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disciplinas:verao2007:exercicios [2007/02/15 17:06] paulojus |
disciplinas:verao2007:exercicios [2007/02/17 20:45] paulojus |
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- Inspect [[http://leg.ufpr.br/geoR/tutorials/Rcruciani.R|an example geoestatistical analysis]] for the hydraulic conductivity data. | - Inspect [[http://leg.ufpr.br/geoR/tutorials/Rcruciani.R|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 $x$. | - (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 $x$. | ||
- | - <latex>$Y_i = \alpha + \beta x_i + Z_i$</latex>, where <latex>$\alpha$</latex>$ and <latex>$\beta$</latex> are parameters and the <latex>$Z_i$</latex> 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 $\sigma_Z^2$. |
- | - <latex>$Y_i = A + B x_i + Z_i$</latex> where the <latex>$Z_i$</latex> 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 <latex>$Y_i$</latex> 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? |
- | - (5) Suppose that <latex>$Y=(Y_1,\ldots,Y_n)$</latex> 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. | + | - (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. |
- (6) 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. <latex> $$ | - (6) Is the following a legitimate correlation function for a one-dimensional spatial process $S(x) : x \in \IR$? Give either a proof or a counter-example. <latex> $$ | ||
\rho(u) = \left\{ | \rho(u) = \left\{ |