1. Produce a plot of the Rongelap data in which a continuous colour scale or grey scale is used to indicate the value of the emission count per unit time at each location, and the two sub-areas with the 5 by 5 sub-grids at 50 metre spacing are shown as insets.
2. Construct a polygonal approximation to the boundary of The Gambia. Construct plots of the malaria data which show the spatial variation in the values of the observed prevalence in each village and of the greenness covariate.
3. Consider the elevation data as a simple regression problem with elevation as the response and north-south location as the explanatory variable. Fit the standard linear regression model using ordinary least squares. Examine the residuals from the linear model, with a view to deciding whether any more sophisticated treatment of the spatial variation in elevation might be necessary.
4. Find a geostatistical data-set which interests you.
   1. What scientific questions are the data intended to address? Do these concern estimation, prediction, or testing?
   2. Identify the study region, the design, -the response and the covariates, if any.
   3. What is the support of each response?
   4. What is the underlying signal?
   5. If you wished to predict the signal throughout the study region, would you choose to interpolate the response data?
5. Load the Paraná data-set from geoR using the command

   data(parana)

   and inspect its documentation using

   help(parana)

   . For these data, consider the same questions as were raised in Exercise 1.4.
6. Read the Chapter 2 of Diggle & Ribeiro (2007) (you can get this chapter here)

1. load the data sets parana, Ksat e ca20 available in geoR using commands such as:

   data(parana)

   and the documentation describing each data set with the help() function

   help(parana)

   Perform exploratory data analysis and build a model you find suitable for each data.
2. In the examples above, would you have othe candidate models for each data-set?
3. Inspect an example geoestatistical analysis for the hydraulic conductivity data.
4. Consider the following two models for a set of responses, $Y_i : i=1,\ldots,n$ associated with a sequence of positions $x_i: i=1,\ldots,n$ along a one-dimensional spatial axis $x$.
   1. $Y_i = \alpha + \beta x_i + Z_i$, where $\alpha$ and $\beta$ are parameters and the $Z_i$ are mutually independent with mean zero and variance $\sigma_Z^2$.
   2. $Y_i = A + B x_i + Z_i$ where the $Z_i$ are as in (a) but $A$ and $B$ are now random variables, independent of each other and of the $Z_i$, each with mean zero and respective variances $\sigma_A^2$ and $\sigma_B^2$.
      For each of these models, find the mean and variance of $Y_i$ and the covariance between $Y_i$ and $Y_j$ for any $j \neq i$. Given a single realisation of either model, would it be possible to distinguish between them?
5. Suppose that $Y=(Y_1,\ldots,Y_n)$ follows a multivariate Gaussian distribution with ${\rm E}[Y_i]=\mu$ and ${\rm Var}\{Y_i\}=\sigma^2$ and that the covariance matrix of $Y$ can be expressed as $V=\sigma^2 R(\phi)$. Write down the log-likelihood function for $\theta=(\mu,\sigma^2,\phi)$

based on a single realisation of $Y$ and obtain explicit expressions for the maximum likelihood estimators of $\mu$ and $\sigma^2$ when $\phi$ is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when $\phi$ is unknown.

1. Is the following a legitimate correlation function for a one-dimensional spatial process $S(x) : x \in \IR$? Give either a proof or a counter-example.

 $$
    \rho(u) = \left\{
      \begin{array}{rcl}
        1-u & : & 0 \leq u \leq 1 \\
        0 & : & u>1
      \end{array}
    \right.
    $$

1. Consider the following method of simulating a realisation of a one-dimensional spatial process on $S(x) : x \in \IR$, with mean zero, variance 1 and correlation function $\rho(u)$. Choose a set of points $x_i \in \IR : i=1,\ldots,n$. Let $R$ denote the correlation matrix of $S=\{S(x_1),\ldots,S(x_n)\}$. Obtain the singular value decomposition of $R$ as $R = D \Lambda D^\prime$ where $\lambda$ is a diagonal matrix whose non-zero entries are the eigenvalues of $R$, in order from largest to smallest. Let $Y=\{Y_1,\ldots,Y_n\}$ be an independent random sample from the standard Gaussian distribution, ${\rm N}(0,1)$. Then the simulated realisation is

\begin{equation}
      S = D \Lambda^{\frac{1}{2}} Y.
      \label{eqn03:svdexercise}
    \end{equation}

Now investigate how the appearance of your realisation $S$ changes if in (\ref{eqn03:svdexercise}) you replace the diagonal matrix $\Lambda$ by truncated form in which you replace the last $k$ eigenvalues by zeros.