ARMA process estimationΒΆ
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:
(1)ΒΆ
where and
is the frequency value.
The Whittle log-likelihood writes:
(2)ΒΆ
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)ΒΆ
where maximizes:
(4)ΒΆ
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)ΒΆ
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.