Feature screening for metric space-valued responses based on Fréchet regression with its applications

In various applications, we need to handle more general types of responses, such as distributional data and matrix-valued data, rather than a scalar variable. When the dimension of predictors is ultrahigh, it is necessarily important to identify the relevant predictors for such complex types of responses. For example, in our Alzheimer's disease neuroimaging study, we need to select the relevant single nucleotide polymorphisms out of 582 591 candidates for the distribution of voxel-level intensities in each of 42 brain regions. To this end, we propose a new sure independence screening (SIS) procedure for general metric space-valued responses based on global Fréchet regression, termed as Fréchet-SIS. The marginal general residual sum of squares is utilized to serve as a marginal utility for evaluating the importance of predictors, where only a distance between data objects is needed. We theoretically show that the proposed Fréchet-SIS procedure enjoys the sure screening property under mild regularity conditions. Monte Carlo simulations are conducted to demonstrate its excellent finite-sample performance. In Alzheimer's disease neuroimaging study, we identify important genes that correlate with brain activity across different stages of the disease and brain regions. In addition, we also include an economic case study to illustrate our proposal.