# ARMA process estimation¶

*best*model that fits the data and estimate the corresponding coefficients. The

*best*model is considered with respect to the criteria (corrected Akaike Information Criterion), defined by:

where is half the number of points of the time grid of the process sample (if the data are a process sample) or in a block of the time series (if the data are a time series).

Two other criteria are computed for each order :

the AIC criterion:

and the BIC criterion:

*BIC*criterion leads to a model that gives a better prediction; the

*AIC*criterion selects the best model that fits the given data; the criterion improves the previous one by penalizing a too high order that would artificially fit to the data.

(1)¶

where and is the frequency value.

The Whittle log-likelihood writes:

(2)¶

where:

is the non parametric estimate of the spectral density, expressed in the Fourier space (frequencies in instead of ). By default the Welch estimator is used.

is the Fourier frequency, , with the largest integer .

We estimate the scalar coefficients by maximizing the log-likelihood function. The corresponding equations lead to the following relation:

(3)¶

where maximizes:

(4)¶

The Whitle estimation requires that:

the determinant of the eigenvalues of the companion matrix associated to the polynomial are outside the unit disc,, which guarantees the stationarity of the process;

the determinant of the eigenvalues of the companion matrix associated to the polynomial are outside the unit disc, which guarantees the invertibility of the process.

**Multivariate estimation**

The likelihood of writes:

(5)¶

where , and where denotes the determinant.

The difficulty arises from the great size () of which is a dense matrix in the general case. [mauricio1995] proposes an efficient algorithm to evaluate the likelihood function. The main point is to use a change of variable that leads to a block-diagonal sparse covariance matrix.

The multivariate Whittle estimation requires that:

the determinant of the eigenvalues of the companion matrix associated to the polynomial are outside the unit disc, which guarantees the stationarity of the process;

the determinant of the eigenvalues of the companion matrix associated to the polynomial are outside the unit disc, which guarantees the invertibility of the process.

API:

See

`WhittleFactory`

See

`WelchFactory`

See

`ARMA`

Examples: