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>

void	g13aac (const double x[], Integer nx, Integer d, Integer ds, Integer s, double xd[], Integer nxd, NagError fail)

The function may be called by the names: g13aac, nag_tsa_uni_diff or nag_tsa_diff.

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 double Input

On entry: the undifferenced time series,

x_{i}

, for

i = 1, 2, \dots, n

2: $nx$ – Integer Input

On entry:

n

, the number of values in the undifferenced time series.

Constraint:

nx > d + (ds \times s)

3: $d$ – Integer Input

On entry:

d

, the order of non-seasonal differencing.

Constraint:

d \geq 0

4: $ds$ – Integer Input

On entry:

D

, the order of seasonal differencing.

Constraint:

ds \geq 0

5: $s$ – Integer Input

On entry:

s

, the seasonality.

Constraints:

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

6: $xd [nx]$ – double Output

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

g13aac is not threaded in any implementation.

9 Further Comments

The time taken by 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.

g13aa: FL CL CPP AD

NAG CL Interfaceg13aac (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

NAG CL Interface
g13aac (uni_diff)