NAG Library Function Document

nag_tsa_transf_filter (g13bbc)

void	nag_tsa_transf_filter (const double y[], Integer ny, Nag_TransfOrder transfv, Nag_ArimaOrder arimas, const double par[], Integer npar, double cy, double b[], Integer nb, NagError *fail)

3 Description

From a given series

y_{1}, y_{2}, \dots, y_{n}

a new series

b_{1}, b_{2}, \dots, b_{n}

is calculated using a supplied (filtering) transfer function model according to the equation

b_{t} = δ_{1} b_{t - 1} + δ_{2} b_{t - 2} + \dots + δ_{p} b_{t - p} + ω_{0} y_{t - b} - ω_{1} y_{t - b - 1} - \dots - ω_{q} y_{t - b - q} .

(1)

As in the use of nag_tsa_arma_filter (g13bac), large transient errors may arise in the early values of

b_{t}

due to ignorance of

y_{t}

for

t < 0

, and two possibilities are allowed.

(i)	The equation (1) is applied from $t = 1 + b + q, \dots, n$ so all terms in $y_{t}$ on the right-hand side of (1) are known, the unknown set of values $b_{t}$ for $t = b + q, \dots, b + q + 1 - p$ being taken as zero.
(ii)	The unknown values of $y_{t}$ for $t \leq 0$ are estimated by backforecasting exactly as for nag_tsa_arma_filter (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 doubleInput

On entry: the

Q_{y}^{'}

backforecasts starting with backforecast at time

1 - Q_{y}^{'}

to backforecast at time

0

followed by the time series starting at time

1

, where

Q_{y}^{'} = arimas . q + arimas . bigq \times 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: $ny$ – IntegerInput

On entry: the total number of backforecasts and time series data points in array y.

Constraint:

ny \geq \max (1 + Q_{y}^{'}, 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.

Constraints:

$transfv . nag_b \geq 0$ ;
$transfv . nag_q \geq 0$ ;
$transfv . nag_p \geq 0$ .

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:

p – Integer
d – IntegerInput
q – IntegerInput
bigp – IntegerInput
bigd – IntegerInput
bigq – IntegerInput
s – IntegerInput: 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.

Constraints:

$arimas . p \geq 0$ ;
$arimas . d \geq 0$ ;
$arimas . q \geq 0$ ;
$arimas . bigp \geq 0$ ;
$arimas . bigd \geq 0$ ;
$arimas . bigq \geq 0$ ;
$arimas . s \geq 0$ ;
$arimas . s \neq 1$ ;
if $arimas . s = 0$ , $arimas . bigp + arimas . bigd + arimas . bigq = 0$ ;
if $arimas . s \neq 0$ , $arimas . bigp + arimas . bigd + arimas . bigq \neq 0$ .

5: $par [npar]$ – const doubleInput

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: $npar$ – IntegerInput

On entry: the total number of parameters held in array par.

Constraints:

if $arimas is not NULL$ , $npar = transfv . nag_q + transfv . nag_p + 1$ ;
if $arimas is NULL$ , $npar = transfv . nag_q + transfv . nag_p + 1 + arimas . p + arimas . q + arimas . bigp + arimas . bigq$ .

7: $cy$ – doubleInput

On entry: if the ARIMA model is known (i.e.,

arimas is NULL

), cy must specify the constant term of the ARIMA model for the time series. If this model is not known (i.e.,

arimas is not NULL

) then cy is not used.

8: $b [nb]$ – doubleOutput

On exit: the filtered output series. If the ARIMA model for the time series was known, and hence

Q_{y}^{'}

backforecasts were supplied in y, then b contains

Q_{y}^{'}

‘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: $nb$ – 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 is known.

Constraints:

if $arimas is not NULL$ , $nb \geq ny$ ;
if $arimas is NULL$ , $nb \geq ny + \max (transfv . nag_b + transfv . nag_q, transfv . nag_p)$ .

10: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE: The array b is too small. Minimum required size: $〈value〉$ .
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONSTRAINT: On entry, $arimas = 〈value〉$ .
Constraint: $arimas . bigd \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . bigp \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . bigq \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . d \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . p \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . q \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . s \neq 1$ .

On entry, $arimas = 〈value〉$ .
Constraint: $arimas . s \geq 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: if $arimas . s = 0$ , $arimas . bigp + arimas . bigd + arimas . bigq = 0$ .

On entry, $arimas = 〈value〉$ .
Constraint: if $arimas . s \neq 0$ , $arimas . bigp + arimas . bigd + arimas . bigq \neq 0$ .

On entry, $transfv = 〈value〉$ .
Constraint: $transfv . nag_b \geq 0$ .

On entry, $transfv = 〈value〉$ .
Constraint: $transfv . nag_p \geq 0$ .

On entry, $transfv = 〈value〉$ .
Constraint: $transfv . nag_q \geq 0$ .
NE_INT: On entry, npar is inconsistent with transfv and arimas: $npar = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_MODEL_PARAMS: A supplied model has invalid parameters.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_SINGULAR: The matrix used to solve for starting values for MA is singular.
NE_TIME_SERIES: 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

nag_tsa_transf_filter (g13bbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_tsa_transf_filter (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 nag_tsa_transf_filter (g13bbc). The total size of these arrays amounts to

K

Integer elements and

K \times (K + 2)

double elements, where

K = transfv . nag_p + arimas . p + arimas . d + (arimas . bigp + arimas . bigd) \times arimas . s

The time taken by nag_tsa_transf_filter (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 nag_tsa_multi_inp_model_forecast (g13bjc). The backforecasts are inserted at the start of the series and nag_tsa_transf_filter (g13bbc) is called to perform the filtering.

NAG Library Function Documentnag_tsa_transf_filter (g13bbc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_tsa_transf_filter (g13bbc)