Abstract: Dimension reduction provides a useful tool for analyzing high-dimensional data. The recently developed envelope method is a parsimonious version of the classical multivariate regression model that identifies a minimal reducing subspace of the responses. However, existing envelope methods assume an independent error structure in the model. While the assumption of independence is convenient, it does not address the additional complications associated with spatial or temporal correlations in the data. Therefore, we propose a Spatial Envelope method for dimension reduction in the presence of dependencies across space. We study the asymptotic properties of the proposed estimators and show that the asymptotic variance of the estimated regression coefficients under the spatial envelope model is smaller than that of the traditional maximum likelihood estimation. Furthermore, we present a computationally efficient approach for inferences. The efficacy of the proposed method is investigated through simulation studies and an analysis of an Air Quality Standard data set provided by the US Environmental Protection Agency.

Key words and phrases: Dimension reduction, grassmanian manifold, matern covariance function, spatial dependency.