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disciplinas:verao2007:exercicios [2007/02/17 22:37] paulojus |
disciplinas:verao2007:exercicios [2007/02/17 22:48] paulojus |
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| - (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. | ||
