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# NAG Toolbox: nag_tsa_multi_inputmod_update (g13bg)

## Purpose

nag_tsa_multi_inputmod_update (g13bg) accepts a series of new observations of an output time series and any associated input time series, for which a multi-input model is already fully specified, and updates the ‘state set’ information for use in constructing further forecasts.
The previous specification of the multi-input model will normally have been obtained by using nag_tsa_multi_inputmod_estim (g13be) to estimate the relevant transfer function and ARIMA parameters. The supplied state set will originally have been produced by nag_tsa_multi_inputmod_estim (g13be) (or possibly nag_tsa_multi_inputmod_forecast (g13bj)), but may since have been updated by nag_tsa_multi_inputmod_update (g13bg).

## Syntax

[sttf, xxyn, res, ifail] = g13bg(sttf, mr, mt, para, xxyn, kzef, 'nsttf', nsttf, 'nser', nser, 'npara', npara, 'nnv', nnv)
[sttf, xxyn, res, ifail] = nag_tsa_multi_inputmod_update(sttf, mr, mt, para, xxyn, kzef, 'nsttf', nsttf, 'nser', nser, 'npara', npara, 'nnv', nnv)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 22: nnv was made optional

## Description

The multi-input model is specified in Description in nag_tsa_multi_inputmod_estim (g13be). The form of these equations required to update the state set is as follows:
 $zt=δ1zt-1+δ2zt-2+⋯+δpzt-p+ω0xt-b-ω1xt-b-1-⋯-ωqxt-b-q$
the transfer models which generate input component values ${z}_{i,t}$ from one or more inputs ${x}_{i,t}$,
 $nt=yt-z1,t-z2,t-⋯-zm,t$
which generates the output noise component from the output ${y}_{t}$ and the input components, and
 $wt =∇d∇sDnt-c et =wt-Φ1wt-s-Φ2wt-2×s-⋯-ΦPwt-P×s+Θ1et-s+Θ2et-2×s+⋯+ΘQet-Q×s at =et-ϕ1et-1-ϕ2et-2-⋯-ϕpet-p+θ1at-1+θ2at-2+⋯+θqat-q$
the ARIMA model for the output noise which generates the residuals ${a}_{t}$.
The state set (as also given in Description in nag_tsa_multi_inputmod_estim (g13be)) is the collection of terms
 $zn+1-k,xn+1-k,nn+1-k,wn+1-k,en+1-k and an+1-k$
for $k=1$ up to the maximum lag associated with each of these series respectively, in the above model equations. $n$ is the latest time point of the series from which the state set has been generated.
The function accepts further values of the series ${y}_{\mathit{t}}$, ${x}_{1,\mathit{t}},{x}_{2,\mathit{t}},\dots ,{x}_{m,\mathit{t}}$, for $\mathit{t}=n+1,\dots ,n+l$, and applies the above model equations over this time range, to generate new values of the various model components, noise series and residuals. The state set is reconstructed, corresponding to the latest time point $n+l$, the earlier values being discarded.
The set of residuals corresponding to the new observations may be of use in checking that the new observations conform to the previously fitted model. The components of the new observations of the output series which are due to the various inputs, and the noise component, are also optionally returned.
The parameters of the model are not changed in this function.

## References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{sttf}\left({\mathbf{nsttf}}\right)$ – double array
The nsttf values in the state set before updating as returned by nag_tsa_multi_inputmod_estim (g13be) or nag_tsa_multi_inputmod_forecast (g13bj), or a previous call to nag_tsa_multi_inputmod_update (g13bg).
2:     $\mathrm{mr}\left(7\right)$int64int32nag_int array
The orders vector $\left(p,d,q,P,D,Q,s\right)$ of the ARIMA model for the output noise component.
$p$, $q$, $P$ and $Q$ refer respectively to the number of autoregressive $\left(\varphi \right)$, moving average $\left(\theta \right)$, seasonal autoregressive $\left(\Phi \right)$ and seasonal moving average $\left(\Theta \right)$ parameters.
$d$, $D$ and $s$ refer respectively to the order of non-seasonal differencing, the order of seasonal differencing, and the seasonal period.
Constraints:
• $p$, $d$, $q$, $P$, $D$, $Q$, $s\ge 0$;
• $p+q+P+Q>0$;
• $s\ne 1$;
• if $s=0$, $P+D+Q=0$;
• if $s>1$, $P+D+Q>0$.
3:     $\mathrm{mt}\left(4,{\mathbf{nser}}\right)$int64int32nag_int array
The transfer function model orders $b$, $p$ and $q$ of each of the input series. The data for input series $i$ are held in column $i$. Row 1 holds the value ${b}_{i}$, row 2 holds the value ${q}_{i}$ and row 3 holds the value ${p}_{i}$. For a simple input, ${b}_{i}={q}_{i}={p}_{i}=0$.
Row 4 holds the value ${r}_{i}$, where ${r}_{i}=1$ for a simple input and ${r}_{i}=2\text{​ or ​}3$ for a transfer function input. When ${r}_{i}=1$ any nonzero contents of rows 1, 2 and 3 of column $i$ are ignored. The choice of ${r}_{i}=2$ or ${r}_{i}=3$ is an option for use in model estimation and does not affect the operation of nag_tsa_multi_inputmod_update (g13bg).
Constraint: ${\mathbf{mt}}\left(4,\mathit{i}\right)=1$, $2$ or $3$, for $\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$.
4:     $\mathrm{para}\left({\mathbf{npara}}\right)$ – double array
Estimates of the multi-input model parameters as returned by nag_tsa_multi_inputmod_estim (g13be). These are in order, firstly the ARIMA model parameters: $p$ values of $\varphi$ parameters, $q$ values of $\theta$ parameters, $P$ values of $\Phi$ parameters and $Q$ values of $\Theta$ parameters. These are followed by the transfer function model parameter values ${\omega }_{0},{\omega }_{1},\dots ,{\omega }_{{q}_{1}}$, ${\delta }_{1},{\delta }_{2},\dots ,{\delta }_{{p}_{1}}$ for the first of any input series and similarly for each subsequent input series. The final component of para is the value of the constant $c$.
5:     $\mathrm{xxyn}\left(\mathit{ldxxyn},{\mathbf{nser}}\right)$ – double array
ldxxyn, the first dimension of the array, must satisfy the constraint $\mathit{ldxxyn}\ge {\mathbf{nnv}}$.
The nnv new observation sets being used to update the state set. Column $i$ contains the values of input series $\mathit{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$. Column ${\mathbf{nser}}$ contains the values of the output series. Consecutive rows correspond to increasing time sequence.
6:     $\mathrm{kzef}$int64int32nag_int scalar
Must not be set to $0$, if the values of the input component series ${z}_{t}$ and the values of the output noise component ${n}_{t}$ are to overwrite the contents of xxyn on exit, and must be set to $0$ if xxyn is to remain unchanged on exit.

### Optional Input Parameters

1:     $\mathrm{nsttf}$int64int32nag_int scalar
Default: the dimension of the array sttf.
The exact number of values in the state set array sttf as returned by nag_tsa_multi_inputmod_estim (g13be) or nag_tsa_multi_inputmod_forecast (g13bj).
2:     $\mathrm{nser}$int64int32nag_int scalar
Default: the dimension of the array mt and the second dimension of the array xxyn. (An error is raised if these dimensions are not equal.)
The total number of input and output series. There may be any number of input series (including none), but only one output series.
3:     $\mathrm{npara}$int64int32nag_int scalar
Default: the dimension of the array para.
The exact number of $\varphi$, $\theta$, $\Phi$, $\Theta$, $\omega$, $\delta$ and $c$ parameters. ($c$ must be included whether its value was previously estimated or was set fixed.)
4:     $\mathrm{nnv}$int64int32nag_int scalar
Default: the first dimension of the array xxyn.
The number of new observation sets being used to update the state set, each observation set consisting of a value of the output series and the associated values of each of the input series at a particular time point.

