We propose a class of fully process-based low-rank spatially-varying cross-covariance matrices that produce non-degenerate spatial processes and that effectively capture non-stationary covariances among the multiple outcomes. We provide theoretical and modeling insight into these constructions and elucidate certain implications of some common structural assumptions in building cross-covariance matrices. We also propose low rank version of cross-covariance functions using predictive process class of models, popularly employed in spatial statistics to handle large datasets. Predictive process is obtained by projecting the parent Gaussian process onto a space spanned by a set of basis functions. An efficient model to choose those basis functions and have been proposed.
Being a low rank model, predictive process often loses spatial information which might lead to spurious inferences. In the Chapter of this thesis, this loss of information has been quantified and model based adjustments have been suggested. Proposed models have been validated with carefully designed simulation studies. Finally, they have been employed to analyze interesting ecological datasets. Our framework has been able to produce substantive inferential tools such as maps of non-stationary cross-covariances that constitute the premise of further mechanistic modeling and hitherto not been easily available for environmental scientists and ecologists.