g13aa 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.

Syntax

C#
public static void g13aa( double[] x, int nx, int nd, int nds, int ns, double[] xd, out int nxd, out int ifail )

Visual Basic
Public Shared Sub g13aa ( _ x As Double(), _ nx As Integer, _ nd As Integer, _ nds As Integer, _ ns As Integer, _ xd As Double(), _ <OutAttribute> ByRef nxd As Integer, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g13aa ( _
	x As Double(), _
	nx As Integer, _
	nd As Integer, _
	nds As Integer, _
	ns As Integer, _
	xd As Double(), _
	<OutAttribute> ByRef nxd As Integer, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g13aa( array<double>^ x, int nx, int nd, int nds, int ns, array<double>^ xd, [OutAttribute] int% nxd, [OutAttribute] int% ifail )

F#
static member g13aa : x : float[] * nx : int * nd : int * nds : int * ns : int * xd : float[] * nxd : int byref * ifail : int byref -> unit

Parameters

x: Type: array<System..::..Double>[]()[][]
An array of size [nx]
On entry: the undifferenced time series, $x_{i}$ , for $i = 1, 2, \dots, n$ .

nx: Type: System..::..Int32
On entry: $n$ , the number of values in the undifferenced time series.

Constraint: $nx > nd + (nds \times ns)$ .

nd: Type: System..::..Int32
On entry: $d$ , the order of non-seasonal differencing.

Constraint: $nd \geq 0$ .

nds: Type: System..::..Int32
On entry: $D$ , the order of seasonal differencing.

Constraint: $nds \geq 0$ .

ns

Type: System..::..Int32

On entry:

s

, the seasonality.

Constraints:

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

xd: Type: array<System..::..Double>[]()[][]
An array of size [nx]
On exit: the differenced values in elements $0$ to $nxd - 1$ , and reconstitution data in the remainder of the array.

nxd: Type: System..::..Int32%
On exit: the number of differenced values in the array xd.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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.

References

None.

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$nd < 0$ ,
or	$nds < 0$ ,
or	$ns < 0$ ,
or	$ns = 0$ when $nds > 0$ .

$ifail = 2$

On entry,

nx \leq nd + (nds \times ns)

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The computations are believed to be stable.

Parallelism and Performance

None.

Further Comments

The time taken by g13aa is approximately proportional to

(nd + nds) \times nx

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.

Example program (C#): g13aae.cs

Example program data: g13aae.d

Example program results: g13aae.r