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disciplinas:verao2007:exercicios [2007/02/17 23:42] paulojus |
disciplinas:verao2007:exercicios [2007/02/18 20:16] (atual) paulojus |
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| ===== Exercícios ===== | ===== Exercícios ===== | ||
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| ==== Semana 1 ==== | ==== Semana 1 ==== | ||
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| <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 <latex>$S(x) : x \in R$</latex>, with mean zero, variance 1 and correlation function <m>rho(u)</m>. Choose a set of points <latex>$x_i \in \R : i=1,\ldots,n$</latex>. Let <m>R</m> denote the correlation matrix of <latex>$S=\{S(x_1),\ldots,S(x_n)\}$</latex>. Obtain the singular value decomposition of <m>R</m> as <latex>$R = D \Lambda D^\prime$</latex> where <m>Lambda</m> is a diagonal matrix whose non-zero entries are the eigenvalues of <m>R</m>, in order from largest to smallest. Let <latex>$Y=\{Y_1,\ldots,Y_n\}$</latex> be an independent random sample from the standard Gaussian distribution, <latex>${\rm N}(0,1)$</latex>. Then the simulated realisation is <latex>$S = D \Lambda^{\frac{1}{2}} Y$</latex> | - (7) Consider the following method of simulating a realisation of a one-dimensional spatial process on <latex>$S(x) : x \in R$</latex>, with mean zero, variance 1 and correlation function <m>rho(u)</m>. Choose a set of points <latex>$x_i \in \R : i=1,\ldots,n$</latex>. Let <m>R</m> denote the correlation matrix of <latex>$S=\{S(x_1),\ldots,S(x_n)\}$</latex>. Obtain the singular value decomposition of <m>R</m> as <latex>$R = D \Lambda D^\prime$</latex> where <m>Lambda</m> is a diagonal matrix whose non-zero entries are the eigenvalues of <m>R</m>, in order from largest to smallest. Let <latex>$Y=\{Y_1,\ldots,Y_n\}$</latex> be an independent random sample from the standard Gaussian distribution, <latex>${\rm N}(0,1)$</latex>. 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 <m>S</m> on (a discrete approximation to) the unit interval <m>(0,1)</m>. | + | - (7) Write an ''R'' function to simulate realisations using the above method for any specified set of points <m>x_i</m> and a range of correlation functions of your choice. Use your function to simulate a realisation of <m>S</m> on (a discrete approximation to) the unit interval <m>(0,1)</m>. |
| - (7) Now investigate how the appearance of your realisation <m>S</m> changes if in the equation above you replace the diagonal matrix <m>Lambda</m> by truncated form in which you replace the last <m>k</m> eigenvalues by zeros. | - (7) Now investigate how the appearance of your realisation <m>S</m> changes if in the equation above you replace the diagonal matrix <m>Lambda</m> by truncated form in which you replace the last <m>k</m> eigenvalues by zeros. | ||
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| - (15) Obtain simulations from the Poison model as shown in Figure 4.1 of the text book for the course. | - (15) Obtain simulations from the Poison model as shown in Figure 4.1 of the text book for the course. | ||
| - (15) Try to reproduce or mimic the results shown in Figure 4.2 of the text book for the course simulating a data set and obtaining a similar data-analysis. **Note:** for the example in the book we have used //set.seed(34)//. | - (15) Try to reproduce or mimic the results shown in Figure 4.2 of the text book for the course simulating a data set and obtaining a similar data-analysis. **Note:** for the example in the book we have used //set.seed(34)//. | ||
| - | - (16) Reproduce the simulated binomial data shown in Figure 4.6. Use the package //geoRglm// in conjunction with priors of your choice to obtain predictive distributions for the signal $S(x)$ at locations <latex>$x=(0.6, 0.6)$</latex> and <latex>$x=(0.9, 0.5)$</latex>. Compare the predictive inferences which you obtained in the previous exercise with those obtained by fitting a linear Gaussian model to the empirical logit transformed data, <m>log{(y+0.5)/(n-y+0.5)}</m>. Compare the results of the two previous analysis and comment generally. | + | - (16) Reproduce the simulated binomial data shown in Figure 4.6. Use the package //geoRglm// in conjunction with priors of your choice to obtain predictive distributions for the signal <m>S(x)</m> at locations <latex>$x=(0.6, 0.6)$</latex> and <latex>$x=(0.9, 0.5)$</latex>. Compare the predictive inferences which you obtained in the previous exercise with those obtained by fitting a linear Gaussian model to the empirical logit transformed data, <m>log{(y+0.5)/(n-y+0.5)}</m>. Compare the results of the two previous analysis and comment generally. |
| ==== Semana 5 ==== | ==== Semana 5 ==== | ||
