ARMA stochastic process¶
(1)¶
where the coefficients of the recurrence are matrix in
and
is white noise
discretized on the same time grid as the process
.
The coefficients
form the Auto Regressive (AR) part of the model, while the
coefficients
the
Moving Average (MA) part.
We introduce the homogeneous system associated to (1):
(2)¶
To get stationary solutions of (1), it is necessary to get its characteristic polynomial defined in (3):
(3)¶
Thus the solutions of (2) are of the form
where the
are the roots of
the polynomials
defined in (3) and
is a polynomials of degree the order of the root
:
The processes decrease with time if and only if
the modulus of all the components of the roots
are
less than 1:
(4)¶
Once given the coefficients of the model , we
evaluate the roots of the polynomials
and checks
the previous condition (4). The roots
, are the
eigenvalues of the matrix
which writes in dimension
as:
(5)¶
and in dimension 1:
(6)¶
The matrix is known to be the companion matrix.
It is important to note that:
when asking for a realization of the stationary process modeled by
, one has to obtain a realization that does not depend on the current state of the process;
whereas, when one asks for a possible future extending a particular current state of the process, the realization of the model must depend on that current sate.
How to proceed to respect these constraints?
If we note and
two distinct solutions of (1)
associated to two distinct initial states, then, the process
is solution of the homogeneous equation associated to (1) and
then decreases with time under the condition (4). Let us note
the number such that:
(7)¶
where the are the roots of the polynomials (3) and
is the precision of the computer (
). Then, after
instants, the process
has
disappeared, which means that the processes
and
do not
differ any more. As a conclusion, after
instants, the
realization of the ARMA process does not depend any more on the
initial state.
That is why, when making a realization of the ARMA model, we perform
a thermalization step that simply consists in realizing the
model upon additional instants, erasing the
first values and finally only retaining the other
ones. That step ensures that the realization of the process does not
depend on the initial state.
By default, the number is evaluated according to
(7) by the method computeNThermalization. The User could get
access to it with the method getNThermalization and can change the
value with the method setNThermalization. (In order to give back to
its default value, it is necessary to re-use the
method computeNThermalization).
On the contrary, in the context of getting a possible future from a
specified current state, the User should care that the number of
additional instants on which he wants to extend the
process, is such that
because beyond
, the future has no link with the present.
More precisely, after
instants, such that:
(8)¶
where the are the components of the covariance matrix
of the white noise
, the influence of the
initial state is of same order than the influence of the white noise.
Let us note that when the ARMA model is created without specifying the current state, we automatically proceed to a thermalization step at the creation of the ARMA object.
Before asking for the generation of a possible future, the user has to specify the current state of the ARMA model, thanks to the creation method that takes into account the current state. In that case, we do not proceed to the thermalization step.
As an ARMA model is a stochastic process, the object ARMA inherits the methods of the Process object. Thus, it is possible to get its marginal processes, its time grid, its dimension and to get several realizations at a time of the process.
API:
See
ARMA
See
ARMACoefficients
See
ARMAState
Examples: