  Dimension reduction is an important tool for analyzing high-dimensional data.
The predictor envelope is a method of dimension reduction for regression that
assumes certain linear combinations of the predictors are immaterial to the
regression. The method can result in substantial gains in estimation efficiency
and prediction accuracy over traditional maximum likelihood and least squares
estimates. While predictor envelopes have been developed and studied for
independent data, no work has been done adapting predictor envelopes to spatial
data. In this work, the predictor envelope is adapted to a popular spatial
model to form the spatial predictor envelope (SPE). Maximum likelihood
estimates for the SPE are derived, along with asymptotic distributions for the
estimates given certain assumptions, showing the SPE estimates to be
asymptotically more efficient than estimates of the original spatial model. The
effectiveness of the proposed model is illustrated through simulation studies
and the analysis of a geo-chemical data set.
