Network Functional Linear Regression

We consider a large-scale network where a scalar response and a functional predictor are observed for each individual.To incorporate the network information and to depict the dynamic impact of the functional predictor on the response of each individual, we investigate a network functional linear model.The model assumes that each individual's response can be explained by a linear combination of the responses of the neighbors and a functional regression of the individual.We first approximate functional regression coefficient by a finite representation based on functional principal component analysis technique (FPCA) and then introduce a novel least-squares type of procedure to estimate the coefficients after dimension reduction.In addition, we introduce two modified BIC-type criteria for choosing the number of principal components.We study the convergence rates of the functional regression coefficients and establish the asymptotic normality of the network autoregression coefficients, as well as the consistency of the model selection procedures.Extensive simulation studies are conducted to evaluate the finite sample performance of our proposed method.Finally, we illustrate the usefulness of our method by applying it to two applications.