates the connection between the distribution of protein abundance in the cerebrospinal fluid and the distribution of brain regional volume. Discrepancies among cognitive normal subjects, patients with mild cognitive impairment, and Alzheimer’s disease (AD) are identified and the findings are in line with existing knowledge about AD.
