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disciplinas:verao2007:exercicios [2007/02/17 22:32] paulojus |
disciplinas:verao2007:exercicios [2007/02/17 22:48] 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, <m>Y_i : i=1, ... ,n</m> associated with a sequence of positions <m>x_i: i=1,...,n</m> along a one-dimensional spatial axis <m>x</m>. | - (4) Consider the following two models for a set of responses, <m>Y_i : i=1, ... ,n</m> associated with a sequence of positions <m>x_i: i=1,...,n</m> along a one-dimensional spatial axis <m>x</m>. | ||
| - | - <m>Y_{i} = alpha + beta x_{i} + Z_{i}</m>, where <m>alpha</m> and <m>beta</m> are parameters and the <m>Z_{i}</m> are mutually independent with mean zero and variance <m>sigma^2 _{Z}</m>. | + | - <m>Y_{i} = alpha + beta x_{i} + Z_{i}</m>, where <m>alpha</m> and <m>beta</m> are parameters and the <m>Z_{i}</m> are mutually independent with mean zero and variance <m>sigma^2_{Z}</m>. |
| - <m>Y_i = A + B x_i + Z_i</m> 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? | - <m>Y_i = A + B x_i + Z_i</m> 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. | - (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. | + | - (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.\\ |
| - | <m> rho(u) = delim{lbrace}{matrix{2}{1}{{1-u : 0 <= u <= 1}{0 : u>1}}}{}</m> | + | <m> rho(u) = delim{lbrace}{matrix{2}{1}{{1-u : 0 <= u <= 1}{0 : u>1}}}{} </m>\\ |
| - | - (7) 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 S = D \Lambda^{\frac{1}{2}} Y | + | - (7) 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 <latex>$S = D \Lambda^{\frac{1}{2}} Y$</latex> |
| - (7) 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 $(0,1)$. | - (7) 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 $(0,1)$. | ||
| - (7) 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. | - (7) 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. | ||
