NAG Library Routine Document

G13BGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G13BGF accepts a series of new observations of an output time series and any associated input time series, for which a multi-input model is already fully specified, and updates the ‘state set’ information for use in constructing further forecasts.

The previous specification of the multi-input model will normally have been obtained by using G13BEF to estimate the relevant transfer function and ARIMA parameters. The supplied state set will originally have been produced by G13BEF (or possibly G13BJF), but may since have been updated by G13BGF.

2 Specification

SUBROUTINE G13BGF (

STTF, NSTTF, MR, NSER, MT, PARA, NPARA, NNV, XXYN, LDXXYN, KZEF, RES, WA, IWA, IFAIL)

INTEGER	NSTTF, MR(7), NSER, MT(4,NSER), NPARA, NNV, LDXXYN, KZEF, IWA, IFAIL
REAL (KIND=nag_wp)	STTF(NSTTF), PARA(NPARA), XXYN(LDXXYN,NSER), RES(NNV), WA(IWA)

3 Description

The multi-input model is specified in Section 3 in G13BEF. The form of these equations required to update the state set is as follows:

z_{t} = δ_{1} z_{t - 1} + δ_{2} z_{t - 2} + \dots + δ_{p} z_{t - p} + ω_{0} x_{t - b} - ω_{1} x_{t - b - 1} - \dots - ω_{q} x_{t - b - q}

the transfer models which generate input component values

z_{i, t}

from one or more inputs

x_{i, t}

n_{t} = y_{t} - z_{1, t} - z_{2, t} - \dots - z_{m, t}

which generates the output noise component from the output

y_{t}

and the input components, and

\begin{array}{l} w_{t} & = \nabla^{d} \nabla_{s}^{D} n_{t} - c \\ e_{t} & = w_{t} - Φ_{1} w_{t - s} - Φ_{2} w_{t - 2 \times s} - \dots - Φ_{P} w_{t - P \times s} + Θ_{1} e_{t - s} + Θ_{2} e_{t - 2 \times s} + \dots + Θ_{Q} e_{t - Q \times s} \\ a_{t} & = e_{t} - ϕ_{1} e_{t - 1} - ϕ_{2} e_{t - 2} - \dots - ϕ_{p} e_{t - p} + θ_{1} a_{t - 1} + θ_{2} a_{t - 2} + \dots + θ_{q} a_{t - q} \end{array}

the ARIMA model for the output noise which generates the residuals

a_{t}

The state set (as also given in Section 3 in G13BEF) is the collection of terms

z_{n + 1 - k}, x_{n + 1 - k}, n_{n + 1 - k}, w_{n + 1 - k}, e_{n + 1 - k} and a_{n + 1 - k}

for

k = 1

up to the maximum lag associated with each of these series respectively, in the above model equations.

n

is the latest time point of the series from which the state set has been generated.

The routine accepts further values of the series

y_{t}

x_{1, t}, x_{2, t}, \dots, x_{m, t}

, for

t = n + 1, \dots, n + l

, and applies the above model equations over this time range, to generate new values of the various model components, noise series and residuals. The state set is reconstructed, corresponding to the latest time point

n + l

, the earlier values being discarded.

The set of residuals corresponding to the new observations may be of use in checking that the new observations conform to the previously fitted model. The components of the new observations of the output series which are due to the various inputs, and the noise component, are also optionally returned.

The parameters of the model are not changed in this routine.

4 References

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

5 Parameters

1: STTF(NSTTF) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the NSTTF values in the state set before updating as returned by G13BEF or G13BJF, or a previous call to G13BGF.

On exit: the state set values after updating.

2: NSTTF – INTEGERInput

On entry: the exact number of values in the state set array STTF as returned by G13BEF or G13BJF.

3: MR( $7$ ) – INTEGER arrayInput

On entry: the orders vector

(p, d, q, P, D, Q, s)

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:

$p$ , $d$ , $q$ , $P$ , $D$ , $Q$ , $s \geq 0$ ;
$p + q + P + Q > 0$ ;
$s \neq 1$ ;
if $s = 0$ , $P + D + Q = 0$ ;
if $s > 1$ , $P + D + Q > 0$ .

4: NSER – INTEGERInput

On entry: the total number of input and output series. There may be any number of input series (including none), but only one output series.

