NAG Library Function Document
nag_tsa_arma_filter (g13bac)
1
Purpose
nag_tsa_arma_filter (g13bac) filters a time series by an ARIMA model.
2
Specification
#include <nag.h> |
#include <nagg13.h> |
void |
nag_tsa_arma_filter (const double y[],
Integer ny,
Nag_ArimaOrder *arimaf,
Nag_ArimaOrder *arimav,
const double par[],
Integer npar,
double cy,
double b[],
Integer nb,
NagError *fail) |
|
3
Description
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 function 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 function 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
or
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
nag_tsa_multi_inp_model_forecast (g13bjc)
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 function must however have the appropriate constant for the forward series.
The series is then supplied to the function, and a corresponding set of values returned for .
4
References
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
5
Arguments
- 1:
– const doubleInput
-
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.
Constraint:
.
- 3:
– Nag_ArimaOrder *Input
-
On entry: the orders for the filtering ARIMA model as a 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
arimaf must specify the orders vector
, respectively, of the ARIMA model for the output noise component.
, , and refer, respectively, to the number of autoregressive (), moving average (), seasonal autoregressive () and seasonal moving average () parameters.
, and refer, respectively, to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.
Constraints:
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- if , ;
- if , .
- 4:
– Nag_ArimaOrder *Input
-
On entry: if available, the orders for the ARIMA model for the series as a 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
, respectively, of the ARIMA model for the output noise component.
, , and refer, respectively, to the number of autoregressive (), moving average (), seasonal autoregressive () and seasonal moving average () parameters.
, and 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
arimav should be set to a
NULL pointer.
Constraints:
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- ;
- if , ;
- if , .
- 5:
– const doubleInput
-
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.
Constraints:
- if , ;
- if , .
Note: the first constraint (i.e.,
) on the orders of the filtering model, in argument
arimav, ensures that
.
- 7:
– doubleInput
-
On entry: if the ARIMA model is known ,
cy must specify the constant term of the ARIMA model for the time series. If this model is not known ,
cy is not used.
- 8:
– doubleOutput
-
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
.
- 9:
– IntegerInput
-
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 was known.
Constraints:
- if , ;
- if , .
Where
- ;
- ;
- .
- 10:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_ARRAY_SIZE
-
The array
b is too small. Minimum required size:
.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONSTRAINT
-
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint: .
Constraint:
if , .
Constraint:
if , .
Constraint:
if , .
Constraint:
if , .
On entry,
npar is inconsistent with
arimaf and
arimav:
.
- NE_INIT_FILTER
-
The initial values of the filtered series are indeterminate for the given models.
- NE_INT
-
On entry,
ny is too small to carry out requested filtering:
.
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_ORDERS_ARIMA
-
On entry,
arimaf or
arimav is invalid.
The orders vector for the ARIMA model is invalid.
- NE_ORDERS_FILTER
-
The orders vector for the filtering model is invalid.
7
Accuracy
Accuracy and stability are high except when the MA parameters are close to the invertibility boundary.
8
Parallelism and Performance
nag_tsa_arma_filter (g13bac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_tsa_arma_filter (g13bac) 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.
If an ARIMA model is supplied,
local workspace arrays
of fixed
lengths are
allocated internally by nag_tsa_arma_filter (g13bac). The total size of
these arrays
amounts to Integer
elements and double
elements, where
.
The time taken by
nag_tsa_arma_filter (g13bac) is approximately proportional to
with an appreciable fixed increase if an ARIMA model is supplied for the time series.
10
Example
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
nag_tsa_multi_inp_model_forecast (g13bjc).
The backforecasts are inserted at the start of the series and
nag_tsa_arma_filter (g13bac) is called to perform the calculations.
10.1
Program Text
Program Text (g13bace.c)
10.2
Program Data
Program Data (g13bace.d)
10.3
Program Results
Program Results (g13bace.r)