g05ph 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 g05ph.

Syntax

C#
public static void g05ph( int mode, int n, double xmean, int ip, double[] phi, int iq, double[] theta, double avar, double[] r, G05..::..G05State g05state, out double var, double[] x, out int ifail )

Visual Basic
Public Shared Sub g05ph ( _ mode As Integer, _ n As Integer, _ xmean As Double, _ ip As Integer, _ phi As Double(), _ iq As Integer, _ theta As Double(), _ avar As Double, _ r As Double(), _ g05state As G05..::..G05State, _ <OutAttribute> ByRef var As Double, _ x As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g05ph ( _
	mode As Integer, _
	n As Integer, _
	xmean As Double, _
	ip As Integer, _
	phi As Double(), _
	iq As Integer, _
	theta As Double(), _
	avar As Double, _
	r As Double(), _
	g05state As G05..::..G05State, _
	<OutAttribute> ByRef var As Double, _
	x As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g05ph( int mode, int n, double xmean, int ip, array<double>^ phi, int iq, array<double>^ theta, double avar, array<double>^ r, G05..::..G05State^ g05state, [OutAttribute] double% var, array<double>^ x, [OutAttribute] int% ifail )

Visual C++

public:
static void g05ph(
	int mode, 
	int n, 
	double xmean, 
	int ip, 
	array<double>^ phi, 
	int iq, 
	array<double>^ theta, 
	double avar, 
	array<double>^ r, 
	G05..::..G05State^ g05state, 
	[OutAttribute] double% var, 
	array<double>^ x, 
	[OutAttribute] int% ifail
)

F#
static member g05ph : mode : int * n : int * xmean : float * ip : int * phi : float[] * iq : int * theta : float[] * avar : float * r : float[] * g05state : G05..::..G05State * var : float byref * x : float[] * ifail : int byref -> unit

static member g05ph : 
        mode : int * 
        n : int * 
        xmean : float * 
        ip : int * 
        phi : float[] * 
        iq : int * 
        theta : float[] * 
        avar : float * 
        r : float[] * 
        g05state : G05..::..G05State * 
        var : float byref * 
        x : float[] * 
        ifail : int byref -> unit

Parameters

mode

Type: System..::..Int32

On entry: a code for selecting the operation to be performed by the method.

$mode = 0$: Set up reference vector only.
$mode = 1$: Generate terms in the time series using reference vector set up in a prior call to g05ph.
$mode = 2$: Set up reference vector and generate terms in the time series.

Constraint:

mode = 0

1

2

n: Type: System..::..Int32
On entry: $n$ , the number of observations to be generated.

Constraint: $n \geq 0$ .

xmean: Type: System..::..Double
On entry: the mean of the time series.

ip: Type: System..::..Int32
On entry: $p$ , the number of autoregressive coefficients supplied.

Constraint: $ip \geq 0$ .

phi: Type: array<System..::..Double>[]()[][]
An array of size [ip]
On entry: the autoregressive coefficients of the model, $ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}$ .

iq: Type: System..::..Int32
On entry: $q$ , the number of moving average coefficients supplied.

Constraint: $iq \geq 0$ .

theta: Type: array<System..::..Double>[]()[][]
An array of size [iq]
On entry: the moving average coefficients of the model, $θ_{1}, θ_{2}, \dots, θ_{q}$ .

avar: Type: System..::..Double
On entry: $σ^{2}$ , the variance of the Normal perturbations.

Constraint: $avar \geq 0.0$ .

r: Type: array<System..::..Double>[]()[][]
An array of size [lr]
On entry: if $mode = 1$ , the reference vector from the previous call to g05ph.
On exit: the reference vector.

g05state: Type: NagLibrary..::..G05..::..G05State
An Object of type G05.G05State.

var: Type: System..::..Double%
On exit: the proportion of the variance of a term in the series that is due to the moving-average (error) terms in the model. The smaller this is, the nearer is the model to non-stationarity.

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: contains the next $n$ observations from the time series.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

Let the vector

x_{t}

, denote a time series which is assumed to follow an autoregressive moving average (ARMA) model of the form:

\begin{matrix} x_{t} - μ = & ϕ_{1} (x_{t - 1} - μ) + ϕ_{2} (x_{t - 2} - μ) + \dots + ϕ_{p} (x_{t - p} - μ) + \\ ε_{t} - θ_{1} ε_{t - 1} - θ_{2} ε_{t - 2} - \dots - θ_{q} ε_{t - q} \end{matrix}

where

ε_{t}

, is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance

σ^{2}

. The parameters

\{ϕ_{i}\}

, for

i = 1, 2, \dots, p

, are called the autoregressive (AR) parameters, and

\{θ_{j}\}

, for

j = 1, 2, \dots, q

, the moving average (MA) parameters. The parameters in the model are thus the

p

ϕ

values, the

q

θ

values, the mean

μ

and the residual variance

σ^{2}

g05ph sets up a reference vector containing initial values corresponding to a stationary position using the method described in Tunnicliffe–Wilson (1979). The method 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 g05ph may be called again to generate a realization of

x_{n + 1}, x_{n + 2}, \dots

, etc. See the description of the parameter mode in [Parameters] for details.

One of the initialization methods (G05KFF not in this release) (for a repeatable sequence if computed sequentially) or (G05KGF not in this release) (for a non-repeatable sequence) must be called prior to the first call to g05ph.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$: On entry, $mode \neq 0$ , $1$ or $2$ .

$ifail = 2$: On entry, $n < 0$ .

$ifail = 4$: On entry, $ip < 0$ .

$ifail = 5$: phi does not define a stationary autoregressive process.

$ifail = 6$: On entry, $iq < 0$ .

$ifail = 8$: On entry, $avar < 0.0$ .

$ifail = 9$: Either r has been corrupted or the value of ip or iq is not the same as when r was set up in a previous call to g05ph with $mode = 0$ or $2$ .

$ifail = 10$: On entry, $lr < ip + iq + 6 + \max (ip, iq + 1)$ .

$ifail = 11$

On entry,

state vector was not initialized or has been corrupted.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

Any errors in the reference vector's initial values should be very much smaller than the error term; see Tunnicliffe–Wilson (1979).

Parallelism and Performance

None.

Further Comments

The time taken by g05ph is essentially of order

{(ip)}^{2}

The ARMA model for a time series can also be written as:

(x_{n} - E) = A_{1} (x_{n - 1} - E) + \dots + A_{NA} (x_{n - NA} - E) + B_{1} a_{n} + \dots + B_{NB} a_{n - NB + 1}

where

$x_{n}$ is the observed value of the time series at time $n$ ,
$NA$ is the number of autoregressive parameters, $A_{i}$ ,
$NB$ 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 [Description] by:

$B_{1}^{2} = σ^{2}$ ,
$B_{i + 1} = - θ_{i} σ = - θ_{i} B_{1}, i = 1, 2, \dots, q$ ,
$NB = q + 1$ ,
$E = μ$ ,
$A_{i} = ϕ_{i}, i = 1, 2, \dots, p$ ,
$NA = p$ .

Example

Example program (C#): g05phe.cs

Example program data: g05phe.d

Example program results: g05phe.r