To answer this question, provide your R code, R outputs, images and explanations where required.

Use data s256i from the package geoR. This is the simulated data set with the coordinates of data locations and numeric data at them.

a. Produce a sample variogram on the interval [0,1] using 20 bins.

(0.5 mark)

b. Fit the spherical variogram to the sample variogram by using ordinary least squares. Use the initial values (1, 0.5) and nugget = 0.5.

(0.5 mark)

c. Consider the location (1, 0.5). Plot locations of the data in black and this location in red in the same image.

(0.5 mark)

d. Use the kriging method to compute the predicted value and the variance at the point (1, 0.5). Round the answers with 4 decimal places.

(0.5 mark)

e. Perform a prediction(kriging) on a grid covering the area [0,2]x[0,2]. Plot the result.

(0.5 mark)

f. Explain the obtained plot.

(1 mark)

g. To prepare your data for cross-validation, use the R commands

> a <- as.data.frame(s256i$data)

> s <- SpatialPointsDataFrame(s256i$coords, a, proj4string=CRS(projargs=as.character(NA)), match.ID=TRUE)

> v.fit <- as.vgm.variomodel(ols.n)

where ols.n is the variogram fitted by the ordinary least squares method.

Cross-validate your model by using leave-one-out cross-validation and a bubble plot of the result.

(0.5 mark)

h. Explain the obtained plot.

(1 mark)

i. Cross-validate your model by using 10-fold cross-validation and a bubble plot of the result. Explain the obtained plot. and differences with leave-one-out cross-validation from f and g.

(1 mark)

j. Optimize the monitoring network using the criterion of minimum mean kriging variances. Which data point has the maximum mean kriging variance?

(0.5 mark)

k. Which data point has the second maximal mean kriging variance?

(0.5 mark)

l. Plot the sorted mean kriging variances. Explain the obtained plot.

(0.5 mark)

m. Plotpoints with top and bottom 5% mean kriging variance (use different point shapes for top and bottom points).

(0.5 mark)