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disciplinas:verao2007:exercicios [2007/02/17 23:29] paulojus |
disciplinas:verao2007:exercicios [2007/02/17 23:30] paulojus |
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| - | - (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 $x$ 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 $\phi$ 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 | ||
