  Gaussian process is a theoretically appealing model for nonparametric
analysis, but its computational cumbersomeness hinders its use in large scale
and the existing reduced-rank solutions are usually heuristic. In this work, we
propose a novel construction of Gaussian process as a projection from fixed
discrete frequencies to any continuous location. This leads to a valid
stochastic process that has a theoretic support with the reduced rank in the
spectral density, as well as a high-speed computing algorithm. Our method
provides accurate estimates for the covariance parameters and concise form of
predictive distribution for spatial prediction. For non-stationary data, we
adopt the mixture framework with a customized spectral dependency structure.
This enables clustering based on local stationarity, while maintains the joint
Gaussianness. Our work is directly applicable in solving some of the challenges
in the spatial data, such as large scale computation, anisotropic covariance,
spatio-temporal modeling, etc. We illustrate the uses of the model via
simulations and an application on a massive dataset.
