Bi-Level Nonstationary Kernels for Online Gaussian Process Regression

In this work, we consider realtime mapping of spatial phenomena using Gaussian processes (GP). For the case that the spatial field changes in variability over different areas, stationary GPs whose kernel hyperparameters are constant may be insufficient in capturing local behaviors. To accurately model and represent data with evolving variability, we propose to use a Gibbs kernel where the length scale is allowed to vary over space. The length scale is modeled as a secondary Gaussian process, and we formulate a maximum a posteriori solution to estimate training values for the length scale GP. Combining this kernel with online sparse methods and nearest neighbor approximations allows us to control computational complexity involved in training and inference, making our approach suitable for real-time applications. The resulting bi-level Gaussian process framework efficiently processes streaming data and maintains good prediction accuracy as measured by standardized mean squared error. Empirical analysis on a bathymetric dataset that has been acquired by an autonomous underwater vehicle suggests that our framework produces comparable predictive performance with respect to recent online GP regression methods while maintaining acceptable computational speed for streaming applications.