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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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, element $〈value〉$ of id is greater than or equal to n.

On entry, element $〈value〉$ of id is less than zero.

On entry, $tr [〈value〉] = 〈value〉$ .
Constraint: $tr [i] = - 1$ , $0$ or $1$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_TRANSFORMATION: On entry, one (or more) of the transformations requested is invalid.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

nag_tsa_multi_diff (g13dlc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

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}] .

10

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 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 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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