NAG Library Routine Document
g05pjf (times_mv_varma)
1
Purpose
g05pjf generates a realization of a multivariate time series from a vector autoregressive moving average (VARMA) model. The realization may be continued or a new realization generated at subsequent calls to g05pjf.
2
Specification
Fortran Interface
Subroutine g05pjf ( |
mode, n, k, xmean, ip, phi, iq, theta, var, ldvar, r, lr, state, x, ldx, ifail) |
Integer, Intent (In) | :: | mode, n, k, ip, iq, ldvar, lr, ldx | Integer, Intent (Inout) | :: | state(*), ifail | Real (Kind=nag_wp), Intent (In) | :: | xmean(k), phi(k*k*ip), theta(k*k*iq), var(ldvar,k) | Real (Kind=nag_wp), Intent (Inout) | :: | r(lr), x(ldx,n) |
|
C Header Interface
#include <nagmk26.h>
void |
g05pjf_ (const Integer *mode, const Integer *n, const Integer *k, const double xmean[], const Integer *ip, const double phi[], const Integer *iq, const double theta[], const double var[], const Integer *ldvar, double r[], const Integer *lr, Integer state[], double x[], const Integer *ldx, Integer *ifail) |
|
3
Description
Let the vector
, denote a
-dimensional time series which is assumed to follow a vector autoregressive moving average (VARMA) model of the form:
where
, is a vector of
residual series assumed to be Normally distributed with zero mean and covariance matrix
. The components of
are assumed to be uncorrelated at non-simultaneous lags. The
's and
's are
by
matrices of parameters.
, for
, are called the autoregressive (AR) parameter matrices, and
, for
, the moving average (MA) parameter matrices. The parameters in the model are thus the
by
-matrices, the
by
-matrices, the mean vector
and the residual error covariance matrix
. Let
where
denotes the
by
identity matrix.
The model
(1) must be both stationary and invertible. The model is said to be stationary if the eigenvalues of
lie inside the unit circle and invertible if the eigenvalues of
lie inside the unit circle.
For
the VARMA model
(1) is recast into state space form and a realization of the state vector at time zero computed. For all other cases the routine computes a realization of the pre-observed vectors
,
, from
(1), see
Shea (1988). This realization is then used to generate a sequence of successive time series observations. Note that special action is taken for pure MA models, that is for
.
At your request a new realization of the time series may be generated more efficiently using the information in a reference vector created during a previous call to
g05pjf. See the description of the argument
mode in
Section 5 for details.
The routine returns a realization of
. On a successful exit, the recent history is updated and saved in the array
r so that
g05pjf may be called again to generate a realization of
, etc. See the description of the argument
mode in
Section 5 for details.
Further computational details are given in
Shea (1988). Note, however, that
g05pjf uses a spectral decomposition rather than a Cholesky factorization to generate the multivariate Normals. Although this method involves more multiplications than the Cholesky factorization method and is thus slightly slower it is more stable when faced with ill-conditioned covariance matrices. A method of assigning the AR and MA coefficient matrices so that the stationarity and invertibility conditions are satisfied is described in
Barone (1987).
One of the initialization routines
g05kff (for a repeatable sequence if computed sequentially) or
g05kgf (for a non-repeatable sequence) must be called prior to the first call to
g05pjf.
4
References
Barone P (1987) A method for generating independent realisations of a multivariate normal stationary and invertible ARMA process J. Time Ser. Anal. 8 125–130
Shea B L (1988) A note on the generation of independent realisations of a vector autoregressive moving average process J. Time Ser. Anal. 9 403–410
5
Arguments
- 1: – IntegerInput
-
On entry: a code for selecting the operation to be performed by the routine.
- Set up reference vector and compute a realization of the recent history.
- Generate terms in the time series using reference vector set up in a prior call to g05pjf.
- Combine the operations of and .
- A new realization of the recent history is computed using information stored in the reference vector, and the following sequence of time series values are generated.
If
or
, you must ensure that the reference vector
r and the values of
k,
ip,
iq,
xmean,
phi,
theta,
var and
ldvar have not been changed between calls to
g05pjf.
Constraint:
, , or .
- 2: – IntegerInput
-
On entry: , the number of observations to be generated.
Constraint:
.
- 3: – IntegerInput
-
On entry: , the dimension of the multivariate time series.
Constraint:
.
- 4: – Real (Kind=nag_wp) arrayInput
-
On entry: , the vector of means of the multivariate time series.
- 5: – IntegerInput
-
On entry: , the number of autoregressive parameter matrices.
Constraint:
.
- 6: – Real (Kind=nag_wp) arrayInput
-
On entry: must contain the elements of the
autoregressive parameter matrices of the model,
. If
phi is considered as a three-dimensional array, dimensioned as
, the
th element of
would be stored in
; that is,
must be set equal to the
th element of
, for
,
and
.
Constraint:
the elements of
phi must satisfy the stationarity condition.
