void	nag_tsa_multi_cross_corr (Nag_CovOrCorr matrix, Integer k, Integer n, Integer m, const double w[], double wmean[], double r0[], double r[], NagError *fail)

3

Description

Let

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

, for

t = 1, 2, \dots, n

, denote

n

observations of a vector of

k

time series. The sample cross-covariance matrix at lag

l

is defined to be the

k

k

matrix

\hat{C} (l)

, whose (

i, j

)th element is given by

{\hat{C}}_{i j} (l) = \frac{1}{n} \sum_{t = l + 1}^{n} (w_{i (t - l)} - {\bar{w}}_{i}) (w_{j t} - {\bar{w}}_{j}), l = 0, 1, 2, \dots, m, ​ i = 1, 2, \dots, k ​ and ​ j = 1, 2, \dots, k,

where

{\bar{w}}_{i}

and

{\bar{w}}_{j}

denote the sample means for the

i

th and

j

th series respectively. The sample cross-correlation matrix at lag

l

is defined to be the

k

k

matrix

\hat{R} (l)

, whose

(i, j)

th element is given by

{\hat{R}}_{i j} (l) = \frac{{\hat{C}}_{i j} (l)}{\sqrt{{\hat{C}}_{i i} (0) {\hat{C}}_{j j} (0)}}, l = 0, 1, 2, \dots, m, ​ i = 1, 2, \dots, k ​ and ​ j = 1, 2, \dots, k .

The number of lags,

m

, is usually taken to be at most

n / 4

W_{t}

follows a vector moving average model of order

q

, then it can be shown that the theoretical cross-correlation matrices

(R (l))

are zero beyond lag

q

. In order to help spot a possible cut-off point, the elements of

\hat{R} (l)

are usually compared to their approximate standard error of 1/

\sqrt{n}

. For further details see, for example, Wei (1990).

The function uses a single pass through the data to compute the means and the cross-covariance matrix at lag zero. The cross-covariance matrices at further lags are then computed on a second pass through the data.

4

References

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

West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

5

Arguments

1: $matrix$ – Nag_CovOrCorrInput

On entry: indicates whether the cross-covariance or cross-correlation matrices are to be computed.

$matrix = Nag_AutoCov$: The cross-covariance matrices are computed.
$matrix = Nag_AutoCorr$: The cross-correlation matrices are computed.

Constraint:

matrix = Nag_AutoCov

Nag_AutoCorr

2: $k$ – IntegerInput

On entry:

k

, the dimension of the multivariate time series.

Constraint:

k \geq 1

3: $n$ – IntegerInput

On entry:

n

, the number of observations in the series.

Constraint:

n \geq 2

4: $m$ – IntegerInput

On entry:

m

, the number of cross-correlation (or cross-covariance) matrices to be computed. If in doubt set

m = 10

. However it should be noted that m is usually taken to be at most

n / 4

Constraint:

1 \leq m < n

5: $w [k \times n]$ – const doubleInput

On entry:

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

must contain the value for series

i

at time

t

, for

i = 1, 2, \dots, k

and

t = 1, 2, \dots, n

6: $wmean [k]$ – doubleOutput

On exit: the means,

{\bar{w}}_{i}

, for

i = 1, 2, \dots, k

7: $r0 [k \times k]$ – doubleOutput

On exit: if

matrix = Nag_AutoCov

r0 [(j - 1) k + i - 1]

contains the

(i, j)

th element of the sample cross-covariance matrix.

matrix = Nag_AutoCorr

r0 [(j - 1) k + i - 1]

i \neq j

contains the

(i, j)

th element of the sample cross-correlation matrix and

r0 [(i - 1) k + i - 1]

contains the standard deviation of the

i

th series.

8: $r [k \times k \times m]$ – doubleOutput

On exit: if

matrix = Nag_AutoCov

r [(l - 1) k^{2} + (j - 1) k + i - 1]

contains the

(i, j)

th element of the sample cross-covariance matrix at lag

l

matrix = Nag_AutoCorr

, then it contains the

(i, j)

th element of the sample cross-correlation matrix lag

l

, for

l = 1, 2, \dots, m

i = 1, 2, \dots, k

and

j = 1, 2, \dots, k

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: $n \geq 2$ .
NE_INT_2: On entry, $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $m \geq 1$ and $m < n$ .
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_ZERO_VARIANCE: On entry, at least one of the series is such that all its elements are practically identical giving zero (or near zero) variance.

7

Accuracy

For a discussion of the accuracy of the one-pass algorithm used to compute the sample cross-covariances at lag zero see West (1979). For the other lags a two-pass algorithm is used to compute the cross-covariances; the accuracy of this algorithm is also discussed in West (1979). The accuracy of the cross-correlations will depend on the accuracy of the computed cross-covariances.

8

Parallelism and Performance

nag_tsa_multi_cross_corr (g13dmc) 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 time taken is roughly proportional to

m n k^{2}

10

Example

This program computes the sample cross-correlation matrices of two time series of length

48

, up to lag

10

. It also prints the cross-correlation matrices together with plots of symbols indicating which elements of the correlation matrices are significant. Three * represent significance at the

0.5

% level, two * represent significance at the 1% level and a single * represents significance at the 5% level. The * are plotted above or below the line depending on whether the elements are significant in the positive or negative direction.

NAG Library Function Document

nag_tsa_multi_cross_corr (g13dmc)

▸▿ 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