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Changes between Version 1 and Version 2 of Ticket #664


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Timestamp:
Mar 23, 2009, 2:28:39 PM (16 years ago)
Author:
Víctor de Buen Remiro
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  • Ticket #664 – Description

    v1 v2  
    88ARMA and variance parameters are changing in each iteration of BSR, and, if there are missing values or non linear filters matrices [[LatexEquation(Y $$)]] and [[LatexEquation(X $$)]] could also change in each iteration. So, an ARIMA decomposition and filter and a Cholesky decomposition is needed in order to simulate linear block. Even if [[LatexEquation(X $$)]] is very sparse, after applying ARIMA filter it could become dense and process will be too slow.
    99
    10 I propose a preconditioning method to save a lot of time calculating and storing all blocks of a simulation only for one of K iterations and using them to generate aproximations that will be refinated in this fast way.
     10I propose a preconditioning method to save a lot of time calculating and storing all blocks of a simulation only for one of a lot of iterations and using them to generate aproximations that will be refinated inside an internal Metropolis-Hastings chain.
     11
     12The main idea behind this method is that, when external BSR Gibbs chain is already in convergence phase, difference of expected values between two different iterations should tend to zero. So, the simulator of a past iteration should be a good candidate generator for distribution of current iteration.
    1113
    1214Let be the last full calculated and stored system
    … …  
    1719[[LatexEquation(e' \sim N\left(0,\sigma'^{2} I\right) $$)]][[BR]]
    1820
    19  1. Since this system has been previously decomposed is very fast to generate a vector [[LatexEquation(\beta' $$)]] matching it
     21Following these steps for each Metropolis-Hastings step we can generate a candidate of vector [[LatexEquation(\beta' $$)]] calculating simultaneously its exact density:
     22 1. Since this system has been previously decomposed is very fast to generate a pre-candidate vector [[LatexEquation(\beta' $$)]] matching it
    2023 1. The corresponding ARIMA noise is simply [[BR]][[LatexEquation(z' = Y' - X' \beta' $$)]]
    2124 1. By means of Almagro method it's posible to calculate residuals [[LatexEquation(e' $$)]] and initial values [[LatexEquation(u' $$)]] that solve difference equation [[BR]] [[LatexEquation(e'_t = \frac{\phi'\left(B\right)}{\theta'\left(B\right)} z'_t $$)]]
    22  1. Then we can purpose residuals and initial values for current system as [[BR]][[LatexEquation(e = \frac{\sigma}{\sigma'}e' $$)]] [[LatexEquation(u = \frac{\sigma}{\sigma'}u' $$)]]
    23  1. ARIMA noise for current system becomwes simply [[BR]][[LatexEquation(z_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e_t $$)]]
     25 1. Then we can purpose residuals and initial values for current system as [[BR]][[LatexEquation(e \frac{\sigma}{\sigma'}e' $$)]] [[LatexEquation(u = \frac{\sigma}{\sigma'}u' $$)]]
     26 1. Le be [[BR]] [[LatexEquation( \epsilon=\frac{1}{\sigma}e $$)]] [[BR]] standarized multinormal residuals
     27 1. By Bayes theorem conditional density [[LatexEquation(\beta|\epsilon $$)]] is proportional to conditional density [[LatexEquation(\epsilon|\beta $$)]] which logarithm can be calculated directly as [[BR]] [[LatexEquation(-\frac{T}{2}\log\left(2\pi\right)-\frac{1}{2}\sum_{t=1}^{T}\epsilon_{t}^{2}$$)]]
     28 1. ARIMA noise for current system becomes simply [[BR]][[LatexEquation(z_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e_t $$)]]
    2429 1. Then, we can solve sparse linear system [[BR]][[LatexEquation(Y = X \beta + z $$)]]
    25  1. El vector resultante se aceptará si se cumplen las restricciones [[BR]] [[LatexEquation(A \beta <= a $$)]][[BR]] y de lo contrario se repite el proceso entero.
     30 1. If resulting vector doesn' match constraining inequations [[BR]] [[LatexEquation(A \beta <= a $$)]][[BR]] density will be toggled to [[LatexEquation(A -\infty $$)]] in order to force rejection.
    2631
    27 Cuando el número de repeticiones se hace insoportable se debe reemplazar el sistema almacenado.
     32The number of internal Metropolis-Hastings non rejected iterations must be a parameter of user configuration.
     33
     34When number of rejected iterations growns is the moment of remake full calculations and store a new preconditioner regression.