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Changes between Version 10 and Version 11 of Ticket #664


Ignore:
Timestamp:
Mar 25, 2009, 7:07:58 PM (16 years ago)
Author:
Víctor de Buen Remiro
Comment:

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  • Ticket #664 – Description

    v10 v11  
    2525 1. The corresponding ARIMA noise is simply [[BR]][[LatexEquation(z' = Y' - X' \beta' $$)]]
    2626 1. By means of Almagro method it's posible to calculate residuals [[LatexEquation(e' $$)]] and initial values [[LatexEquation(u' $$)]] that solve difference equation [[BR]] [[LatexEquation(e'_t = \frac{\phi'\left(B\right)}{\theta'\left(B\right)} z'_t $$)]]
    27  1. Then we can purpose residuals and initial values for current system as [[BR]][[LatexEquation(e = \frac{\sigma}{\sigma'}e' $$)]] [[BR]] [[LatexEquation(u = \frac{\sigma}{\sigma'}u' $$)]]
    28  1. Let be the standarized multinormal residuals [[BR]] [[LatexEquation( \epsilon=\frac{1}{\sigma}e $$)]] [[BR]]
    29  1. Since [[LatexEquation(\beta $$)]] are determined by [[LatexEquation(\epsilon $$)]] by means of a full range linear equation, their densities have constant ratio, and logarithm of density of candidate vector can be calculated directly as [[BR]] [[LatexEquation(cte -\frac{T}{2}\log\left(2\pi\right)-\frac{1}{2}\sum_{t=1}^{T}\epsilon_{t}^{2}$$)]]
    30  1. ARIMA noise for current system becomes simply [[BR]][[LatexEquation(z_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e_t $$)]]
    31  1. Then, we can solve sparse linear system [[BR]][[LatexEquation(X \beta = Y-z $$)]]
     27 1. Let be the standarized multinormal residuals [[BR]] [[LatexEquation( \epsilon' =\frac{1}{\sigma'}e' $$)]] [[BR]]
     28 1. Since [[LatexEquation(\beta' $$)]] are determined by [[LatexEquation(\epsilon' $$)]] by means of a full range linear equation, their densities have constant ratio, and logarithm of density of candidate vector can be calculated directly as [[BR]] [[LatexEquation(cte -\frac{T}{2}\log\left(2\pi\right)-\frac{1}{2}\sum_{t=1}^{T}\epsilon'_{t}^{2}$$)]]
     29 1. Then we can propose residuals and initial values for current system as [[BR]][[LatexEquation(e^{*} = \frac{\sigma}{\sigma'}e' $$)]] [[BR]] [[LatexEquation(u = \frac{\sigma}{\sigma'}u' $$)]]
     30 1. ARIMA noise for current system becomes simply [[BR]][[LatexEquation(z^{*}_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e^{*}_t $$)]]
     31 1. In order to get the refined candidate, we will take minimum residuals solution of sparse linear system  [[BR]][[LatexEquation(X \beta = Y-z^{*} $$)]]
     32 1. In order to calculate its exact density, it will be obtained the ARIMA noise [[BR]][[LatexEquation(z = Y - X \beta $$)]] to get standarized residuals from ARIMA equations.
    3233 1. If resulting vector doesn't match constraining inequations [[BR]] [[LatexEquation(A \beta <= a $$)]][[BR]] density will be toggled to [[LatexEquation(-\infty $$)]] in order to force rejection.
    3334