- 7: – IntegerInput
-
On entry: , the number of moving average parameter matrices.
Constraint:
.
- 8: – Real (Kind=nag_wp) arrayInput
-
On entry: must contain the elements of the
moving average parameter matrices of the model,
. If
theta is considered as a three-dimensional array, dimensioned as
theta(
k,
k,
iq), the
th element of
would be stored in
; that is,
must be set equal to the
th element of
, for
,
and
.
Constraint:
the elements of
theta must be within the invertibility region.
- 9: – Real (Kind=nag_wp) arrayInput
-
On entry: must contain the ()th element of , for and . Only the lower triangle is required.
Constraint:
the elements of
var must be such that
is positive semidefinite.
- 10: – IntegerInput
-
On entry: the first dimension of the array
var as declared in the (sub)program from which
g05pjf is called.
Constraint:
.
- 11: – Real (Kind=nag_wp) arrayCommunication Array
-
On entry: if
or
, the array
r as output from the previous call to
g05pjf must be input without any change.
If
or
, the contents of
r need not be set.
On exit: information required for any subsequent calls to the routine with
or
. See
Section 9.
- 12: – IntegerInput
-
On entry: the dimension of the array
r as declared in the (sub)program from which
g05pjf is called.
Constraints:
- if , ;
- if , .
Where and if , , or if , .
See
Section 9 for some examples of the required size of the array
r.
- 13: – Integer arrayCommunication Array
Note: the actual argument supplied
must be the array
state supplied to the initialization routines
g05kff or
g05kgf.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
- 14: – Real (Kind=nag_wp) arrayOutput
-
On exit: will contain a realization of the th component of , for and .
- 15: – IntegerInput
-
On entry: the first dimension of the array
x as declared in the (sub)program from which
g05pjf is called.
Constraint:
.
- 16: – IntegerInput/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
-
On entry, .
Constraint: , , or .
-
On entry, .
Constraint: .
-
On entry, .
Constraint: .
-
On entry, .
Constraint: .
-
On entry, the AR parameters are outside the stationarity region.
-
On entry, .
Constraint: .
-
On entry, the moving average parameter matrices are such that the model is non-invertible.
-
On entry, the covariance matrix
var is not positive semidefinite to
machine precision.
-
On entry, and .
Constraint: .
-
k is not the same as when
r was set up in a previous call.
Previous value of
and
.
-
On entry,
lr is not large enough,
: minimum length required
.
-
On entry,
state vector has been corrupted or not initialized.
-
On entry, and .
Constraint: .
-
An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues of the matrices used to test for stationarity or invertibility.
-
The reference vector cannot be computed because the AR parameters are too close to the boundary of the stationarity region.
-
An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues of the covariance matrix.
-
An excessive number of iterations were required by the NAG routine used to evaluate the eigenvalues stored in the reference vector.
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The accuracy is limited by the matrix computations performed, and this is dependent on the condition of the parameter and covariance matrices.
8
Parallelism and Performance
g05pjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05pjf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
Note that, in reference to , g05pjf will permit moving average parameters on the boundary of the invertibility region.
The elements of
r contain amongst other information details of the spectral decompositions which are used to generate future multivariate Normals. Note that these eigenvectors may not be unique on different machines. For example the eigenvectors corresponding to multiple eigenvalues may be permuted. Although an effort is made to ensure that the eigenvectors have the same sign on all machines, differences in the signs may theoretically still occur.
The following table gives some examples of the required size of the array
r, specified by the argument
lr, for
or
, and for various values of
and
.
|
|
|
|
|
|
|
|
0 |
1 |
2 |
3 |
|
|
|
13 |
20 |
31 |
46 |
|
0 |
36 |
56 |
92 |
144 |
|
|
85 |
124 |
199 |
310 |
|
|
|
19 |
30 |
45 |
64 |
|
1 |
52 |
88 |
140 |
208 |
|
|
115 |
190 |
301 |
448 |
p |
|
|
35 |
50 |
69 |
92 |
|
2 |
136 |
188 |
256 |
340 |
|
|
397 |
508 |
655 |
838 |
|
|
|
57 |
76 |
99 |
126 |
|
3 |
268 |
336 |
420 |
520 |
|
|
877 |
1024 |
1207 |
1426 |
Note that
g13dxf may be used to check whether a VARMA model is stationary and invertible.
The time taken depends on the values of , and especially and .
10
Example
This program generates two realizations, each of length
, from the bivariate AR(1) model
with
and
The pseudorandom number generator is initialized by a call to
g05kff. Then, in the first call to
g05pjf,
in order to set up the reference vector before generating the first realization. In the subsequent call
and a new recent history is generated and used to generate the second realization.
10.1
Program Text
Program Text (g05pjfe.f90)
10.2
Program Data
Program Data (g05pjfe.d)
10.3
Program Results
Program Results (g05pjfe.r)