NAG Library Function Document

nag_tsa_diff (g13aac)

+− 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

nag_tsa_diff (g13aac) 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

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_diff (const double x[], Integer nx, Integer d, Integer ds, Integer s, double xd[], Integer nxd, NagError fail)

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] – const doubleInput

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 > d + (ds \times s)

3: d – IntegerInput

On entry:

d

, the order of non-seasonal differencing.

Constraint:

d \geq 0

4: ds – IntegerInput

On entry:

D

, the order of seasonal differencing.

Constraint:

ds \geq 0

5: s – IntegerInput

On entry:

s

, the seasonality.

Constraints:

if $ds > 0$ , $s > 0$ ;
if $ds = 0$ , $s \geq 0$ .

6: xd[nx] – doubleOutput

On exit: the differenced values in elements

0

nxd - 1

, and reconstitution data in the remainder of the array.

7: nxd – Integer *Output

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

8: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $d = ⟨value⟩$ .
Constraint: $d \geq 0$ .
On entry, $ds = ⟨value⟩$ .
Constraint: $ds \geq 0$ .
On entry, $s = ⟨value⟩$ .
Constraint: $s \geq 0$ .
NE_INT_2: On entry, $ds = ⟨value⟩$ .
Constraint: if $s = 0$ , $ds \leq 0$ .
NE_INT_4: On entry, $nx = ⟨value⟩$ , $d = ⟨value⟩$ , $ds = ⟨value⟩$ and $s = ⟨value⟩$ .
Constraint: $nx > d + (ds \times s)$ .
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.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_tsa_diff (g13aac) is approximately proportional to

(d + ds) \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 Function Documentnag_tsa_diff (g13aac)

+− 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_diff (g13aac)