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| Ambos lados da revisão anterior Revisão anterior Próxima revisão | Revisão anterior Próxima revisão Ambos lados da revisão seguinte | ||
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disciplinas:verao2007:exercicios [2007/01/30 14:28] paulojus |
disciplinas:verao2007:exercicios [2007/02/15 09:26] paulojus |
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| Linha 38: | Linha 38: | ||
| - 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 $(0,1)$. | - 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 $(0,1)$. | ||
| - 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. | - 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. | ||
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| ==== Semana 3 ==== | ==== Semana 3 ==== | ||
| - | - Fit a model to the surface elevation data assuming a linear trend model on the coordinates and a Matérn correlation function with parameter kappa=2.5. | + | - Fit a model to the surface elevation data assuming a linear trend model on the coordinates and a Matérn correlation function with parameter kappa=2.5. Use the fitted model as the true model and perform a simulation study (i.e. simulate from this model) to compare parameter estimation based on maximum likelihood, restricted maximum likelihood and variograms. |
| - | Use the fitted model as the true model and perform a simulation study (i.e. simulate from this model) to compare parameter estimation based on maximum likelihood, restricted maximum likelihood and variograms. | + | |
| - Simulate 200 points in the unit square from the Gaussian model without measurement error, constant mean equals to zero, unit variance and exponential correlation function with $\phi=0.25$ and anisotropy parameters $(\psi_A=\pi/3, \psi_R=2)$. Obtain parameter estimates (using maximum likelihood): | - Simulate 200 points in the unit square from the Gaussian model without measurement error, constant mean equals to zero, unit variance and exponential correlation function with $\phi=0.25$ and anisotropy parameters $(\psi_A=\pi/3, \psi_R=2)$. Obtain parameter estimates (using maximum likelihood): | ||
| * assuming a isotropic model | * assuming a isotropic model | ||
| - | * try to estimate the anisotropy parameters | + | * try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for $\phi_R=4$. |
| - | Compare the results and repeat the exercise for $\phi_R=4$. | + | |
| - Consider a stationary trans-Gaussian model with known transformation function $h(\cdot)$, let $x$ be an arbitrary | - Consider a stationary trans-Gaussian model with known transformation function $h(\cdot)$, let $x$ be an arbitrary | ||
| location within the study region and define $T=h^{- 1}{S(x)}$. Find explicit expressions for ${\rm P}(T>c|Y)$ where | location within the study region and define $T=h^{- 1}{S(x)}$. Find explicit expressions for ${\rm P}(T>c|Y)$ where | ||
| $Y=(Y_1,...,Y_n)$ denotes the observed measurements on the untransformed scale and: | $Y=(Y_1,...,Y_n)$ denotes the observed measurements on the untransformed scale and: | ||
| - | * $h(u)=u$ | + | * <latex>$h(u)=u$</latex> |
| * $h(u) = \log u$ | * $h(u) = \log u$ | ||
| - | * $h(u) = \sqrt{u}$ | + | * $h(u) = \sqrt{u}$. |
| - Analyse the Paraná data-set or any other data set of your choice assuming priors obtaining: | - Analyse the Paraná data-set or any other data set of your choice assuming priors obtaining: | ||
| * a map of the predicted values over the area | * a map of the predicted values over the area | ||
