naginterfaces.library.rand.times_​arma

naginterfaces.library.rand.times_arma(mode, n, xmean, phi, theta, avar, comm, statecomm)

times_arma 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 times_arma.

For full information please refer to the NAG Library document for g05ph

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g05/g05phf.html

Parameters

modeint

A code for selecting the operation to be performed by the function.

$mode=0$

Set up reference vector only.

$mode=1$

Generate terms in the time series using reference vector set up in a prior call to times_arma.

$mode=2$

Set up reference vector and generate terms in the time series.

nint

$n$, the number of observations to be generated.

xmeanfloat

The mean of the time series.

phifloat, array-like, shape $\left(\text{ip}\right)$

The autoregressive coefficients of the model, ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_p$.

thetafloat, array-like, shape $\left(\text{iq}\right)$

The moving average coefficients of the model, ${\theta }_1,{\theta }_2,\dots ,{\theta }_q$.

avarfloat

${\sigma }^2$, the variance of the Normal perturbations.

commdict, communication object, modified in place

Communication structure for the reference vector.

If $mode=1$, this argument must have been initialized by a prior call to times_arma.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

varfloat

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.

xNone or float, ndarray, shape $\left(n\right)$

Contains the next $n$ observations from the time series.

Raises

NagValueError

(errno $1$)

On entry, $mode=⟨value⟩$.

Constraint: $mode=0$, $1$ or $2$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $4$)

On entry, $\text{ip}=⟨value⟩$.

Constraint: $\text{ip}\ge 0$.

(errno $5$)

On entry, the AR parameters are outside the stationarity region.

(errno $6$)

On entry, $\text{iq}=⟨value⟩$.

Constraint: $\text{iq}\ge 0$.

(errno $8$)

On entry, $avar=⟨value⟩$.

Constraint: $avar\ge 0.0$.

(errno $9$)

Reference vector $comm$[‘r’] has been corrupted or not initialized correctly.

(errno $9$)

$\text{ip}$ or $\text{iq}$ is not the same as when $comm$[‘r’] was set up in a previous call.

Previous value of $\text{ip}=⟨value⟩$ and $\text{ip}=⟨value⟩$.

Previous value of $\text{iq}=⟨value⟩$ and $\text{iq}=⟨value⟩$.

(errno $11$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

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

$$\begin{array}{c}\begin{array}{cc}{x}_t-\mu =& {\varphi }_1(x_{t-1}-\mu )+{\varphi }_2(x_{t-2}-\mu )+\cdots +{\varphi }_p(x_{t-p}-\mu )+\\ & {ϵ}_t-{\theta }_1{ϵ}_{t-1}-{\theta }_2{ϵ}_{t-2}-\cdots -{\theta }_q{ϵ}_{t-q}\end{array}\end{array}$$

where ${ϵ}_t$, 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 $\text{i}=1,2,\dots ,p$, are called the autoregressive (AR) parameters, and $\left\{{\theta }_j\right\}$, for $\text{j}=1,2,\dots ,q$, 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 ${\sigma }^2$.

times_arma 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 $comm$[‘r’] so that times_arma 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 Parameters for details.

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to times_arma.

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