NAG Library Function Document

nag_tsa_arma_filter (g13bac)

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

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

, a new series

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

is calculated using a supplied (filtering) ARIMA model. This model will be one which has previously been fitted to a series

x_{t}

with residuals

a_{t}

. The equations defining

b_{t}

in terms of

y_{t}

are very similar to those by which

a_{t}

is obtained from

x_{t}

. The only dissimilarity is that no constant correction is applied after differencing. This is because the series

y_{t}

is generally distinct from the series

x_{t}

with which the model is associated, though

y_{t}

may be related to

x_{t}

. Whilst it is appropriate to apply the ARIMA model to

y_{t}

so as to preserve the same relationship between

b_{t}

and

a_{t}

as exists between

y_{t}

and

x_{t}

, the constant term in the ARIMA model is inappropriate for

y_{t}

. The consequence is that

b_{t}

will not necessarily have zero mean.

The equations are precisely:

w_{t} = \nabla^{d} \nabla_{s}^{D} y_{t},

(1)

the appropriate differencing of

y_{t}

; both the seasonal and non-seasonal inverted autoregressive operations are then applied,

u_{t} = w_{t} - Φ_{1} w_{t - s} - \dots - Φ_{P} w_{t - s \times P}

(2)

v_{t} = u_{t} - ϕ_{1} u_{t - 1} - \dots - ϕ_{p} u_{t - p}

(3)

followed by the inverted moving average operations

z_{t} = v_{t} + Θ_{1} z_{t - s} + \dots + Θ_{Q} z_{t - s \times Q}

(4)

b_{t} = z_{t} + θ_{1} b_{t - 1} + \dots + θ_{q} b_{t - q} .

(5)

Because the filtered series value

b_{t}

depends on present and past values

y_{t}, y_{t - 1}, \dots

, there is a problem arising from ignorance of

y_{0}, y_{- 1}, \dots

which particularly affects calculation of the early values

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

, 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

v_{t}

being defined for

t = (1 + d + s \times D + s \times P), \dots, n

. 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

b_{t}

(ii)

The unknown values

y_{0}, y_{- 1}, \dots

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

y_{t}

For efficiency, you are asked to supply both this ARIMA model for

y_{t}

and a limited number of backforecasts which are prefixed to the known values of

y_{t}

. Within the function further backforecasts of

y_{t}

, and the series

w_{t}

u_{t}

v_{t}

in (1), (2) and (3) are then easily calculated, and a set of linear equations solved for backforecasts of

z_{t}, b_{t}

for use in (4) and (5) in the case that

q + Q > 0

Even if the best model for

y_{t}

is not available, a very approximate guess such as

y_{t} = c + e_{t}

\nabla y_{t} = e_{t}

can help to reduce the transients substantially.

The backforecasts which need to be prefixed to

y_{t}

are of length

Q_{y}^{'} = q_{y} + s_{y} \times Q_{y}

, where

q_{y}

and

Q_{y}

are the non-seasonal and seasonal moving average orders and

s_{y}

the seasonal period for the ARIMA model of

y_{t}

. Thus you need not carry out the backforecasting exercise if

Q_{y}^{'} = 0

. Otherwise, the series

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

should be reversed to obtain

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

and nag_tsa_multi_inp_model_forecast (g13bjc) should be used to forecast

Q_{y}^{'}

values,

{\hat{y}}_{0}, \dots, {\hat{y}}_{1 - Q_{y}^{'}}

. The ARIMA model used is that fitted to

y_{t}

(as a forward series) except that, if

d_{y} + D_{y}

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

y_{t}

supplied to the filtering function must however have the appropriate constant for the forward series.

The series

{\hat{y}}_{1 - Q_{y}^{'}}, \dots, {\hat{y}}_{0}, y_{1}, \dots, y_{n}

is then supplied to the function, and a corresponding set of values returned for

b_{t}

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}^{'} = arimav . q + arimav . bigq \times arimav . 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: arimaf – 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 $(p, d, q, P, D, Q, s)$ , 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.

Constraints:

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

4: arimav – 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 $(p, d, q, P, D, Q, s)$ , 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 arimav should be set to a NULL pointer.

Constraints:

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

5: par[npar] – 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: npar – IntegerInput

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

Constraints:

if $arimav is not NULL$ , $npar = arimaf . p + arimaf . q + arimaf . bigp + arimaf . bigq$ ;
if $arimav is NULL$ , $npar = arimaf . p + arimaf . q + arimaf . bigp + arimaf . bigq + arimav . p + arimav . q + arimav . bigp + arimav . bigq$ .

Note: the first constraint (i.e.,

arimaf . p + arimaf . q + arimaf . bigp + arimaf . bigq > 0

) on the orders of the filtering model, in argument arimav, ensures that

npar > 0

7: cy – 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 , 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 was known.

Constraints:

if $arimav is NULL$ , $nb \geq ny + \max (K_{3}, K_{1} + K_{2})$ ;
if $arimav is not NULL$ , $nb \geq ny$ .

Where

$K_{1} = arimaf . p + arimaf . bigp \times arimaf . s$ ;
$K_{2} = arimaf . d + arimaf . bigd \times arimaf . s$ ;
$K_{3} = arimaf . q + arimaf . bigq \times arimaf . s$ .

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.
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, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . bigd \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . bigp \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . bigq \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . d \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . p + arimaf . q + arimaf . bigp + arimaf . bigq > 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . p \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . q \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . s \neq 1$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: $arimaf . s \geq 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: if $arimaf . s = 0$ , $arimaf . bigp + arimaf . bigd + arimaf . bigq = 0$ .
On entry, $arimaf = ⟨value⟩$ .
Constraint: if $arimaf . s \neq 0$ , $arimaf . bigp + arimaf . bigd + arimaf . bigq \neq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . bigd \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . bigp \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . bigq \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . d \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . p \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . q \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . s \neq 1$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: $arimav . s \geq 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: if $arimav . s = 0$ , $arimav . bigp + arimav . bigd + arimav . bigq = 0$ .
On entry, $arimav = ⟨value⟩$ .
Constraint: if $arimav . s \neq 0$ , $arimav . bigp + arimav . bigd + arimav . bigq \neq 0$ .
On entry, npar is inconsistent with arimaf and arimav: $npar = ⟨value⟩$ .
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: $ny = ⟨value⟩$ .
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.
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

Not applicable.

9 Further Comments

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

K

Integer elements and

K \times (K + 2)

double elements, where

K = arimaf . q + arimaf . bigq \times arimaf . s + arimav . p + arimav . d + (arimav . bigp + arimav . bigd) \times arimav . s

The time taken by nag_tsa_arma_filter (g13bac) is approximately proportional to

ny \times (arimaf . p + arimaf . q + arimaf . bigp + arimaf . bigq),

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

296

. It reads the univariate ARIMA

(4, 0, 2, 0, 0, 0, 0)

model and the ARIMA filtering

(3, 0, 0, 0, 0, 0, 0)

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.

NAG Library Function Documentnag_tsa_arma_filter (g13bac)

+− 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_arma_filter (g13bac)