NAG Library Function Document
nag_rand_varma (g05pjc)
1
Purpose
nag_rand_varma (g05pjc) 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 nag_rand_varma (g05pjc).
2
Specification
#include <nag.h> |
#include <nagg05.h> |
void |
nag_rand_varma (Nag_OrderType order,
Nag_ModeRNG mode,
Integer n,
Integer k,
const double xmean[],
Integer p,
const double phi[],
Integer q,
const double theta[],
const double var[],
Integer pdv,
double r[],
Integer lr,
Integer state[],
double x[],
Integer pdx,
NagError *fail) |
|
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 function 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
nag_rand_varma (g05pjc). See the description of the argument
mode in
Section 5 for details.
The function returns a realization of
. On a successful exit, the recent history is updated and saved in the array
r so that
nag_rand_varma (g05pjc) 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
nag_rand_varma (g05pjc) 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 functions
nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or
nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to
nag_rand_varma (g05pjc).
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:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_ModeRNGInput
-
On entry: a code for selecting the operation to be performed by the function.
- 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 nag_rand_varma (g05pjc).
- 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,
p,
q,
xmean,
phi,
theta,
var and
pdv have not been changed between calls to
nag_rand_varma (g05pjc).
Constraint:
, , or .
- 3:
– IntegerInput
-
On entry: , the number of observations to be generated.
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the dimension of the multivariate time series.
Constraint:
.
- 5:
– const doubleInput
-
On entry: , the vector of means of the multivariate time series.
- 6:
– IntegerInput
-
On entry: , the number of autoregressive parameter matrices.
Constraint:
.
- 7:
– const doubleInput
-
On entry: must contain the elements of the autoregressive parameter matrices of the model, . The th element of is stored in
, for , and .
Constraint:
the elements of
phi must satisfy the stationarity condition.
- 8:
– IntegerInput
-
On entry: , the number of moving average parameter matrices.
Constraint:
.
- 9:
– const doubleInput
-
On entry: must contain the elements of the moving average parameter matrices of the model, . The th element of is stored in
, for , and .
Constraint:
the elements of
theta must be within the invertibility region.
- 10:
– const doubleInput
-
Note: the dimension,
dim, of the array
var
must be at least
.
Where
appears in this document, it refers to the array element
- when ;
- when .
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.
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
var.
Constraint:
.
- 12:
– doubleCommunication Array
-
On entry: if
or
, the array
r as output from the previous call to
nag_rand_varma (g05pjc) 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 function with
or
. See
Section 9.
- 13:
– IntegerInput
-
On entry: the dimension of the array
r.
Constraints:
- if , ;
- if , .
Where and if , , or if , .
See
Section 9 for some examples of the required size of the array
r.
- 14:
– IntegerCommunication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
state in the previous call to
nag_rand_init_repeatable (g05kfc) or
nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
- 15:
– doubleOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: will contain a realization of the th component of , for and .
- 16:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 17:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CLOSE_TO_STATIONARITY
-
The reference vector cannot be computed because the AR parameters are too close to the boundary of the stationarity region.
- NE_INT
-
On entry, .
Constraint: .
On entry,
lr is not large enough,
: minimum length required
.
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_INVALID_STATE
-
On entry,
state vector has been corrupted or not initialized.
- NE_INVERTIBILITY
-
On entry, the moving average parameter matrices are such that the model is non-invertible.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_POS_DEF
-
On entry, the covariance matrix
var is not positive semidefinite to
machine precision.
- NE_PREV_CALL
-
k is not the same as when
r was set up in a previous call.
Previous value of
and
.
- NE_STATIONARY_AR
-
On entry, the AR parameters are outside the stationarity region.
- NE_TOO_MANY_ITER
-
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues of the covariance matrix.
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues of the matrices used to test for stationarity or invertibility.
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues stored in the reference vector.
7
Accuracy
The accuracy is limited by the matrix computations performed, and this is dependent on the condition of the argument and covariance matrices.
8
Parallelism and Performance
nag_rand_varma (g05pjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_rand_varma (g05pjc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
Note that, in reference to
NE_INVERTIBILITY,
nag_rand_varma (g05pjc) 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
nag_tsa_arma_roots (g13dxc) 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
nag_rand_init_repeatable (g05kfc). Then, in the first call to
nag_rand_varma (g05pjc),
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 (g05pjce.c)
10.2
Program Data
Program Data (g05pjce.d)
10.3
Program Results
Program Results (g05pjce.r)