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disciplinas:verao2007:exercicios [2007/02/17 21:22] paulojus |
disciplinas:verao2007:exercicios [2007/02/17 22:31] paulojus |
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- (3) In the examples above, would you have othe //candidate// models for each data-set? | - (3) In the examples above, would you have othe //candidate// models for each data-set? | ||
- Inspect [[http://leg.ufpr.br/geoR/tutorials/Rcruciani.R|an example geoestatistical analysis]] for the hydraulic conductivity data. | - Inspect [[http://leg.ufpr.br/geoR/tutorials/Rcruciani.R|an example geoestatistical analysis]] for the hydraulic conductivity data. | ||
- | - (4) Consider the following two models for a set of responses, $Y_i : i=1,\ldots,n$ associated with a sequence of positions $x_i: i=1,\ldots,n$ along a one-dimensional spatial axis $x$. | + | - (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,\ldots,n</m> along a one-dimensional spatial axis <m>x</m>. |
- | - $Y_i = \alpha + \beta x_i + Z_i$, where $\alpha$ and $\beta$ are parameters and the $Z_i$ are mutually independent with mean zero and variance $\sigma_Z^2$. | + | - <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>sigma²_{Z}</m>. |
- | - $Y_i = A + B x_i + Z_i$ where the $Z_i$ 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 $Z_i$ 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? |
- (5) Suppose that $Y=(Y_1,\ldots,Y_n)$ follows a multivariate Gaussian distribution with ${\rm E}[Y_i]=\mu$ and ${\rm Var}\{Y_i\}=\sigma^2$ and that the covariance matrix of $Y$ can be expressed as $V=\sigma^2 R(\phi)$. Write down the log-likelihood function for $\theta=(\mu,\sigma^2,\phi)$ based on a single realisation of $Y$ and obtain explicit expressions for the maximum likelihood estimators of $\mu$ and $\sigma^2$ when $\phi$ is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when $\phi$ is unknown. | - (5) Suppose that $Y=(Y_1,\ldots,Y_n)$ follows a multivariate Gaussian distribution with ${\rm E}[Y_i]=\mu$ and ${\rm Var}\{Y_i\}=\sigma^2$ and that the covariance matrix of $Y$ can be expressed as $V=\sigma^2 R(\phi)$. Write down the log-likelihood function for $\theta=(\mu,\sigma^2,\phi)$ based on a single realisation of $Y$ and obtain explicit expressions for the maximum likelihood estimators of $\mu$ and $\sigma^2$ when $\phi$ is known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when $\phi$ is unknown. | ||
- | - (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. $$ | + | - (6) Is the following a legitimate correlation function for a one-dimensional spatial process $S(x) : x \in \IR$? Give either a proof or a counter-example. |
- | \rho(u) = \left\{ | + | <m> rho(u) = delim{lbrace}{matrix{2}{1}{{1-u : 0 <= u <= 1}{0 : u>1}}}{}</m> |
- | \begin{array}{rcl} | + | |
- | 1-u & : & 0 \leq u \leq 1 \\ | + | |
- | 0 & : & u>1 | + | |
- | \end{array} | + | |
- | \right. | + | |
- | $$ | + | |
- (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 S = D \Lambda^{\frac{1}{2}} Y | ||
- (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 $(0,1)$. | ||
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* try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for $\phi_R=4$. | * try to estimate the anisotropy parameters \\ Compare the results and repeat the exercise for $\phi_R=4$. | ||
- (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 $h(\cdot)$, 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 <m>T=h^{-1}{S(x)}</m>. 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: | ||
- | * $h(u)=u$ | + | * <m>h(u)=u</m> |
- | * $h(u) = \log u$ | + | * <m>h(u) = \log u</m> |
- | * $h(u) = \sqrt{u}$. | + | * <m>h(u) = sqrt{u}</m>. |
- (11) Analyse the Paraná data-set or any other data set of your choice assuming priors obtaining: | - (11) 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 | ||
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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 $x=(0.6, 0.6)$ and $x=(0.9, 0.5)$. 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, $\log\{(y+0.5)/(n-y+0.5)\}$. 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 $S(x)$ at locations $x=(0.6, 0.6)$ and $x=(0.9, 0.5)$. 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 ==== | ||
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* write down a code to obtain CL parameter estimates for the s100 data set and compare with the ones given by ML and REML. | * write down a code to obtain CL parameter estimates for the s100 data set and compare with the ones given by ML and REML. | ||
- | <m>S(f)(t)=a_{0}+sum{n=1}{+infty}{a_{n} cos(n omega t)+b_{n} sin(n omega t)}</m> | ||
- | <m 8>delim{lbrace}{matrix{3}{1}{{3x-5y+z=0} {sqrt{2}x-7y+8z=0} {x-8y+9z=0}}}{ }</m> | ||
- | <m 32>delim{|}{{1/N} sum{n=1}{N}{gamma(u_n)} - 1/{2 pi} int{0}{2 pi}{gamma(t) dt}}{|} <= epsilon/3</m> | ||