### Output Parameters

1:     $\mathrm{sttf}\left({\mathbf{nsttf}}\right)$ – double array
The state set values after updating.
2:     $\mathrm{xxyn}\left(\mathit{ldxxyn},{\mathbf{nser}}\right)$ – double array
If ${\mathbf{kzef}}=0$, xxyn remains unchanged.
If ${\mathbf{kzef}}\ne 0$, the columns of xxyn hold the corresponding values of the input component series ${z}_{t}$ and the output noise component ${n}_{t}$ in that order.
3:     $\mathrm{res}\left({\mathbf{nnv}}\right)$ – double array
The values of the residual series ${a}_{t}$ corresponding to the new observations of the output series.
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, nsttf is not consistent with the orders in arrays mr and mt.
${\mathbf{ifail}}=2$
 On entry, npara is not consistent with the orders in arrays mr and mt.
${\mathbf{ifail}}=3$
 On entry, ldxxyn is too small.
${\mathbf{ifail}}=4$
 On entry, iwa is too small.
${\mathbf{ifail}}=5$
On entry, one of the ${r}_{\mathit{i}}$, stored in ${\mathbf{mt}}\left(4,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$ does not equal $1$, $2$ or $3$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The computations are believed to be stable.

The time taken by nag_tsa_multi_inputmod_update (g13bg) is approximately proportional to ${\mathbf{nnv}}×{\mathbf{npara}}$.

## Example

This example uses the data described in nag_tsa_multi_inputmod_estim (g13be) in which $40$ observations of an output time series and a single input series were processed. In this example a model which included seasonal differencing of order $1$ was used. The $10$ values of the state set and the $5$ final values of para then obtained are used as input to this program, together with the values of $4$ new observations and the transfer function orders of the input series. The model used is ${\varphi }_{1}=0.5158$, ${\Theta }_{1}=0.9994$, ${\omega }_{0}=8.6343$, ${\delta }_{1}=0.6726$, $c=-0.3172$.
The following are computed and printed out: the updated state set, the residuals ${a}_{t}$ and the values of the components ${z}_{t}$ and the output noise component ${n}_{t}$ corresponding to the new observations.
```function g13bg_example

fprintf('g13bg example results\n\n');

sttf = [ 6.0530;  184.4749;  -80.0885;  -75.1704;  -76.9481;
-81.4749;    0.7776;   -2.6190;   -2.3054;   -1.1963];

% orders and transfer
mr = [int64(1);0;0;0;1;1;4];
mt = [int64(1),0;
0, 0;
1, 0;
3, 0];

% Parameters
para = [ 0.5158;    0.9994;    8.6343;    0.6726;   -0.3172];

% New observations
xxyn = [5.941, 96;
5.386, 95;
5.811, 80;
6.716, 88];

% Artifact
kzef = int64(1);

%Update state set with new observations
[sttf, xxyn, res, ifail] = ...
g13bg( ...
sttf, mr, mt, para, xxyn, kzef);

% Display results
disp('The updated state set');
ns = numel(sttf);
for j = 1:6:ns
fprintf('%10.4f', sttf(j:min(j+5,ns)));
fprintf('\n');
end
fprintf('\nThe residuals (after differencing)\n');
fprintf('%4d%12.4f\n',[[1:numel(res)]; res']);
fprintf('\n    z(t)      n(t)\n');
disp(xxyn);

```
```g13bg example results

The updated state set
6.7160  158.3155  -80.3412  -74.9035  -80.7814  -70.3155
0.8416   -2.0333   -5.8201   10.2810

The residuals (after differencing)
1      1.4586
2     -2.4674
3     -4.7714
4     13.2830

z(t)      n(t)
176.3412  -80.3412
169.9035  -74.9035
160.7814  -80.7814
158.3155  -70.3155

```

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