nag_tsa_uni_diff (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

[xd, nxd, ifail] = g13aa(x, nd, nds, ns, 'nx', nx)

[xd, nxd, ifail] = nag_tsa_uni_diff(x, nd, nds, ns, 'nx', nx)

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.

Parameters

Compulsory Input Parameters

1: $x (nx)$ – double array

The undifferenced time series,

x_{i}

, for

i = 1, 2, \dots, n

2: $nd$ – int64int32nag_int scalar

d

, the order of non-seasonal differencing.

Constraint:

nd \geq 0

3: $nds$ – int64int32nag_int scalar

D

, the order of seasonal differencing.

Constraint:

nds \geq 0

4: $ns$ – int64int32nag_int scalar

s

, the seasonality.

Constraints:

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

Optional Input Parameters

1: $nx$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of values in the undifferenced time series.

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

Output Parameters

1: $xd (nx)$ – double array: The differenced values in elements $1$ to $nxd$ , and reconstitution data in the remainder of the array.
2: $nxd$ – int64int32nag_int scalar: The number of differenced values in the array xd.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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)

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computations are believed to be stable.

Further Comments

The time taken by nag_tsa_uni_diff (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.

Open in the MATLAB editor: g13aa_example

function g13aa_example


fprintf('g13aa example results\n\n');

x = [120;     108;      98;     118;     135;
     131;     118;     125;     121;     100;
      82;      82;      89;      88;      86;
      96;     108;     110;      99;     105];
nd  = int64(2);
nds = int64(1);
ns  = int64(4);

[xd, nxd, ifail] = g13aa( ...
                          x, nd, nds, ns);

% Display results
nx = numel(xd);
fprintf(' Non-seasonal differencing of order %4d\n', nd);
fprintf(' and seasonal differencing of order %4d\n', nds);
fprintf('       are applied with seasonality %4d\n\n',ns);
fprintf('The output array holds %4d values,\n',nx); 
fprintf('    of which the first %4d are differenced values\n\n',nxd);
for j = 1:7:nxd
  fprintf('%7.0f',xd(j:min(j+6,nxd)));
  fprintf('\n');
end
fprintf('\n');
for j = nxd+1:7:nx
  fprintf('%7.0f',xd(j:min(j+6,nx)));
  fprintf('\n');
end

g13aa example results

 Non-seasonal differencing of order    2
 and seasonal differencing of order    1
       are applied with seasonality    4

The output array holds   20 values,
    of which the first   14 are differenced values

    -11    -10     -8      4     12     -2     18
      9     -4     -6     -5     -2    -12      5

      2    -10    -13     17      6    105

PDF version (NAG web site, 64-bit version, 64-bit version)

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