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Opened 17 years ago

Last modified 17 years ago

#662 closed defect

ARMAProcess::Eval.Almagro fails sometimes — at Initial Version

Reported by: Víctor de Buen Remiro Owned by: Víctor de Buen Remiro
Priority: highest Milestone: Numerical methods
Component: Math Version: 1.1.7
Severity: blocker Keywords: ARMA, likelihood evaluation, BSR
Cc:

Description

In ​tol_tests/tol/random_generation/test_0005/test.tol there is a test for random generation of parameters of a linear regression with ARMA noise.

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) $$

It generates a set of independent vectors \beta $$ and it calculates their logarithm of density.

First, the generalized linear regression is normalized by means of Cholesky decomposition of ARMA covariance

\Sigma_{\phi,\theta} = L_{\phi,\theta} L_{\phi,\theta}^{T} $$

Y' = \frac{1}{\sigma} L_{\phi,\theta}^{-1} Y $$
X' = \frac{1}{\sigma} L_{\phi,\theta}^{-1} X $$
z' = \frac{1}{\sigma} L_{\phi,\theta}^{-1} z \sim N\left(0,\sigma^{2} I\right) $$

Y' = X' \beta + z' $$

Calculating Cholesky decomposition

X'^{T} X' = L L^{T} $$

we have that

\mu = L^{-T}L^{-1}X'^{T} Y' $$
\epsilon \sim N\left(0,I\right) $$
\beta = \mu + L^{-T} \epsilon $$

Obviously, logarithm of density of \beta | \phi,\theta $$ is the same than for \epsilon | \phi,\theta $$

logDens\left(\beta| \phi,\theta\right) = logDens\left(\epsilon | \phi,\theta\right) = \sum_{k=1}^n{ \frac{1}{\sqrt{2\pi}}exp\left(\frac{\epsilon_k^2}{2}\right)} $$

Now, if we calculate ARMA noise corresponding for simulated \beta $$

z = Y - X \beta + $$

By Bayes theorem, logarithm of density of z | \beta,\phi,\theta $$ matches

logDens\left(z | \beta,\phi,\theta\right) = cte + logDens\left(\beta| \phi,\theta\right) $$

TOL has two methods for calculate likelihood of an ARMA noise, ARIMALevinsonEval and ARIMAAlmagroEval (or alternatively ARMAProcess::Eval.Almagro). Well, ARIMALevinsonEval matches perfectly the given relation but ARIMAAlmagroEval fails sometimes.

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