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disciplinas:verao2007:exercicios [2007/02/17 23:29] paulojus |
disciplinas:verao2007:exercicios [2007/02/18 20:16] (atual) paulojus |
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| ===== Exercícios ===== | ===== Exercícios ===== | ||
| - | **Atenção**: para visualizar melhor esta página voce deve se ''autenticar'' neste wiki. Para isto clique no botão ''AUTENTICAR'' no canto inferior desta página e entre com o usuário e senha ''verao2007'' | ||
| ==== Semana 1 ==== | ==== Semana 1 ==== | ||
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| - (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 <m>Z_i</m> 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 <m>Z_i</m> are as in (a) but //A// and //B// are now random variables, independent of each other and of the <m>Z_i</m>, each with mean zero and respective variances <latex>$\sigma_A^2$</latex> and <latex>$\sigma_B^2$</latex>.\\ For each of these models, find the mean and variance of <m>Y_i</m> and the covariance between <m>Y_i</m> and <m>Y_j</m> for any <m>j != i</m>. Given a single realisation of either model, would it be possible to distinguish between them? |
| - | - (5) Suppose that <latex>$Y=(Y_1,\ldots,Y_n)$</latex> follows a multivariate Gaussian distribution with <latex>${\rm E}[Y_i]=\mu$</latex> and <latex>${\rm Var}\{Y_i\}=\sigma^2$</latex> and that the covariance matrix of $Y$ can be expressed as $V=\sigma^2 R(\phi)$. Write down the log-likelihood function for <latex>$\theta=(\mu,\sigma^2,\phi)$</latex> based on a single realisation of $Y$ and obtain explicit expressions for the maximum likelihood estimators of $\mu$ and $\sigma^2$ when <m>phi</m> is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when <m>phi</m> is unknown. | + | - (5) Suppose that <latex>$Y=(Y_1,\ldots,Y_n)$</latex> follows a multivariate Gaussian distribution with <latex>${\rm E}[Y_i]=\mu$</latex> and <latex>${\rm Var}\{Y_i\}=\sigma^2$</latex> and that the covariance matrix of <m>Y</m> can be expressed as <m>V=\sigma^2 R(phi)</m>. Write down the log-likelihood function for <latex>$\theta=(\mu,\sigma^2,\phi)$</latex> based on a single realisation of <m>Y</m> and obtain explicit expressions for the maximum likelihood estimators of <m>mu</m> and <m>sigma^2</m> when <m>phi</m> is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when <m>phi</m> is unknown. |
| - (6) Is the following a legitimate correlation function for a one-dimensional spatial process <latex>$S(x) : x \in R$</latex>? Give either a proof or a counter-example.\\ | - (6) Is the following a legitimate correlation function for a one-dimensional spatial process <latex>$S(x) : x \in R$</latex>? 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 <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 $R$ as <latex>$R = D \Lambda D^\prime$</latex> where <m>Lambda<m> is a diagonal matrix whose non-zero entries are the eigenvalues of $R$, 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 $S$ 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 <latex>$\Lambda$</latex> 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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| * assuming a isotropic model | * assuming a isotropic model | ||
| * try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for <m>phi_R=4</m>. | * try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for <m>phi_R=4</m>. | ||
| - | - (10) Consider a stationary trans-Gaussian model with known transformation function $h(\cdot)$, let $x$ be an arbitrary | + | - (10) Consider a stationary trans-Gaussian model with known transformation function <latex>$h(\cdot)$</latex>, let $x$ be an arbitrary |
| - | location within the study region and define <m>T=h^{-1}{S(x)}</m>. Find explicit expressions for <latex>${\rm P}(T>c|Y)$</latex> where <m>Y=(Y_1,...,Y_n)</m> denotes the observed measurements on the untransformed scale and: | + | location within the study region and define <m>T=h^{-1}(S(x))</m>. Find explicit expressions for <latex>${\rm P}(T>c|Y)$</latex> where <m>Y=(Y_1,...,Y_n)</m> denotes the observed measurements on the untransformed scale and: |
| * <m>h(u)=u</m> | * <m>h(u)=u</m> | ||
| * <m>h(u) = log{u}</m> | * <m>h(u) = log{u}</m> | ||
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| ==== Semana 4 ==== | ==== Semana 4 ==== | ||
| - | - (12) Consider the stationary Gaussian model in which <m>Y_i = beta + S(x_i) + Z_i :i=1,...,n</m>, where <m>S(x)</m> is a stationary Gaussian process with mean zero, variance <m>sigma^2</m> and correlation function <m>rho(u)</m>, whilst the <m>Z_i</m> are mutually independent <latex>${\rm N}(0,\tau^2)$</latex> random variables. Assume that all parameters except <m>beta</m> are known. Derive the Bayesian predictive distribution of <m>S(x)</m> for an arbitrary location $x$ when $\beta$ is assigned an improper uniform prior, <m>pi(beta)</m> constant for all real <m>beta</m>. Compare the result with the ordinary kriging formulae. | + | - (12) Consider the stationary Gaussian model in which <m>Y_i = beta + S(x_i) + Z_i :i=1,...,n</m>, where <m>S(x)</m> is a stationary Gaussian process with mean zero, variance <m>sigma^2</m> and correlation function <m>rho(u)</m>, whilst the <m>Z_i</m> are mutually independent <latex>${\rm N}(0,\tau^2)$</latex> random variables. Assume that all parameters except <m>beta</m> are known. Derive the Bayesian predictive distribution of <m>S(x)</m> for an arbitrary location <m>x</m> when <m>beta</m> is assigned an improper uniform prior, <m>pi(beta)</m> constant for all real <m>beta</m>. Compare the result with the ordinary kriging formulae. |
| - | - (13) For the model assumed in the previous exercise, assuming a correlation function parametrised by a scalar parameter $\phi$ obtain the posterior distribution for: | + | - (13) For the model assumed in the previous exercise, assuming a correlation function parametrised by a scalar parameter <m>phi</m> obtain the posterior distribution for: |
| * a normal prior for <m>beta</m> and assuming the remaining parameters are known | * a normal prior for <m>beta</m> and assuming the remaining parameters are known | ||
| * a normal-scaled-inverse-<latex>$\chi^2$</latex> prior for <latex>$(\beta, \sigma^2)$</latex> and assuming the correlation parameter is known | * a normal-scaled-inverse-<latex>$\chi^2$</latex> prior for <latex>$(\beta, \sigma^2)$</latex> and assuming the correlation parameter is known | ||
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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 ==== | ||
