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Chapter Contents
Chapter Introduction
NAG Toolbox

# 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:
Mark 22: nnv has been made optional
.

## Description

The multi-input model is specified in Section [Description] in (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 $zt=δ1zt-1+δ2zt-2+⋯+δpzt-p+ω0xt-b-ω1xt-b-1-⋯-ωqxt-b-q$
the transfer models which generate input component values zi,t${z}_{i,t}$ from one or more inputs xi,t${x}_{i,t}$,
 nt = yt − z1,t − z2,t − ⋯ − zm,t $nt=yt-z1,t-z2,t-⋯-zm,t$
which generates the output noise component from the output yt${y}_{t}$ 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
$wt =∇d∇sDnt-c et =wt-Φ1wt-s-Φ2wt-2×s-⋯-ΦPwt-P×s+Θ1et-s+Θ2et-2×s+⋯+ΘQet-Q×s at =et-ϕ1et-1-ϕ2et-2-⋯-ϕpet-p+θ1at-1+θ2at-2+⋯+θqat-q$
the ARIMA model for the output noise which generates the residuals at${a}_{t}$.
The state set (as also given in Section [Description] in (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 $zn+1-k,xn+1-k,nn+1-k,wn+1-k,en+1-k and an+1-k$
for k = 1$k=1$ up to the maximum lag associated with each of these series respectively, in the above model equations. n$n$ is the latest time point of the series from which the state set has been generated.
The function accepts further values of the series yt${y}_{\mathit{t}}$, x1,t,x2,t,,xm,t${x}_{1,\mathit{t}},{x}_{2,\mathit{t}},\dots ,{x}_{m,\mathit{t}}$, for t = n + 1,,n + l$\mathit{t}=n+1,\dots ,n+l$, 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$n+l$, 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:     sttf(nsttf) – 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:     mr(7$7$) – int64int32nag_int array
The orders vector (p,d,q,P,D,Q,s)$\left(p,d,q,P,D,Q,s\right)$ of the ARIMA model for the output noise component.
p$p$, q$q$, P$P$ and Q$Q$ refer respectively to the number of autoregressive (φ)$\left(\varphi \right)$, moving average (θ)$\left(\theta \right)$, seasonal autoregressive (Φ)$\left(\Phi \right)$ and seasonal moving average (Θ)$\left(\Theta \right)$ parameters.
d$d$, D$D$ and s$s$ refer respectively to the order of non-seasonal differencing, the order of seasonal differencing, and the seasonal period.
Constraints:
• p$p$, d$d$, q$q$, P$P$, D$D$, Q$Q$, s0$s\ge 0$;
• p + q + P + Q > 0$p+q+P+Q>0$;
• s1$s\ne 1$;
• if s = 0$s=0$, P + D + Q = 0$P+D+Q=0$;
• if s > 1$s>1$, P + D + Q > 0$P+D+Q>0$.
3:     mt(4$4$,nser) – int64int32nag_int array
The transfer function model orders b$b$, p$p$ and q$q$ of each of the input series. The data for input series i$i$ are held in column i$i$. Row 1 holds the value bi${b}_{i}$, row 2 holds the value qi${q}_{i}$ and row 3 holds the value pi${p}_{i}$. For a simple input, bi = qi = pi = 0${b}_{i}={q}_{i}={p}_{i}=0$.
Row 4 holds the value ri${r}_{i}$, where ri = 1${r}_{i}=1$ for a simple input and ri = 2​ or ​3${r}_{i}=2\text{​ or ​}3$ for a transfer function input. When ri = 1${r}_{i}=1$ any nonzero contents of rows 1, 2 and 3 of column i$i$ are ignored. The choice of ri = 2${r}_{i}=2$ or ri = 3${r}_{i}=3$ is an option for use in model estimation and does not affect the operation of nag_tsa_multi_inputmod_update (g13bg).
Constraint: mt(4,i) = 1${\mathbf{mt}}\left(4,\mathit{i}\right)=1$, 2$2$ or 3$3$, for i = 1,2,,nser1$\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$.
4:     para(npara) – 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$p$ values of φ$\varphi$ parameters, q$q$ values of θ$\theta$ parameters, P$P$ values of Φ$\Phi$ parameters and Q$Q$ values of Θ$\Theta$ parameters. These are followed by the transfer function model parameter values ω0,ω1,,ωq1${\omega }_{0},{\omega }_{1},\dots ,{\omega }_{{q}_{1}}$, δ1,δ2,,δp1${\delta }_{1},{\delta }_{2},\dots ,{\delta }_{{p}_{1}}$ 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$c$.
5:     xxyn(ldxxyn,nser) – double array
ldxxyn, the first dimension of the array, must satisfy the constraint ldxxynnnv$\mathit{ldxxyn}\ge {\mathbf{nnv}}$.
The nnv new observation sets being used to update the state set. Column i$i$ contains the values of input series i$\mathit{i}$, for i = 1,2,,nser1$\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$. Column nser${\mathbf{nser}}$ contains the values of the output series. Consecutive rows correspond to increasing time sequence.
6:     kzef – int64int32nag_int scalar
Must not be set to 0$0$, if the values of the input component series zt${z}_{t}$ and the values of the output noise component nt${n}_{t}$ are to overwrite the contents of xxyn on exit, and must be set to 0$0$ if xxyn is to remain unchanged on exit.

