Motivated by recent work on studying massive imaging data in various neuroimaging studies, we propose a novel spatially varying coefficient model (SVCM) to capture the varying association between imaging measures in a three-dimensional volume (or two-dimensional surface) with a set of covariates. Two stylized features of neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations. To specifically account for these two features, SVCM includes a measurement model with multiple varying coefficient functions, a jumping surface model for each varying coefficient function, and a functional principal component model. We develop a three-stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations. The estimation procedure includes a fast multiscale adaptive estimation and testing procedure to independently estimate each varying coefficient function, while preserving its edges among different piecewise-smooth regions. We systematically investigate the asymptotic properties (e.g., consistency and asymptotic normality) of the multiscale adaptive parameter estimates. We also establish the uniform convergence rate of the estimated spatial covariance function and its associated eigenvalues and eigenfunctions. Our Monte Carlo simulation and real-data analysis have confirmed the excellent performance of SVCM. Supplementary materials for this article are available online.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Asymptotic normality
- Functional principal component analysis
- Jumping surface model
- Wald test