A Low Rank Gaussian Process Prediction Model for Very Large Datasets

Abstract

The Gaussian process prediction model requires expensive computation to invert the covariance matrix it depends on and also has considerable storage needs. A recent method for very large spatial data known as Fixed Rank Kriging allows for prediction when the Gaussian process prediction model cannot and is easily implemented with less assumptions about the process. However, Fixed Rank Kriging requires the estimation of a matrix which must be positive definite and the original estimation procedure cannot guarantee this property. We present a result that shows when a matrix subtraction of a given form will give a positive definite matrix. Motivated by this result, we propose an iterative Fixed Rank Kriging algorithm that ensures positive definiteness of the matrix required for prediction and show that under mild conditions the algorithm numerically converges. The new Fixed Rank Kriging procedure is implemented to predict missing chlorophyll observations for very large regions of ocean color. Predictions are compared to those made by other well known methods of spatial prediction.