naginterfaces.library.tsa.uni_​arima_​forecast_​state

naginterfaces.library.tsa.uni_arima_forecast_state(st, mr, par, c, rms, nfv)

uni_arima_forecast_state produces forecasts of a time series, given a time series model which has already been fitted to the time series using uni_arima_estim() or uni_arima_estim_easy(). The original observations are not required, since uni_arima_forecast_state uses as input either the original state set produced by uni_arima_estim() or uni_arima_estim_easy() or the state set updated by a series of new observations using uni_arima_update(). Standard errors of the forecasts are also provided.

For full information please refer to the NAG Library document for g13ah

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13ahf.html

Parameters

stfloat, array-like, shape $\left(\text{nst}\right)$

The state set derived from uni_arima_estim() or uni_arima_estim_easy() originally, or as modified using earlier calls of uni_arima_update().

mrint, array-like, shape $\left(7\right)$

The orders vector $(p,d,q,P,D,Q,s)$ of the ARIMA model, in the usual notation.

parfloat, array-like, shape $\left(\text{npar}\right)$

The estimates of the $p$ values of the $\varphi$ parameters, the $q$ values of the $\theta$ parameters, the $P$ values of the $\Phi$ parameters and the $Q$ values of the $\Theta$ parameters which specify the model and which were output originally by uni_arima_estim() or uni_arima_estim_easy().

cfloat

$c$, the value of the model constant. This will have been output by uni_arima_estim() or uni_arima_estim_easy().

rmsfloat

$V$, the residual variance associated with the model.

If uni_arima_estim_easy() was used to estimate the model, $rms$ should be set to $\text{s}/\text{ndf}$, where $\text{s}$ and $\text{ndf}$ were output by uni_arima_estim_easy().

If uni_arima_estim() was used to estimate the model, $rms$ should be set to $\text{s}/\text{icount}\left[4\right]$, where $\text{s}$ and $\text{icount}\left[4\right]$ were output by uni_arima_estim().

nfvint

$L$, the required number of forecasts.

Returns

fvafloat, ndarray, shape $\left(nfv\right)$

$nfv$ forecast values relating to the original undifferenced series.

fsdfloat, ndarray, shape $\left(nfv\right)$

The standard errors associated with each of the $nfv$ forecast values in $fva$.

Raises

NagValueError

(errno $1$)

The orders vector $mr$ is invalid.

(errno $1$)

On entry, $\text{npar}=⟨value⟩$.

Constraint: $\text{npar}=p+q+P+Q$.

(errno $2$)

On entry, $\text{nst}=⟨value⟩$.

Constraint: $\text{nst}=P\times s+D\times s+d+q+max(Q\times s,p)$.

(errno $3$)

On entry, $nfv=⟨value⟩$.

Constraint: $nfv>0$.

(errno $4$)

On entry, $\text{nwa}=⟨value⟩$ and the minimum size $\text{required}=⟨value⟩$.

Constraint: $\text{nwa}\ge 4\times \text{npar}+3\times \text{nst}$.

(errno $5$)

On entry, $rms=⟨value⟩$.

Constraint: $rms\ge 0.0$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The original time series is $x_{\text{t}}$, for $\text{t}=1,2,\dots ,n$ and parameters have been fitted to the model of this time series using uni_arima_estim() or uni_arima_estim_easy().

Forecasts of $x_{\text{t}}$, for $\text{t}=n+1,\dots ,n+L$, are calculated in five stages, as follows:

1. set $a_t=0$ for $t=N+1,N+2,\dots ,N+L$, where $N=n-d-(D\times s)$ is the number of differenced values in the series;

2. calculate the values of $e_{\text{t}}$, for $\text{t}=N+1,\dots ,N+L$, and $e_{\text{t}}={\varphi }_1\times e_{t-1}+\cdots +{\varphi }_p\times e_{t-p}+a_t-{\theta }_1\times a_{t-1}-\cdots -{\theta }_q\times a_{t-q}$;

3. calculate the values of $w_{\text{t}}$, for $\text{t}=N+1,\dots ,N+L$, where $w_t={\Phi }_1\times w_{t-s}+\cdots +{\Phi }_P\times w_{t-s\times P}+e_t-{\Theta }_1\times e_{t-s}-\cdots -{\Theta }_Q\times e_{t-s\times Q}$ and $w_t$ for $t\le N$ are the first $s\times P$ values in the state set, corrected for the constant;

4. add the constant term $c$ to give the differenced series ${\nabla }^d{\nabla }_s^D{x}_{\text{t}}=w_{\text{t}}+c$, for $\text{t}=N+1,\dots ,N+L$;

5. the differencing operations are reversed to reconstitute $x_{\text{t}}$, for $\text{t}=n+1,\dots ,n+L$.

The standard errors of these forecasts are given by $s_{\text{t}}={\left[V\times ({\psi }_0^2+{\psi }_1^2+\cdots +{\psi }_{\text{t}-n-1}^2)\right]}^{1/2}$, for $\text{t}=n+1,\dots ,n+L$, where ${\psi }_0=1$, $V$ is the residual variance of $a_t$, and ${\psi }_j$ is the coefficient expressing the dependence of $x_t$ on $a_{t-j}$.

To calculate ${\psi }_{\text{j}}$, for $\text{j}=1,2,\dots ,(L-1)$, the following device is used.

A copy of the state set is initialized to zero throughout and the calculations outlined above for the construction of forecasts are carried out with the settings $a_{N+1}=1$, and $a_{\text{t}}=0$, for $\text{t}=N+2,\dots ,N+L$.

The resulting quantities corresponding to the sequence $x_{N+1},x_{N+2},\dots ,x_{N+L}$ are precisely $1$, ${\psi }_1,{\psi }_2,\dots ,{\psi }_{L-1}$.

The supplied time series model is used throughout these calculations, with the exception that the constant term $c$ is taken to be zero.