NAG CL Interface
g13bbc (multi_​filter_​transf)

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1 Purpose

g13bbc filters a time series by a transfer function model.

2 Specification

#include <nag.h>
void  g13bbc (const double y[], Integer ny, Nag_TransfOrder *transfv, Nag_ArimaOrder *arimas, const double par[], Integer npar, double cy, double b[], Integer nb, NagError *fail)
The function may be called by the names: g13bbc, nag_tsa_multi_filter_transf or nag_tsa_transf_filter.

3 Description

From a given series y1,y2,,yn a new series b1,b2,,bn is calculated using a supplied (filtering) transfer function model according to the equation
bt=δ1bt-1+δ2bt-2++δpbt-p+ω0yt-b-ω1yt-b-1--ωqyt-b-q. (1)
As in the use of g13bac, large transient errors may arise in the early values of bt due to ignorance of yt for t<0, and two possibilities are allowed.
  1. (i)The equation (1) is applied from t=1+b+q,,n so all terms in yt on the right-hand side of (1) are known, the unknown set of values bt for t=b+q,,b+q+1-p being taken as zero.
  2. (ii)The unknown values of yt for t0 are estimated by backforecasting exactly as for g13bac.

4 References

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

5 Arguments

1: y[ny] const double Input
On entry: the Qy backforecasts starting with backforecast at time 1-Qy to backforecast at time 0 followed by the time series starting at time 1, where Qy=arimas.q+arimas.bigq×arimas.s. If there are no backforecasts either because the ARIMA model for the time series is not known or because it is known but has no moving average terms, then the time series starts at the beginning of y.
2: ny Integer Input
On entry: the total number of backforecasts and time series data points in array y.
Constraint: nymax(1+Qy,npar).
3: transfv Nag_TransfOrder * Input
On entry: the orders of the transfer function model where the triplet (transfv. nag_b, transfv. nag_q, transfv. nag_p) corresponds to the triplet (b,q,p) as described in Section 2.3.1 in the G13 Chapter Introduction.
  • transfv.nag_b0 ;
  • transfv.nag_q0 ;
  • transfv.nag_p0 .
4: arimas Nag_ArimaOrder * Input
On entry: if available, the orders for the filtering ARIMA model for the time series as a pointer to structure of type Nag_ArimaOrder with the following members:
On entry: these seven members of arimas 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.
If no ARIMA model for the series is to be supplied arimas should be set to a NULL pointer.
  • arimas.p0 ;
  • arimas.d0 ;
  • arimas.q0 ;
  • arimas.bigp0 ;
  • arimas.bigd0 ;
  • arimas.bigq0 ;
  • arimas.s0 ;
  • arimas.s1 ;
  • if arimas.s=0, arimas.bigp+arimas.bigd+arimas.bigq=0 ;
  • if arimas.s0, arimas.bigp+arimas.bigd+arimas.bigq0 .
5: par[npar] const double Input
On entry: the parameters of the filtering transfer function model followed by the parameters of the ARIMA model for the time series. In the transfer function model the parameters are in the standard order of MA-like followed by AR-like operator parameters. In the ARIMA model the parameters are in the standard order of non-seasonal AR and MA followed by seasonal AR and MA.
6: npar Integer Input
On entry: the total number of parameters held in array par.
  • if arimasis notNULL, npar=transfv.nag_q+transfv.nag_p+1 ;
  • if arimasisNULL, npar=transfv.nag_q+transfv.nag_p+1+arimas.p+ arimas.q+arimas.bigp+arimas.bigq .
7: cy double Input
On entry: if the ARIMA model is known (i.e., arimasisNULL), cy must specify the constant term of the ARIMA model for the time series. If this model is not known (i.e., arimasis notNULL) then cy is not used.
8: b[nb] double Output
On exit: the filtered output series. If the ARIMA model for the time series was known, and hence Qy backforecasts were supplied in y, then b contains Qy ‘filtered’ backforecasts followed by the filtered series. Otherwise, the filtered series begins at the start of b just as the original series began at the start of y. In either case, if the value of the series at time t is held in y[t-1], then the filtered value at time t is held in b[t-1].
9: nb Integer Input
On entry: the dimension of the array b.
In addition to holding the returned filtered series, b is also used as an intermediate work array if the ARIMA model for the time series is known.
  • if arimasis notNULL, nbny;
  • if arimasisNULL, nbny+max(transfv.nag_b+transfv.nag_q,transfv.nag_p).
10: 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, nb=value and the minimum size required=value.
Constraint: if arimasisNULLarimas is not NULL then nbmax(b,p,q), otherwise nbny.
On entry, argument value had an illegal value.
Constraint: arimas.bigd0 .
Constraint: arimas.bigp0 .
Constraint: arimas.bigq0 .
Constraint: arimas.d0 .
Constraint: arimas.p0 .
Constraint: arimas.q0 .
Constraint: arimas.s1 .
Constraint: arimas.s0 .
Constraint: if arimas.s=0, arimas.bigp+arimas.bigd+arimas.bigq=0 .
Constraint: if arimas.s0, arimas.bigp+arimas.bigd+arimas.bigq0 .
Constraint: transfv.nag_b0 .
Constraint: transfv.nag_p0 .
Constraint: transfv.nag_q0 .
On entry, npar is inconsistent with transfv and arimas: npar=value.
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.
A supplied model has invalid parameters.
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.
The matrix used to solve for starting values for MA is singular.
The supplied time series is too short.

7 Accuracy

Accuracy and stability are high except when the AR-like parameters are close to the invertibility boundary. All calculations are performed in basic precision except for one inner product type calculation which on machines of low precision is performed in additional precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g13bbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13bbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If an ARIMA model is supplied, local workspace arrays of fixed lengths are allocated internally by g13bbc. The total size of these arrays amounts to K Integer elements and K×(K+2) double elements, where K=transfv.nag_p+arimas.p+arimas.d+(arimas.bigp+arimas.bigd)×arimas.s.
The time taken by g13bbc is roughly proportional to the product of the length of the series and number of parameters in the filtering model with appreciable increase if an ARIMA model is supplied for the time series.

10 Example

This example reads a time series of length 296. It reads one univariate ARIMA (1,1,0,0,1,1,12) model for the series and the (0,13,12) filtering transfer function model. 12 initial backforecasts are required and these are calculated by a call to g13bjc. The backforecasts are inserted at the start of the series and g13bbc is called to perform the filtering.

10.1 Program Text

Program Text (g13bbce.c)

10.2 Program Data

Program Data (g13bbce.d)

10.3 Program Results

Program Results (g13bbce.r)