Maximum Likelihood Inference of Spatial Covariance Parameters of Large Data Sets in Geosciences

Abstract

In the last two decades, there has been an increase in the availability of large data sets in geosciences (e.g. data provided by satellites, large data sets collected with modern geophysical techniques, exhaustive sampling campaigns or the increase of historical data like climatological data). One challenging problem for such large data in spatial statistics is in implementing the well- established method of maximum likelihood for the models based on spatial covariances. In these cases, we need to invert a matrix of dimension $n\,\times \,n; n$ being the number of experimental data. The inversion must be performed several times in any numerical maximization procedure which implies that for the number of experimental data around 1000, the application of the full maximum likelihood is impractical. Nevertheless, we can resort to an approximate maximum likelihood estimation method proposed by Vecchia (1988). This procedure uses an approximation to the conditional probability distribution which involves factoring the complete likelihood. In this paper, we investigate a new resampling method to make the Vecchia approximation faster. Also, for the first time, the Vecchia method is applied to random fields in presence of a drift. A simulated random field is used to assess the performance of our method.