NAG CL Interface
g13bgc (multi_​inputmod_​update)

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1 Purpose

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.

2 Specification

#include <nag.h>
void  g13bgc (Nag_ArimaOrder *arimav, Integer nser, Nag_TransfOrder *transfv, const double para[], Integer npara, Integer nnv, double xxyn[], Integer tdxxyn, Integer kzef, Nag_G13_Opt *options, NagError *fail)
The function may be called by the names: g13bgc, nag_tsa_multi_inputmod_update or nag_tsa_multi_inp_update.

3 Description

The multi-input model is specified in Section 3 in g13bec. The form of these equations required to update the state set is as follows:
the transfer models which generate input component values zi,t from one or more inputs xi,t,
which generates the output noise component from the output yt and the input components, and
wt =dsDnt-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.
The state set (as also given in Section 3 in g13bec) 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 yt, x1,t,x2,t,,xm,t, for t=n+1,,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.

4 References

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

5 Arguments

1: arimav Nag_ArimaOrder *
Pointer to structure of type Nag_ArimaOrder with the following members:
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 (ϕ), moving average (θ), seasonal autoregressive (Φ) and seasonal moving average (Θ) parameters.
d, D and s refer, respectively, to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.
2: nser Integer Input
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: transfv Nag_TransfOrder *
Pointer to structure of type Nag_TransfOrder with the following members:
bInteger *Input
qInteger *Input
pInteger *
rInteger *Input
On entry: before use, these member pointers must be allocated memory by calling g13byc which allocates 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 ith element of the allocated memory for each pointer. b[i-1] holds the value bi, transfvq[i-1] holds the value qi and transfvp[i-1] holds the value pi.
For a simple input, bi=qi=pi=0.
r[i-1] holds the value ri, where ri=1 for a simple input, and ri=2 or 3 for a transfer function input.
The choice ri=3 leads to estimation of the pre-period input effects as nuisance parameters, and ri=2 suppresses this estimation. This choice may affect the returned forecasts.
When ri=1, any nonzero contents of the ith element of the memory of b, transfvq and transfvp are ignored.
Constraint: r[i-1]=1, 2 or 3, for i=1,2,,nseries-1
The memory allocated to the members of transfv must be freed by a call to g13bzc.
4: para[npara] const double Input
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 ϕ parameters, q values of θ parameters, P values of Φ parameters and Q values of Θ parameters. These are followed by the transfer function model parameter values ω0,ω1,,ωq1, δ1,δ2,,δp1 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: npara Integer Input
On entry: the exact number of ϕ, θ, Φ, Θ, ω, δ and c parameters. (c must be included whether its value was previously estimated or was set fixed.)
6: nnv Integer Input
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.
7: xxyn[nnv×tdxxyn] double Input/Output
Note: the (i,j)th element of the matrix is stored in xxyn[(i-1)×tdxxyn+j-1].
On entry: the nnv new observation sets being used to update the state set. Column i-1 contains the values of input series i, for i=1,2,,nser-1. Column nser-1 contains the values of the output series. Consecutive rows correspond to increasing time sequence.
On exit: if kzef=0, xxyn remains unchanged.
If kzef0, the columns of xxyn hold the corresponding values of the input component series zt and the output noise component nt in that order.
8: tdxxyn Integer Input
On entry: the stride separating matrix column elements in the array xxyn.
Constraint: tdxxynnser.
9: kzef Integer Input
On entry: must not be set to 0, if the values of the input component series zt and the values of the output noise component nt are to overwrite the contents of xxyn on exit, and must be set to 0 if xxyn is to remain unchanged on exit.
10: options Nag_G13_Opt * Input/Output
On entry: a pointer to a structure of type Nag_G13_Opt as returned by g13bec or g13bjc.
On exit: the structure contains the updated state space information.
11: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, tdxxyn=value.
Constraint: tdxxyn>0.
On entry, tdxxyn=value and nser=value.
Constraint: tdxxynnser.
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.
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.
Values of the members of structures arimav, transfv and options are not compatible.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g13bgc is not threaded in any implementation.

9 Further Comments

The time taken by g13bgc is approximately proportional to nnv×npara.

10 Example

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 ϕ1=0.5158, Θ1=0.9994, ω0=8.6343, δ1=0.6726, c=-0.3172.
The following are computed and printed out: the updated state set, the residuals at and the values of the components zt and the output noise component nt corresponding to the new observations.

10.1 Program Text

Program Text (g13bgce.c)

10.2 Program Data

Program Data (g13bgce.d)

10.3 Program Results

Program Results (g13bgce.r)