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Opened 16 years ago

Last modified 16 years ago

#664 closed enhancement

Preconditioned simulation of BSR linear block — at Initial Version

Reported by: Víctor de Buen Remiro Owned by: Víctor de Buen Remiro
Priority: highest Milestone: Numerical methods
Component: Math Version: 2.0.1
Severity: major Keywords: BSR, linear, ARIMA, preconditioned
Cc:

Description

In each iteration of BSR, we need to simulate vector \beta $$ matching this linear regression with ARIMA noise:

Y = X \beta + z $$
A \beta <= a $$
\phi\left(B\right) z_t = \theta\left(B\right) e_t $$
e \sim N\left(0,\sigma^{2} I\right) $$

ARMA and variance parameters are changing in each iteration of BSR, and, if there are missing values or non linear filters matrices Y $$ and X $$ 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 X $$ is very sparse, after applying ARIMA filter it could become dense and process will be too slow.

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.

Let be the last full calculated and stored system

Y' = X' \beta' + z' $$
\phi'\left(B\right) z'_t = \theta'\left(B\right) e'_t $$
e' \sim N\left(0,\sigma'^{2} I\right) $$

  1. Since this system has been previously decomposed is very fast to generate a vector \beta' $$ matching it
  2. The corresponding ARIMA noise is simply
    z' = Y' - X' \beta' $$
  3. By means of Almagro method it's posible to calculate residuals e' $$ and initial values u' $$ that solve difference equation
    e'_t = \frac{\phi'\left(B\right)}{\theta'\left(B\right)} z'_t $$
  4. Then we can purpose residuals and initial values for current system as
    e = \frac{\sigma}{\sigma'}e' $$ u = \frac{\sigma}{\sigma'}u' $$
  5. ARIMA noise for current system becomwes simply
    z_t = \frac{\theta\left(B\right)}{\phi\left(B\right)} e_t $$
  6. Then, we can solve sparse linear system
    Y = X \beta + z $$
  7. El vector resultante se aceptará si se cumplen las restricciones
    A \beta <= a $$
    y de lo contrario se repite el proceso entero.

Cuando el número de repeticiones se hace insoportable se debe reemplazar el sistema almacenado.

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