$2.(15$ marks$)$ The data in juraZn.csv are the data of the zinc $($ Zn$\backslash ()\backslash )$ concentration in soil at $259$ locations in the Swiss part of the Jura Mountains. The coordinates are given in the columns of Xloc and Yloc.

$($a$)(3$ marks$)$ Read the data into R using read.csv with the option header$=$TRUE. Transform the data into an sf object $($without specifying the CRS since it is unknown$).$ Plot the data using ggplot$()$ with a color bar.

$($b$)(3$ marks$)$ Following $($a$),$ suppose that we treat the process as isotropic. Plot the semivariogram cloud using all $259$ locations, using the variogram$()$ function with the option cloud$=$TRUE.

Hint: Check out the solution to Problem $3($c$)$ in Tutorial $3.$

$($c$)(3$ marks$)$ Following $($b$),$ in the semivariogram cloud, identify all the pairs of observations that have the fitted semivariance greater than $14,000$ with distance less than $0\mathrm{.}25.$ The semivariance can be found from the variable gamma in the output of variogram$().$

$($d$)(3$ marks$)$ Following $($b$),$ plot the empirical semivariogram using the default options of cutoff and bin width. Then plot the empirical semivariogram with $22$ bins from $0$ to $5\mathrm{.}5($i$.$e$.,$ cutoff $\backslash (=5\mathrm{.}5\backslash ),$ width$=5\mathrm{.}5/22).$ If we are interested in using one of these two plots to fit a semivariogram model at short distances, which do you prefer to use? Briefly explain your choice.

$($e$)(3$ marks$)$ Suppose that we want to evaluate whether the assumption of isotropy is appropriate. Plot the empirical semivariogram in the directions of N$,$ NE$,$ E and SE$.$ Comment on the plots.