NAG Library Function Document

nag_tsa_multi_diff (g13dlc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_tsa_multi_diff (g13dlc) differences and/or transforms a multivariate time series.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_multi_diff (Integer k, Integer n, const double z[], const Integer tr[], const Integer id[], const double delta[], double w[], Integer nd, NagError fail)

3 Description

For certain time series it may first be necessary to difference the original data to obtain a stationary series before calculating autocorrelations, etc. This function also allows you to apply either a square root or a log transformation to the original time series to stabilize the variance if required.

If the order of differencing required for the

i

th series is

d_{i}

, then the differencing operator is defined by

δ_{i} (B) = 1 - δ_{i 1} B - δ_{i 2} B^{2} - \dots - δ_{i d_{i}} B^{d_{i}}

, where

B

is the backward shift operator; that is,

B Z_{t} = Z_{t - 1}

. Let

d

denote the maximum of the orders of differencing,

d_{i}

, over the

k

series. The function computes values of the differenced/transformed series

W_{t} = {(w_{1 t}, w_{2 t}, \dots, w_{k t})}^{T}

, for

t = d + 1, \dots, n

, as follows:

w_{i t} = δ_{i} (B) z_{i t}^{*}, i = 1, 2, \dots, k

where

z_{i t}^{*}

are the transformed values of the original

k

-dimensional time series

Z_{t} = {(z_{1 t}, z_{2 t}, \dots, z_{k t})}^{T}

The differencing parameters

δ_{i j}

, for

i = 1, 2, \dots, k

and

j = 1, 2, \dots, d_{i}

, must be supplied by you. If the

i

th series does not require differencing, then

d_{i} = 0

4 References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley

5 Arguments

1: k – IntegerInput

On entry:

k

, the dimension of the multivariate time series.

Constraint:

k \geq 1

2: n – IntegerInput

On entry:

n

, the number of observations in the series, prior to differencing.

Constraint:

n \geq 1

3: z[ $k \times n$ ] – const doubleInput

On entry:

z [(t - 1) k + i - 1]

must contain the

i

th series at time

t

, for

t = 1, 2, \dots, n

and

i = 1, 2, \dots, k

4: tr[k] – const IntegerInput

On entry:

tr [i - 1]

indicates whether the

i

th series is to be transformed, for

i = 1, 2, \dots, k

$tr [i - 1] = - 1$: A square root transformation is used.
$tr [i - 1] = 0$: No transformation is used.
$tr [i - 1] = 1$: A log transformation is used.

Constraint:

tr [i - 1] = - 1

0

1

, for

i = 1, 2, \dots, k

5: id[k] – const IntegerInput

On entry: the order of differencing for each series,

d_{1}, d_{2}, \dots, d_{k}

Constraint:

0 \leq id [i] < n

, for

i = 0, 1, \dots, k - 1

6: delta[ $\dim$ ] – const doubleInput

Note: the dimension, dim, of the array delta must be at least

k \times \max (1, d)

, where

d = \max (id [i - 1])

On entry: if

id [i - 1] > 0

then

delta [(j - 1) k + i - 1]

must be set to

δ_{i j}

, for

j = 1, 2, \dots, d_{l}

and

i = 1, 2, \dots, k

7: w[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array w must be at least

k \times (n - d)

, where

d = \max (id [i - 1])

On exit:

w [(t - 1) k + i - 1]

contains the value of

w_{i, t + d}

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n - d

8: nd – Integer *Output

On exit: the number of differenced values,

n - d

, in the series, where

d = \max (id [i - 1])

9: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 1$ .; On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INT_ARRAY: On entry, $k = 〈value〉$ , $id [i] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $0 \leq id [i] < n$ , for $i = 0, 1, \dots, k - 1$ .; On entry, $tr [i - 1] = 〈value〉$ and $k = 〈value〉$ .
Constraint: $tr [i - 1] = - 1$ , $0$ or $1$ , for $i = 1, 2, \dots, k$ .
NE_INT_ARRAY_ELEM_CONS: On entry, element $〈value〉$ of id is greater than or equal to n.; On entry, element $〈value〉$ of id is less than zero.
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.
NE_TRANSFORMATION: On entry, one (or more) of the transformations requested is invalid.

7 Accuracy

The computations are believed to be stable.

8 Further Comments

The same differencing operator does not have to be applied to all the series. For example, suppose we have

k = 2

, and wish to apply the second-order differencing operator

\nabla^{2}

to the first series and the first-order differencing operator

\nabla

to the second series:

\begin{array}{l} w_{1 t} & = \nabla^{2} z_{1 t} = {(1 - B)}^{2} z_{1 t} = (1 - 2 B + B^{2}) z_{1 t},   and \\ w_{2 t} & = \nabla z_{2 t} = (1 - B) z_{2 t} . \end{array}

Then

d_{1} = 2, d_{2} = 1

d = \max (d_{1}, d_{2}) = 2

, and

delta = [\begin{array}{r} δ_{11} & δ_{12} \\ δ_{21} \end{array}] = [\begin{array}{r} 2 & - 1 \\ 1 \end{array}] .

9 Example

A program to difference (non-seasonally) each of two time series of length

48

. No transformation is to be applied to either of the series.

NAG Library Function Documentnag_tsa_multi_diff (g13dlc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_tsa_multi_diff (g13dlc)