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disciplinas:verao2007:exercicios [2007/01/30 14:25] paulojus |
disciplinas:verao2007:exercicios [2007/01/30 14:30] paulojus |
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- Load the Paraná data-set from geoR using the command <code R>data(parana)</code> and inspect its documentation using <code R>help(parana)</code>. For these data, consider the same questions as were raised in Exercise 1.4. | - Load the Paraná data-set from geoR using the command <code R>data(parana)</code> and inspect its documentation using <code R>help(parana)</code>. For these data, consider the same questions as were raised in Exercise 1.4. | ||
- Read the Chapter 2 of Diggle & Ribeiro (2007) (you can get {{disciplinas:pdf:chapter2.pdf|this chapter here}}) | - Read the Chapter 2 of Diggle & Ribeiro (2007) (you can get {{disciplinas:pdf:chapter2.pdf|this chapter here}}) | ||
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- $Y_i = \alpha + \beta x_i + Z_i$, where $\alpha$ and $\beta$ are parameters and the $Z_i$ are mutually independent with mean zero and variance $\sigma_Z^2$. | - $Y_i = \alpha + \beta x_i + Z_i$, where $\alpha$ and $\beta$ are parameters and the $Z_i$ are mutually independent with mean zero and variance $\sigma_Z^2$. | ||
- $Y_i = A + B x_i + Z_i$ 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? | - $Y_i = A + B x_i + Z_i$ 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? | ||
- | - 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 $\theta=(\mu,\sigma^2,\phi)$ | + | - 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. |
- | 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. | + | |
- 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. <code> $$ | - Is the following a legitimate correlation function for a one-dimensional spatial process $S(x) : x \in \IR$? Give either a proof or a counter-example. <code> $$ | ||
\rho(u) = \left\{ | \rho(u) = \left\{ | ||
Linha 35: | Linha 35: | ||
\right. | \right. | ||
$$ </code> | $$ </code> | ||
- | - 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 | + | - 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 <code> S = D \Lambda^{\frac{1}{2}} Y </code> |
- | <code> S = D \Lambda^{\frac{1}{2}} Y </code> | + | - Write an ''R'' function to simulate realisations using the above method for any specified set of points $x_i$ and a range of correlation functions of your choice. Use your function to simulate a realisation of $S$ on (a discrete approximation to) the unit interval $(0,1)$. |
- | - Write an ''R'' function to simulate realisations using the above method for any specified set of points $x_i$ and a range of correlation functions of your choice. Use your function to simulate a realisation of $S$ on (a discrete approximation to) the unit interval $(0,1)$. | + | - Now investigate how the appearance of your realisation $S$ changes if in the equation above you replace the diagonal matrix $\Lambda$ by truncated form in which you replace the last $k$ eigenvalues by zeros. |
- | - Now investigate how the appearance of your realisation $S$ changes if in the equation above you replace the diagonal matrix $\Lambda$ by truncated form in which you replace the last $k$ eigenvalues by zeros. | + | |
- | ==== Semana 3 ==== | ||
- | - Fit a model to the surface elevation data assuming a linear trend model on the coordinates and a Matérn correlation function with parameter kappa=2.5. | + | |
- | Use the fitted model as the true model and perform a simulation study (i.e. simulate from this model) to compare parameter estimation based on maximum likelihood, restricted maximum likelihood and variograms. | + | |
- | - Simulate 200 points in the unit square from the Gaussian model without measurement error, constant mean equals to zero, unit variance and exponential correlation function with $\phi=0.25$ and anisotropy parameters $(\psi_A=\pi/3, \psi_R=2)$. Obtain parameter estimates (using maximum likelihood): | + | |
- | * assuming a isotropic model | + | |
- | * try to estimate the anisotropy parameters | + | ==== Semana 3 ==== |
- | Compare the results and repeat the exercise for $\phi_R=4$. | + | - Fit a model to the surface elevation data assuming a linear trend model on the coordinates and a Matérn correlation function with parameter kappa=2.5. Use the fitted model as the true model and perform a simulation study (i.e. simulate from this model) to compare parameter estimation based on maximum likelihood, restricted maximum likelihood and variograms. |
- | - Consider a stationary trans-Gaussian model with known transformation function $h(\cdot)$, let $x$ be an arbitrary | + | - Simulate 200 points in the unit square from the Gaussian model without measurement error, constant mean equals to zero, unit variance and exponential correlation function with $\phi=0.25$ and anisotropy parameters $(\psi_A=\pi/3, \psi_R=2)$. Obtain parameter estimates (using maximum likelihood): |
+ | * assuming a isotropic model | ||
+ | * try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for $\phi_R=4$. | ||
+ | - Consider a stationary trans-Gaussian model with known transformation function $h(\cdot)$, let $x$ be an arbitrary | ||
location within the study region and define $T=h^{- 1}{S(x)}$. Find explicit expressions for ${\rm P}(T>c|Y)$ where | location within the study region and define $T=h^{- 1}{S(x)}$. Find explicit expressions for ${\rm P}(T>c|Y)$ where | ||
$Y=(Y_1,...,Y_n)$ denotes the observed measurements on the untransformed scale and: | $Y=(Y_1,...,Y_n)$ denotes the observed measurements on the untransformed scale and: | ||
* $h(u)=u$ | * $h(u)=u$ | ||
* $h(u) = \log u$ | * $h(u) = \log u$ | ||
- | * $h(u) = \sqrt{u}$ | + | * $h(u) = \sqrt{u}$. \\ |
- | - Analyse the Paraná data-set or any other data set of your choice assuming priors obtaining: | + | - Analyse the Paraná data-set or any other data set of your choice assuming priors obtaining: |
* a map of the predicted values over the area | * a map of the predicted values over the area | ||
* a map of the predicted std errors over the area | * a map of the predicted std errors over the area |