g13bgc 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 g13bec to estimate the relevant transfer function and ARIMA parameters. The supplied state set will originally have been produced by g13bec (or possibly g13bjc), but may since have been updated by g13bgc.
the ARIMA model for the output noise which generates the residuals ${a}_{t}$.
The state set (as also given in Section 3 in g13bec) is the collection of terms
$${z}_{n+1-k},{x}_{n+1-k},{n}_{n+1-k},{w}_{n+1-k},{e}_{n+1-k}\text{\hspace{1em} and \hspace{1em}}{a}_{n+1-k}$$
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.
4References
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
5Arguments
1: $\mathbf{arimav}$ – Nag_ArimaOrder *
Pointer to structure of type Nag_ArimaOrder with the following members:
p – Integer
d – IntegerInput
q – IntegerInput
bigp – IntegerInput
bigd – IntegerInput
bigq – IntegerInput
s – IntegerInput
On entry: these seven members of arimav must specify the orders vector $(p,d,q,P,D,Q,s)$, respectively, of the ARIMA model for the output noise component.
$p$, $q$, $P$ and $Q$ refer, respectively, to the number of autoregressive ($\varphi $), moving average ($\theta $), seasonal autoregressive ($\Phi $) and seasonal moving average ($\Theta $) parameters.
$d$, $D$ and $s$ refer, respectively, to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.
2: $\mathbf{nser}$ – IntegerInput
On entry: the total number of input and output series. There may be any number of input series (including none), but only one output series.
3: $\mathbf{transfv}$ – Nag_TransfOrder *
Pointer to structure of type Nag_TransfOrder with the following members:
b – Integer *Input
q – Integer *Input
p – Integer *
r – Integer *Input
On entry: before use, these member pointers must be allocated memory by calling g13byc which allocates ${\mathbf{nseries}}-1$ elements to each pointer. The memory allocated to these pointers must be given the transfer function model orders $b$, $q$ and $p$ of each of the input series. The order parameters for input series $i$ are held in the $i$th element of the allocated memory for each pointer. $\mathbf{b}\left[i-1\right]$ holds the value ${b}_{i}$, $\mathbf{transfv}\mathbf{\to}\mathbf{q}\left[i-1\right]$ holds the value ${q}_{i}$ and $\mathbf{transfv}\mathbf{\to}\mathbf{p}\left[i-1\right]$ holds the value ${p}_{i}$.
For a simple input, ${b}_{i}={q}_{i}={p}_{i}=0$.
$\mathbf{r}\left[i-1\right]$ holds the value ${r}_{i}$, where ${r}_{i}=1$ for a simple input, and ${r}_{i}=2$ or $3$ for a transfer function input.
The choice ${r}_{i}=3$ leads to estimation of the pre-period input effects as nuisance parameters, and ${r}_{i}=2$ suppresses this estimation. This choice may affect the returned forecasts.
When ${r}_{i}=1$, any nonzero contents of the $i$th element of the memory of $\mathbf{b}$, $\mathbf{transfv}\mathbf{\to}\mathbf{q}$ and $\mathbf{transfv}\mathbf{\to}\mathbf{p}$ are ignored.
Constraint:
$\mathbf{r}\left[\mathit{i}-1\right]=1$, $2$ or $3$, for $\mathit{i}=1,2,\dots ,{\mathbf{nseries}}-1$
The memory allocated to the members of transfv must be freed by a call to g13bzc.
On entry: estimates of the multi-input model parameters as returned by g13bec. 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: $\mathbf{npara}$ – IntegerInput
On entry: 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.)
6: $\mathbf{nnv}$ – IntegerInput
On entry: 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.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{xxyn}}\left[(i-1)\times {\mathbf{tdxxyn}}+j-1\right]$.
On entry: the nnv new observation sets being used to update the state set. Column $i-1$ contains the values of input series
$\mathit{i}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nser}}-1$. Column ${\mathbf{nser}}-1$ contains the values of the output series. Consecutive rows correspond to increasing time sequence.
On exit: 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.
8: $\mathbf{tdxxyn}$ – IntegerInput
On entry: the stride separating matrix column elements in the array xxyn.
On entry: 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.
On entry: a pointer to a structure of type Nag_G13_Opt as returned by g13becorg13bjc.
On exit: the structure contains the updated state space information.
11: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{tdxxyn}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{tdxxyn}}>0$.
NE_INT_2
On entry, ${\mathbf{tdxxyn}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nser}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tdxxyn}}\ge {\mathbf{nser}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_STRUCT_CORRUPT
Values of the members of structures arimav, transfv and options are not compatible.
7Accuracy
The computations are believed to be stable.
8Parallelism and Performance
g13bgc is not threaded in any implementation.
9Further Comments
The time taken by g13bgc is approximately proportional to ${\mathbf{nnv}}\times {\mathbf{npara}}$.
10Example
This example uses the data described in g13bec 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.