NAG Library Routine Document

g13aaf carries out non-seasonal and seasonal differencing on a time series. Information which allows the original series to be reconstituted from the differenced series is also produced. This information is required in time series forecasting.

2

Specification

Fortran Interface

Subroutine g13aaf (

x, nx, nd, nds, ns, xd, nxd, ifail)

Integer, Intent (In)	::	nx, nd, nds, ns
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nxd
Real (Kind=nag_wp), Intent (In)	::	x(nx)
Real (Kind=nag_wp), Intent (Out)	::	xd(nx)

C Header Interface

#include <nagmk26.h>

void	g13aaf_ (const double x[], const Integer nx, const Integer nd, const Integer nds, const Integer ns, double xd[], Integer nxd, Integer ifail)

3

Description

Let

\nabla^{d} \nabla_{s}^{D} x_{i}

be the

i

th value of a time series

x_{i}

, for

i = 1, 2, \dots, n

after non-seasonal differencing of order

d

and seasonal differencing of order

D

(with period or seasonality

s

). In general,

$\nabla^{d} \nabla_{s}^{D} x_{i}$	$=$	$\nabla^{d - 1} \nabla_{s}^{D} x_{i + 1} - \nabla^{d - 1} \nabla_{s}^{D} x_{i}$	$d > 0$
$\nabla^{d} \nabla_{s}^{D} x_{i}$	$=$	$\nabla^{d} \nabla_{s}^{D - 1} x_{i + s} - \nabla^{d} \nabla_{s}^{D - 1} x_{i}$	$D > 0$

Non-seasonal differencing up to the required order

d

is obtained using

$\nabla^{1} x_{i}$	$=$	$x_{i + 1} - x_{i}$	for $i = 1, 2, \dots, (n - 1)$
$\nabla^{2} x_{i}$	$=$	$\nabla^{1} x_{i + 1} - \nabla^{1} x_{i}$	for $i = 1, 2, \dots, (n - 2)$
$⋮$
$\nabla^{d} x_{i}$	$=$	$\nabla^{d - 1} x_{i + 1} - \nabla^{d - 1} x_{i}$	for $i = 1, 2, \dots, (n - d)$

Seasonal differencing up to the required order

D

is then obtained using

$\nabla^{d} \nabla_{s}^{1} x_{i}$	$=$	$\nabla^{d} x_{i + s} - \nabla^{d} x_{i}$	for $i = 1, 2, \dots, (n - d - s)$
$\nabla^{d} \nabla_{s}^{2} x_{i}$	$=$	$\nabla^{d} \nabla_{s}^{1} x_{i + s} - \nabla^{d} \nabla_{s}^{1} x_{i}$	for $i = 1, 2, \dots, (n - d - 2 s)$
$⋮$
$\nabla^{d} \nabla_{s}^{D} x_{i}$	$=$	$\nabla^{d} \nabla_{s}^{D - 1} x_{i + s} - \nabla^{d} \nabla_{s}^{D - 1} x_{i}$	for $i = 1, 2, \dots, (n - d - D \times s)$

Mathematically, the sequence in which the differencing operations are performed does not affect the final resulting series of

m = n - d - D \times s

values.

4

References

None.

5

Arguments

1: $x (nx)$ – Real (Kind=nag_wp) arrayInput

On entry: the undifferenced time series,

x_{i}

, for

i = 1, 2, \dots, n

2: $nx$ – IntegerInput

On entry:

n

, the number of values in the undifferenced time series.

Constraint:

nx > nd + (nds \times ns)

3: $nd$ – IntegerInput

On entry:

d

, the order of non-seasonal differencing.

Constraint:

nd \geq 0

4: $nds$ – IntegerInput

On entry:

D

, the order of seasonal differencing.

Constraint:

nds \geq 0

5: $ns$ – IntegerInput

On entry:

s

, the seasonality.

Constraints:

if $nds > 0$ , $ns > 0$ ;
if $nds = 0$ , $ns \geq 0$ .

6: $xd (nx)$ – Real (Kind=nag_wp) arrayOutput

On exit: the differenced values in elements

1

nxd

, and reconstitution data in the remainder of the array.

7: $nxd$ – IntegerOutput

On exit: the number of differenced values in the array xd.

8: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $nd = 〈value〉$ .
Constraint: $nd \geq 0$ .

On entry, $nds = 〈value〉$ .
Constraint: if $ns = 0$ , $nds \leq 0$ .

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

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

$ifail = 2$: On entry, $nx = 〈value〉$ , $nd = 〈value〉$ , $nds = 〈value〉$ and $ns = 〈value〉$ .
Constraint: $nx > nd + (nds \times ns)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

g13aaf is not threaded in any implementation.

9

Further Comments

The time taken by g13aaf is approximately proportional to

(nd + nds) \times nx

10

Example

This example reads in a set of data consisting of

20

observations from a time series. Non-seasonal differencing of order

2

and seasonal differencing of order

1

(with seasonality of

4

) are applied to the input data, giving an output array holding

14

differenced values and

6

values which can be used to reconstitute the output array.

NAG Library Routine Document

g13aaf (uni_diff)

▸▿ 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

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