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disciplinas:verao2007:exercicios [2007/02/17 22:19]
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disciplinas:verao2007:exercicios [2007/02/18 20:16] (atual)
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Linha 18: Linha 18:
  
   - (3) load the data sets ''​parana'',​ ''​Ksat''​ e ''​ca20''​ available in ''​geoR'' ​ using commands such as:  <code R>​data(parana)</​code>​ and the documentation describing each data set with the ''​help()''​ function <code R>​help(parana)</​code>​ Perform exploratory data analysis and build a model you find suitable for each data.   - (3) load the data sets ''​parana'',​ ''​Ksat''​ e ''​ca20''​ available in ''​geoR'' ​ using commands such as:  <code R>​data(parana)</​code>​ and the documentation describing each data set with the ''​help()''​ function <code R>​help(parana)</​code>​ Perform exploratory data analysis and build a model you find suitable for each data.
-  - (3) In the examples above, would you have othe //​candidate//​ models for each data-set? ​+  - (3) In the examples above, would you have other //​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, ​$Y_i : i=1,\ldots,nassociated with a sequence of positions ​$x_i: i=1,\ldots,nalong 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,...,n</​m> ​along a one-dimensional spatial axis <m>x</m>
-    - $Y_i \alpha + \beta x_i Z_i$, where $\alphaand $\betaare 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>​sigma^2_{Z}</​m>​
-    - $Y_i = A + B x_i + Z_iwhere the $Z_iare 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 $Y_iand the covariance between ​$Y_iand $Y_jfor any $\neq i$. 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>!= i</m>. 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 $Ycan 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 $Yand obtain explicit expressions for the maximum likelihood estimators of $\muand $\sigma^2when $\phiis known. Discuss how you would use these expressions to find maximum likelihood estimators numerically when $\phiis unknown. +  - (5) Suppose that <​latex>​$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 $S(x) : x \in \IR$? Give either a proof or a counter-example. ​$$ +  - (6) Is the following a legitimate correlation function for a one-dimensional spatial process ​<​latex>​$S(x) : x \in R$</​latex>​? Give either a proof or a counter-example.\\  
-  \rho(u) = \left\{ +<​m> ​rho(u) = delim{lbrace}{matrix{2}{1}{{1-u : 0 <= <= 1}{  u>1}}}{} </m>\
-    \begin{array}{rcl} +  - (7) Consider the following method of simulating a realisation of a one-dimensional spatial process on <​latex>​$S(x) : x \in R$</​latex>​, with mean zero, variance 1 and correlation function ​<m>rho(u)</m>. Choose a set of points ​<​latex>​$x_i \in \: i=1,​\ldots,​n$</​latex>​. Let <m>R</​m> ​denote the correlation matrix of <​latex>​$S=\{S(x_1),​\ldots,​S(x_n)\}$</​latex>​. Obtain the singular value decomposition of <m>R</​m> ​as <​latex>​$R = D \Lambda D^\prime$</​latex> ​where <​m>​Lambda</​m> ​is a diagonal matrix whose non-zero entries are the eigenvalues of <m>R</m>, in order from largest to smallest. Let <​latex>​$Y=\{Y_1,​\ldots,​Y_n\}$</​latex> ​be an independent random sample from the standard Gaussian distribution, ​<​latex>​${\rm N}(0,1)$</​latex>​. Then the simulated realisation is <​latex>​$S = D \Lambda^{\frac{1}{2}} Y$</​latex> ​ 
-      ​1-u \leq \leq \\ +  - (7) Write an ''​R''​ function to simulate realisations using the above method for any specified set of points ​<m>x_i</​m> ​and a range of correlation functions of your choice. Use your function to simulate a realisation of <m>S</​m> ​on (a discrete approximation to) the unit interval ​<m>(0,1)</m>
-      ​u>1 +  - (7) Now investigate how the appearance of your realisation ​<m>S</​m> ​changes if in the equation above you replace the diagonal matrix ​<m>Lambda</​m> ​by truncated form in which you replace the last <m>k</​m> ​eigenvalues by zeros.
-    \end{array} +
-  ​\right. +
-$$  +
-  - (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 $Rdenote the correlation matrix of $S=\{S(x_1),​\ldots,​S(x_n)\}$. Obtain the singular value decomposition of $Ras $R = D \Lambda D^\prime$ 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_iand a range of correlation functions of your choice. Use your function to simulate a realisation of $Son (a discrete approximation to) the unit interval ​$(0,1)$+
-  - (7) Now investigate how the appearance of your realisation ​$Schanges if in the equation above you replace the diagonal matrix ​$\Lambdaby truncated form in which you replace the last $keigenvalues by zeros.+
  
