# Fit Gaussian random field for spatial data

I'm dealing with spatial data where the response variable is the gas concentration. In addition, I've the x,y-coordinate values, and another covariates. I'm thinking to fit a Gaussian random field model as follows:

$$Z(s) = X(s) \beta + \epsilon(s)$$

where $[Z(s) : s \in R^2]$, $X(s)$ is a set of p covariates associated with each site s, $\beta$ is a p-dimensional vector of coefficients. I already estimated the covariance function for the error term, but now I'm having trouble on how to proceed. I'm using R, but so far, unfortunately, I've not come across any package where I can include covariates in the model.

I'm still learning spatial statistics, so if you've any hint, I would truly appreciate it.

• There is a package in R called R-INLA (Integrated Nested Laplace Approximation) which is increasingly being used to model latent Gaussian Random Fields (with covariates). See a nice paper here which has some R code in the supplement to get going. There are also several other papers on the INLA webpage here several of which have R code. INLA is is very computationally fast, but it may be a bit black-boxy for some. Regardless, it is the state of the art as far as fitting latent Gaussian models. – gregory_britten Jul 10 '13 at 17:54
• I should add: I don't think R-INLA is on CRAN yet, so you can get the package and documentation from the website I posted above; i.e. here – gregory_britten Jul 10 '13 at 17:57
• If by "Gaussian Random Field" you mean that the distribution of concentration follows a gaussian, then I would like to commend a "Gaussian Mixture Model". If it is the real world, then likely your initial conditions are not perfectly spherical, so your distribution is not perfectly gaussian. Such data, as long as it does not have discontinuities, can often be well fit using gaussian mixtures. These tools are mature - they are fast, they work as they are supposed to. As a MatLab guy, I use the built in "gmdistribution" tools. I cannot direct you in "R". – EngrStudent - Reinstate Monica Jul 10 '13 at 19:01