Problem 1

The covariogram is chosen to be a gaussian model. Assume \[ Y \sim \textrm{N}(10, \Sigma(\sigma^2, \phi))\] where \(\Sigma(\sigma^2, \phi)_{ij} = \sigma^2 \exp(-(d_{ij}/\phi)^2)\) with \(\sigma^2 = 2\) and \(\phi = 5\). The grid is a \(50 \times 30\) equally spaced grid on the US map.

True Covariogram and Semi-variogramTrue Covariogram and Semi-variogram

Simulated Gaussian Spatial Process

Problem 2

The estimated mean parameter is 10.2652628.

Subset of Simulated Data

Empirical Semi-variogram

Problem 3

Using initial values of \(\sigma^2_{(1)} = 2.5\) and \(\phi_{(1)} = 5.5\), the gaussian model is fitted.

Empirical Semi-variogram with the fitted and true variogram

The estimated parameter is \(\hat\sigma^2 = ``2.0681803``\) and \(\hat\phi = ``5.1847943``\). The parameter estimates are quite close. From the plot, we can see it fits the true semi-covariogram quite well.

The semivariogram on 4 directions is plotted. From the way we built the model, we expect it to be isotropic.

The empirical semi-variogram on 4 directions is estimated. From the plot we can see that when the distance is less than 6, the process seems to be isotropic (the four lines are almost indentical). But when distrance larger than 6, the difference becomes larger.