5: MT( $4$ ,NSER) – INTEGER arrayInput

On entry: the transfer function model orders

b

p

and

q

of each of the input series. The data for input series

i

are held in column

i

. Row 1 holds the value

b_{i}

, row 2 holds the value

q_{i}

and row 3 holds the value

p_{i}

. For a simple input,

b_{i} = q_{i} = p_{i} = 0

Row 4 holds the value

r_{i}

, where

r_{i} = 1

for a simple input and

r_{i} = 2 ​ or ​ 3

for a transfer function input. When

r_{i} = 1

any nonzero contents of rows 1, 2 and 3 of column

i

are ignored. The choice of

r_{i} = 2

r_{i} = 3

is an option for use in model estimation and does not affect the operation of G13BGF.

Constraint:

MT (4, i) = 1

2

3

, for

i = 1, 2, \dots, NSER - 1

6: PARA(NPARA) – REAL (KIND=nag_wp) arrayInput

On entry: estimates of the multi-input model parameters as returned by G13BEF. These are in order, firstly the ARIMA model parameters:

p

values of

ϕ

parameters,

q

values of

θ

parameters,

P

values of

Φ

parameters and

Q

values of

Θ

parameters. These are followed by the transfer function model parameter values

ω_{0}, ω_{1}, \dots, ω_{q_{1}}

δ_{1}, δ_{2}, \dots, δ_{p_{1}}

for the first of any input series and similarly for each subsequent input series. The final component of PARA is the value of the constant

c

7: NPARA – INTEGERInput

On entry: the exact number of

ϕ

θ

Φ

Θ

ω

δ

and

c

parameters. (

c

must be included whether its value was previously estimated or was set fixed.)

8: NNV – INTEGERInput

On entry: the number of new observation sets being used to update the state set, each observation set consisting of a value of the output series and the associated values of each of the input series at a particular time point.

9: XXYN(LDXXYN,NSER) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the NNV new observation sets being used to update the state set. Column

i

contains the values of input series

i

, for

i = 1, 2, \dots, NSER - 1

. Column

NSER

contains the values of the output series. Consecutive rows correspond to increasing time sequence.

On exit: if

KZEF = 0

, XXYN remains unchanged.

KZEF \neq 0

, the columns of XXYN hold the corresponding values of the input component series

z_{t}

and the output noise component

n_{t}

in that order.

10: LDXXYN – INTEGERInput

On entry: the first dimension of the array XXYN as declared in the (sub)program from which G13BGF is called.

Constraint:

LDXXYN \geq NNV

11: KZEF – INTEGERInput

On entry: must not be set to

0

, if the values of the input component series

z_{t}

and the values of the output noise component

n_{t}

are to overwrite the contents of XXYN on exit, and must be set to

0

if XXYN is to remain unchanged on exit.

12: RES(NNV) – REAL (KIND=nag_wp) arrayOutput

On exit: the values of the residual series

a_{t}

corresponding to the new observations of the output series.

13: WA(IWA) – REAL (KIND=nag_wp) arrayWorkspace

14: IWA – INTEGERInput

On entry: the dimension of the array WA as declared in the (sub)program from which G13BGF is called.

Constraint:

IWA \geq NNV + 2 \times NSTTF + \max (NNV, NSTTF) + \max (NNV, ncc)

, where

ncc = 4 \times (p + q + P + Q)

15: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

NSTTF is not consistent with the orders in arrays MR and MT.

$IFAIL = 2$

On entry,

NPARA is not consistent with the orders in arrays MR and MT.

$IFAIL = 3$

On entry,

LDXXYN is too small.

$IFAIL = 4$

On entry,

IWA is too small.

$IFAIL = 5$: On entry, one of the $r_{i}$ , stored in $MT (4, i)$ , for $i = 1, 2, \dots, NSER - 1$ does not equal $1$ , $2$ or $3$ .

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The time taken by G13BGF is approximately proportional to

NNV \times NPARA

9 Example

This example uses the data described in G13BEF in which

40

observations of an output time series and a single input series were processed. In this example a model which included seasonal differencing of order

1

was used. The

10

values of the state set and the

5

final values of PARA then obtained are used as input to this program, together with the values of

4

new observations and the transfer function orders of the input series. The model used is

ϕ_{1} = 0.5158

Θ_{1} = 0.9994

ω_{0} = 8.6343

δ_{1} = 0.6726

c = - 0.3172

The following are computed and printed out: the updated state set, the residuals

a_{t}

and the values of the components

z_{t}

and the output noise component

n_{t}

corresponding to the new observations.

NAG Library Routine DocumentG13BGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G13BGF