NAG CL Interface
g05phc (times_arma)
1
Purpose
g05phc generates a realization of a univariate time series from an autoregressive moving average (ARMA) model. The realization may be continued or a new realization generated at subsequent calls to g05phc.
2
Specification
void 
g05phc (Nag_ModeRNG mode,
Integer n,
double xmean,
Integer ip,
const double phi[],
Integer iq,
const double theta[],
double avar,
double r[],
Integer lr,
Integer state[],
double *var,
double x[],
NagError *fail) 

The function may be called by the names: g05phc, nag_rand_times_arma or nag_rand_arma.
3
Description
Let the vector
${x}_{t}$, denote a time series which is assumed to follow an autoregressive moving average (ARMA) model of the form:
where
${\epsilon}_{t}$, is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance
${\sigma}^{2}$. The parameters
$\left\{{\varphi}_{i}\right\}$, for
$\mathit{i}=1,2,\dots ,p$, are called the autoregressive (AR) parameters, and
$\left\{{\theta}_{j}\right\}$, for
$\mathit{j}=1,2,\dots ,q$, the moving average (MA) parameters. The parameters in the model are thus the
$p$ $\varphi $ values, the
$q$ $\theta $ values, the mean
$\mu $ and the residual variance
${\sigma}^{2}$.
g05phc sets up a reference vector containing initial values corresponding to a stationary position using the method described in
Tunnicliffe–Wilson (1979). The function can then return a realization of
${x}_{1},{x}_{2},\dots ,{x}_{n}$. On a successful exit, the recent history is updated and saved in the reference vector
r so that
g05phc may be called again to generate a realization of
${x}_{n+1},{x}_{n+2},\dots $, etc. See the description of the argument
mode in
Section 5 for details.
One of the initialization functions
g05kfc (for a repeatable sequence if computed sequentially) or
g05kgc (for a nonrepeatable sequence) must be called prior to the first call to
g05phc.
4
References
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
Tunnicliffe–Wilson G (1979) Some efficient computational procedures for high order ARMA models J. Statist. Comput. Simulation 8 301–309
5
Arguments

1:
$\mathbf{mode}$ – Nag_ModeRNG
Input

On entry: a code for selecting the operation to be performed by the function.
 ${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$
 Set up reference vector only.
 ${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$
 Generate terms in the time series using reference vector set up in a prior call to g05phc.
 ${\mathbf{mode}}=\mathrm{Nag\_InitializeAndGenerate}$
 Set up reference vector and generate terms in the time series.
Constraint:
${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$, $\mathrm{Nag\_GenerateFromReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of observations to be generated.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{xmean}$ – double
Input

On entry: the mean of the time series.

4:
$\mathbf{ip}$ – Integer
Input

On entry: $p$, the number of autoregressive coefficients supplied.
Constraint:
${\mathbf{ip}}\ge 0$.

5:
$\mathbf{phi}\left[{\mathbf{ip}}\right]$ – const double
Input

On entry: the autoregressive coefficients of the model, ${\varphi}_{1},{\varphi}_{2},\dots ,{\varphi}_{p}$.

6:
$\mathbf{iq}$ – Integer
Input

On entry: $q$, the number of moving average coefficients supplied.
Constraint:
${\mathbf{iq}}\ge 0$.

7:
$\mathbf{theta}\left[{\mathbf{iq}}\right]$ – const double
Input

On entry: the moving average coefficients of the model, ${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{q}$.

8:
$\mathbf{avar}$ – double
Input

On entry: ${\sigma}^{2}$, the variance of the Normal perturbations.
Constraint:
${\mathbf{avar}}\ge 0.0$.

9:
$\mathbf{r}\left[{\mathbf{lr}}\right]$ – double
Communication Array

On entry: if ${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$, the reference vector from the previous call to g05phc.
On exit: the reference vector.

10:
$\mathbf{lr}$ – Integer
Input

On entry: the dimension of the array
r.
Constraint:
${\mathbf{lr}}\ge {\mathbf{ip}}+{\mathbf{iq}}+6+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ip}},{\mathbf{iq}}+1\right)$.

11:
$\mathbf{state}\left[\mathit{dim}\right]$ – Integer
Communication Array
Note: the dimension,
$\mathit{dim}$, 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.

12:
$\mathbf{var}$ – double *
Output

On exit: the proportion of the variance of a term in the series that is due to the movingaverage (error) terms in the model. The smaller this is, the nearer is the model to nonstationarity.

13:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – double
Output

On exit: contains the next $n$ observations from the time series.

14:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ip}}\ge 0$.
On entry, ${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{iq}}\ge 0$.
On entry,
lr is not large enough,
${\mathbf{lr}}=\u2329\mathit{\text{value}}\u232a$: minimum length required
$\text{}=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_STATE

On entry,
state vector has been corrupted or not initialized.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_PREV_CALL

ip or
iq is not the same as when
r was set up in a previous call.
Previous value of
${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{ip}}=\u2329\mathit{\text{value}}\u232a$.
Previous value of
${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{iq}}=\u2329\mathit{\text{value}}\u232a$.
 NE_REAL

On entry, ${\mathbf{avar}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{avar}}\ge 0.0$.
 NE_REF_VEC

Reference vector
r has been corrupted or not initialized correctly.
 NE_STATIONARY_AR

On entry, the AR parameters are outside the stationarity region.
7
Accuracy
Any errors in the reference vector's initial values should be very much smaller than the error term; see
Tunnicliffe–Wilson (1979).
8
Parallelism and Performance
g05phc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 implementationspecific information.
The time taken by g05phc is essentially of order ${\left({\mathbf{ip}}\right)}^{2}$.
Note: The reference vector,
r, contains a copy of the recent history of the series. If attempting to reinitialize the series by calling
g05kfc or
g05kgc a call to
g05phc with
${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$ must also be made. In the repeatable case the calls to
g05phc should be performed in the same order (at the same point(s) in simulation) every time
g05kfc is used. When the generator state is saved and restored using the argument
state, the time series reference vector must be saved and restored as well.
The ARMA model for a time series can also be written as:
where
 ${x}_{n}$ is the observed value of the time series at time $n$,
 $\mathit{NA}$ is the number of autoregressive parameters, ${A}_{i}$,
 $\mathit{NB}$ is the number of moving average parameters, ${B}_{i}$,
 $E$ is the mean of the time series,
and
 ${a}_{t}$ is a series of independent random Standard Normal perturbations.
This is related to the form given in
Section 3 by:
 ${B}_{1}^{2}={\sigma}^{2}$,
 ${B}_{i+1}={\theta}_{i}\sigma ={\theta}_{i}{B}_{1}\text{, \hspace{1em}}i=1,2,\dots ,q$,
 $\mathit{NB}=q+1$,
 $E=\mu $,
 ${A}_{i}={\varphi}_{i}\text{, \hspace{1em}}i=1,2,\dots ,p$,
 $\mathit{NA}=p$.
10
Example
This example generates values for an autoregressive model given by
where
${\epsilon}_{t}$ is a series of independent random Normal perturbations with variance
$1.0$. The random number generators are initialized by
g05kfc and then
g05phc is called to initialize a reference vector and generate a sample of ten observations.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results