Fast Spatial Interpolation using Sparse Gaussian Processes
1 July 2005
Ben Ingram, Lehel Csato, David Evans
The estimation of the natural ambient radioactivity in this entry to the Spatial Interpolation Comparison 2004 (SIC2004) uses Gaussian processes (GP's) to predict the underlying dispersal process. GP's enable us to predict easily levels of radioactivity at previously unseen locations and in addition they allow us to assess the uncertainty in the predicted value. To speed up computation time, which is cubic in the number of examples, a sequential, sparse implementation of the Gaussian process inference (SSGP) was used together with a Gaussian observational noise assumption. The examination of the available data led to a covariance function which is a mixture of exponential and squared-exponential functions. The mixture was chosen so that it incorporates both the local ambiguity in the data and also at the same time it captures the larger-scale variation of the observations.
A further characteristic of the competition was the availability of 10 days of "prior observations". The individual sets comprised of data that was recorded at the same time but at different locations across Germany. Consequently in the modelling stage we assumed that the underlying process governing the observations was invariant across the 10 days of "prior observations", but that each day, i.e. each dataset, should be treated independently. These prior observations were used to infer the parameters of the covariance function, which are called hyper-parameters. Conforming to the Bayesian inference method, the exact values of the hyper-parameters were not fixed; rather we assigned a probability distribution to these parameters, giving a two level inference scheme. The inferred values of the hyper-parameters were used to set up a prior GP, and the same two level inference was employed when evaluating the method on the released SIC2004 dataset.
We show that the sparse approximation with the SSGP method can be used instead of the conventional GP's without significant loss in accuracy leading to a greatly reduced calculation time. We also compare the SSGP performance with standard machine learning techniques: the SSGP results compare favourably to the other benchmark techniques.
Volume 1, No. 2, August 2005