The routine may be called by the names g13baf or nagf_tsa_multi_filter_arima.
From a given series , a new series is calculated using a supplied (filtering) ARIMA model. This model will be one which has previously been fitted to a series with residuals . The equations defining in terms of are very similar to those by which is obtained from . The only dissimilarity is that no constant correction is applied after differencing. This is because the series is generally distinct from the series with which the model is associated, though may be related to . Whilst it is appropriate to apply the ARIMA model to so as to preserve the same relationship between and as exists between and , the constant term in the ARIMA model is inappropriate for . The consequence is that will not necessarily have zero mean.
The equations are precisely:
the appropriate differencing of ; both the seasonal and non-seasonal inverted autoregressive operations are then applied,
followed by the inverted moving average operations
Because the filtered series value depends on present and past values , there is a problem arising from ignorance of which particularly affects calculation of the early values , causing ‘transient errors’. The routine allows two possibilities.
(i)The equations (1), (2) and (3) are applied from successively later time points so that all terms on their right-hand sides are known, with being defined for . Equations (4) and (5) are then applied over the same range, taking any values on the right-hand side associated with previous time points to be zero.
This procedure may still however result in unacceptably large transient errors in early values of .
(ii)The unknown values are estimated by backforecasting. This requires that an ARIMA model distinct from that which has been supplied for filtering, should have been previously fitted to .
For efficiency, you are asked to supply both this ARIMA model for and a limited number of backforecasts which are prefixed to the known values of . Within the routine further backforecasts of , and the series , , in (1), (2) and (3) are then easily calculated, and a set of linear equations solved for backforecasts of for use in (4) and (5) in the case that .
Even if the best model for is not available, a very approximate guess such as
can help to reduce the transients substantially.
The backforecasts which need to be prefixed to are of length , where and are the non-seasonal and seasonal moving average orders and the seasonal period for the ARIMA model of . Thus you need not carry out the backforecasting exercise if . Otherwise, the series should be reversed to obtain and
should be used to forecast values, . The ARIMA model used is that fitted to (as a forward series) except that, if is odd, the constant should be changed in sign (to allow, for example, for the fact that a forward upward trend is a reversed downward trend). The ARIMA model for supplied to the filtering routine must however have the appropriate constant for the forward series.
The series is then supplied to the routine, and a corresponding set of values returned for .
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
1: – Real (Kind=nag_wp) arrayInput
On entry: the backforecasts, starting with backforecast at time to backforecast at time , followed by the time series starting at time , where . 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: – IntegerInput
On entry: the total number of backforecasts and time series data points in array y.
3: – Integer arrayInput
On entry: the orders vector for the filtering model, followed by the orders vector for the ARIMA model for the time series if the latter is known. The orders appear in the standard sequence as given in the G13 Chapter Introduction. If the ARIMA model for the time series is supplied, the routine will assume that the first values of the array y are backforecasts.
the filtering model is restricted in the following ways:
, i.e., filtering by a model which contains only differencing terms is not permitted;
, for ;
if , ;
if , ;
the ARIMA model for the time series is restricted in the following ways:
, for ;
if , ;
if , ;
4: – IntegerInput
On entry: the number of values specified in the array mr. It takes the value if no ARIMA model for the time series is supplied but otherwise it takes the value . Thus nmr acts as an indicator as to whether backforecasting can be carried out.
5: – Real (Kind=nag_wp) arrayInput
On entry: the parameters of the filtering model, followed by the parameters of the ARIMA model for the time series, if supplied. Within each model the parameters are in the standard order of non-seasonal AR and MA followed by seasonal AR and MA.
6: – IntegerInput
On entry: the total number of parameters held in array par.
if , ;
if , .
Note: the first constraint (i.e., ) on the orders of the filtering model, in argument mr, ensures that .
7: – Real (Kind=nag_wp)Input
On entry: if the ARIMA model is known (i.e., ), cy must specify the constant term of the ARIMA model for the time series. If this model is not known (i.e., ), cy is not used.
8: – Real (Kind=nag_wp) arrayOutput
9: – IntegerInput
These arguments are no longer accessed by g13baf. Workspace is provided internally by dynamic allocation instead.
10: – Real (Kind=nag_wp) arrayOutput
On exit: the filtered output series. If the ARIMA model for the time series was known, and hence backforecasts were supplied in y, then b contains ‘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 is held in , then the filtered value at time is held in .
11: – IntegerInput
On entry: the dimension of the array b as declared in the (sub)program from which g13baf is called. 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 was known.
if , ;
if , .
12: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
On entry, .
Constraint: npar must be inconsistent with mr.
On entry, and the minimum size .
On entry, and the minimum size .
Constraint: if then , otherwise .
The orders vector for the filtering model is invalid.
The orders vector for the ARIMA model is invalid.
The initial values of the filtered series are indeterminate for the given models.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
Accuracy and stability are high except when the MA parameters are close to the invertibility boundary.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g13baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13baf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
If an ARIMA model is supplied,
a local workspace array
allocated internally by g13baf. The total size of
amounts to integer
is the expression defined in the description of the argument wa.
The time taken by g13baf is approximately proportional to
with an appreciable fixed increase if an ARIMA model is supplied for the time series.
This example reads a time series of length . It reads the univariate ARIMA model and the ARIMA filtering model for the series. Two initial backforecasts are required and these are calculated by a call to
The backforecasts are inserted at the start of the series and g13baf is called to perform the calculations.