### Optional Input Parameters

1:     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:     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:     npara – int64int32nag_int scalar
Default: The dimension of the array para.
The exact number of φ$\varphi$, θ$\theta$, Φ$\Phi$, Θ$\Theta$, ω$\omega$, δ$\delta$ and c$c$ parameters. (c$c$ must be included whether its value was previously estimated or was set fixed.)
4:     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.

ldxxyn wa iwa

### Output Parameters

1:     sttf(nsttf) – double array
The state set values after updating.
2:     xxyn(ldxxyn,nser) – double array
ldxxynnnv$\mathit{ldxxyn}\ge {\mathbf{nnv}}$.
If kzef = 0${\mathbf{kzef}}=0$, xxyn remains unchanged.
If kzef0${\mathbf{kzef}}\ne 0$, the columns of xxyn hold the corresponding values of the input component series zt${z}_{t}$ and the output noise component nt${n}_{t}$ in that order.
3:     res(nnv) – double array
The values of the residual series at${a}_{t}$ corresponding to the new observations of the output series.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, nsttf is not consistent with the orders in arrays mr and mt.
ifail = 2${\mathbf{ifail}}=2$
 On entry, npara is not consistent with the orders in arrays mr and mt.
ifail = 3${\mathbf{ifail}}=3$
 On entry, ldxxyn is too small.
ifail = 4${\mathbf{ifail}}=4$
 On entry, iwa is too small.
ifail = 5${\mathbf{ifail}}=5$
On entry, one of the ri${r}_{\mathit{i}}$, stored in mt(4,i)${\mathbf{mt}}\left(4,\mathit{i}\right)$, for i = 1,2,,nser1$\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$ does not equal 1$1$, 2$2$ or 3$3$.

## 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

```function nag_tsa_multi_inputmod_update_example
sttf = [6.053;
184.4749;
-80.0885;
-75.1704;
-76.9481;
-81.4749;
0.7776;
-2.619;
-2.3054;
-1.1963];
mr = [int64(1);0;0;0;1;1;4];
mt = [int64(1),0; ...
0,0; ...
1,0; ...
3,0];
para = [0.5158;
0.9994;
8.6343;
0.6726;
-0.3172];
xxyn = [5.941, 96;
5.386, 95;
5.811, 80;
6.716, 88];
kzef = int64(1);
[sttfOut, xxynOut, res, ifail] = nag_tsa_multi_inputmod_update(sttf, mr, mt, para, xxyn, kzef)
```
```

sttfOut =

6.7160
158.3155
-80.3412
-74.9035
-80.7814
-70.3155
0.8416
-2.0333
-5.8201
10.2810

xxynOut =

176.3412  -80.3412
169.9035  -74.9035
160.7814  -80.7814
158.3155  -70.3155

res =

1.4586
-2.4674
-4.7714
13.2830

ifail =

0

```
```function g13bg_example
sttf = [6.053;
184.4749;
-80.0885;
-75.1704;
-76.9481;
-81.4749;
0.7776;
-2.619;
-2.3054;
-1.1963];
mr = [int64(1);0;0;0;1;1;4];
mt = [int64(1),0; ...
0,0; ...
1,0; ...
3,0];
para = [0.5158;
0.9994;
8.6343;
0.6726;
-0.3172];
xxyn = [5.941, 96;
5.386, 95;
5.811, 80;
6.716, 88];
kzef = int64(1);
[sttfOut, xxynOut, res, ifail] = g13bg(sttf, mr, mt, para, xxyn, kzef)
```
```

sttfOut =

6.7160
158.3155
-80.3412
-74.9035
-80.7814
-70.3155
0.8416
-2.0333
-5.8201
10.2810

xxynOut =

176.3412  -80.3412
169.9035  -74.9035
160.7814  -80.7814
158.3155  -70.3155

res =

1.4586
-2.4674
-4.7714
13.2830

ifail =

0

```