  
 ==== Semana 3 ==== ==== Semana 3 ====
-  - (8) 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. +  - (8) Fit a model to the surface elevation data assuming a linear trend model on the coordinates and a Matérn correlation function with parameter ​<m>kappa=2.5</m>.  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. 
-  - (9) 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 $\phi=0.25and anisotropy parameters ​$(\psi_A=\pi/3, \psi_R=2)$. Obtain parameter estimates (using maximum likelihood):​+  - (9) 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 <m>phi=0.25</​m> ​and anisotropy parameters ​<m>(psi_A=pi/​3,​ psi_R=2)</m>. Obtain parameter estimates (using maximum likelihood):​
     * assuming ​ a isotropic model     * assuming ​ a isotropic model
-    * 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 <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 ​$T=h^{- 1}{S(x)}$. 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 <​latex>​${\rm P}(T>​c|Y)$</​latex> ​where <m>Y=(Y_1,​...,​Y_n)</​m> ​denotes the observed measurements on the untransformed scale and:
-$Y=(Y_1,​...,​Y_n)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>
     * <​m>​h(u) = sqrt{u}</​m>​.     * <​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
     * a map of the predicted std errors over the area     * a map of the predicted std errors over the area
-    * a map of the probabilities of being above a certain (arbitrarily) ​choosen ​threshold over the area+    * a map of the probabilities of being above a certain (arbitrarily) ​chosen ​threshold over the area
     * a map of the 10th, 25th, 50th, 75th and 90th percentiles over the area     * a map of the 10th, 25th, 50th, 75th and 90th percentiles over the area
-    * the predictive distribution of the porportion ​of the area with the value of the study variable below a certain threshold. (as a suggestion you can use the 30th percentile of the data as the value of such a threshold)  ​+    * the predictive distribution of the proportion ​of the area with the value of the study variable below a certain threshold. (as a suggestion you can use the 30th percentile of the data as the value of such a threshold)  ​
  
  
 ==== Semana 4 ==== ==== Semana 4 ====
  
-  - (12) Consider the stationary Gaussian model in which $Y_i = \beta + S(x_i) + Z_i :i=1,\ldots,n$, where $S(x)is a stationary Gaussian process with mean zero, variance ​$\sigma^2and correlation function ​$\rho(u)$, whilst the $Z_iare mutually independent ${\rm N}(0,​\tau^2)$ random variables. Assume that all parameters except ​$\betaare known. Derive the Bayesian predictive distribution of $S(x)for an arbitrary location ​$xwhen $\betais assigned an improper uniform prior, ​$\pi(\beta)constant for all real $\beta$. 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 ​$\phiobtain 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 $\betaand assuming the remaining parameters are known +    * a normal prior for <m>beta</​m> ​and assuming the remaining parameters are known 
-    * a normal-scaled-inverse-$chi^2$ prior for $(\beta, \sigma^2)$ 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 
-    * a normal-scaled-inverse-$chi^2prior for $(\beta, \sigma^2|\phi)and assuming a generic prior $p(\phi)for correlation parameter.  +    * a normal-scaled-inverse-<m>chi^2</​m> ​prior for <m>(beta, sigma^2|phi)</​m> ​and assuming a generic prior <m>p(phi)</​m> ​for correlation parameter.  
-  - (14) Analyse ​the Paraná data-set or any other data set of your choice assuming priors for the model parameters and obtaining:+  - (14) Analise ​the Paraná data-set or any other data set of your choice assuming priors for the model parameters and obtaining:
     * the posterior distribution for the model parameters     * the posterior distribution for the model parameters
     * a map of the predictive mean over the area     * a map of the predictive mean over the area
Linha 70: Linha 63:
   - (15) Obtain simulations from the Poison model as shown in Figure 4.1 of the text book for the course.   - (15) Obtain simulations from the Poison model as shown in Figure 4.1 of the text book for the course.
   - (15) Try to reproduce or mimic the results shown in Figure 4.2 of the text book for the course simulating a data set and obtaining a similar data-analysis. **Note:** for the example in the book we have used //​set.seed(34)//​.   - (15) Try to reproduce or mimic the results shown in Figure 4.2 of the text book for the course simulating a data set and obtaining a similar data-analysis. **Note:** for the example in the book we have used //​set.seed(34)//​.
-  - (16) Reproduce the simulated binomial data shown in Figure 4.6. Use the package //geoRglm// in conjunction with priors of your choice to obtain predictive distributions for the signal ​$S(x)at locations $x=(0.6, 0.6)$ 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 ​<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 ====

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