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disciplinas:verao2007:exercicios [2007/02/18 16:51]
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disciplinas:verao2007:exercicios [2007/02/18 20:16] (atual)
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Linha 27: Linha 27:
 <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_iand 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.
  
Linha 63: Linha 63:
   - (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.
  
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