An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach - Discussion on the paper by Lindgren, Rue and Lindström

First, we congratulate the authors for their extremely interesting work that sheds new light on Gaussian Markov random-filed (GMRF) modelling. A connection is established with Matérn Gaussian fields through a weak solution of a linear fractional stochastic partial differential equation (SPDE) driven by a Gaussian white noise process. One of the main theoretical results is stated in theorems 3 and 4, where the authors prove the weak convergence of the finite Hilbert representation of the solution for positive integer values of the parameter α to the continuous solution. Recently Bolin and Lindgren (2011) have characterized a class of non-Gaussian random fields as the solution to a nested SPDE driven by Gaussian white noise. Could the authors comment on the possible further extension of their results to the case of SPDEs driven by more general non-Gaussian laws as stable or Lévy processes, in view of obtaining tools suitable for rates or concentrations, which do not typically follow a normal distribution? When d = 2, the integer α restriction of theorems 3 and 4 implies that also the parameter ν is integer in the Matérn class specification. Within this class, the exponential covariance function, one of the most popular in many applied fields, corresponds to the value ν = 0.5. Is the exponential covariance model definitively excluded by the approach proposed? In the discussion the authors recognize that “the approach comes with an implementation and pre-processing cost for setting up the models, as it involves the SPDE solution, triangulations and GMRF representations”. We would be interested in some more comments on the possible qualifications of this cost from an applied statistical point of view. A more detailed comparison of the GMRF solution to competing modelling approaches from a predictive perspective would also be of considerable interest. Here we mainly refer to low-rank (Banerjee et al., 2008; Cressie and Johannesson, 2008; Eidsvik et al., 2010; Crainiceanu et al.,2008) and process convolution methods (Higdon, 1998; Higdon et al., 1999) when estimation is carried out with the same technique (either integrated nested Laplace approximation or Markov chain Monte Carlo methods). Incidentally we think that the geographical interpretation of figure 6(b) would benefit from a more detailed description.