Exploratory analysis

First of all, some visual tools is used to explore the data. The data is plotted on original scale as well as on a first order trend.

From the first set of plots, we can see from the partial plots that the data is highly correlated with the coordinates. After we removed the first order trend (plot2), this correlation is greatly eliminated. Therefore in the following modeling procedures, we will always remove the first order trend first.

Fitting semivariogram and model comparison

Again a first order trend is assumed when fitting the semivariogram. The parameters for the trend is \(\hat\beta_0 = 607.770661\), \(\hat\beta_1 = -1.278442\) and \(\hat\beta_2 = -1.138741\). The empirical semivariogram is plotted.

## variog: computing omnidirectional variogram

Then we tried to fit a model to the semivariogram. We considered exponential model and gaussian models, both with a first order trend, and with or without nugget effect. The results are summarised in the following table.

model AIC
exponential with nugget effect 930.392205792884
exponential with nugget effect 929.732456215654
gaussian with nugget effect singular
gaussian with nugget effect 946.099836017318

Finally the exponential model without nugget effect is chosen. The model fit to the empirical semivariogram like this.

Universal kriging

Fianlly we want to do a universal kriging on 100 points on each axis, that’s 10000 points in the domain. The model is fit using the first order trend and semivariogram in model 2.

## krige.conv: model with mean given by a 1st order polynomial on the coordinates
## krige.conv: Kriging performed using global neighbourhood

From the prediction plots, we can see that the south-western corner are higher than the north-eastern corner. The prediction error are higer on the north-western corner because that’s where we have very few data points.