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NAG Library Function Document

nag_tsa_cross_corr (g13bcc)

▸▿ 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

nag_tsa_cross_corr (g13bcc) calculates cross-correlations between two time series.

2 Specification

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_cross_corr (const double x[], const double y[], Integer nxy, Integer nl, double s, double r0, double r[], double stat, NagError fail)

3 Description

Given two series

x_{1}, x_{2}, \dots, x_{n}

and

y_{1}, y_{2}, \dots, y_{n}

the function calculates the cross-correlations between

x_{t}

and lagged values of

y_{t}

r_{x y} (l) = \frac{\sum_{t = 1}^{n - l} (x_{t} - \bar{x}) (y_{t + l} - \bar{y})}{n s_{x} s_{y}}, l = 0, 1, \dots, L

where

\bar{x} = \frac{\sum_{t = 1}^{n} x_{t}}{n}

s_{x}^{2} = \frac{\sum_{t = 1}^{n} {(x_{t} - \bar{x})}^{2}}{n}

and similarly for

y

The ratio of standard deviations

s_{y} / s_{x}

is also returned, and a portmanteau statistic is calculated:

stat = n \sum_{l = 1}^{L} r_{x y} {(l)}^{2} .

Provided

n

is large,

L

much less than

n

, and both

x_{t}, y_{t}

are samples of series whose true autocorrelation functions are zero, then, under the null hypothesis that the true cross-correlations between the series are zero, stat has a

χ^{2}

-distribution with

L

degrees of freedom. Values of stat in the upper tail of this distribution provide evidence against the null hypothesis.

4 References

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

5 Arguments

1: $x [nxy]$ – const doubleInput: On entry: the $n$ values of the $x$ series.
2: $y [nxy]$ – const doubleInput: On entry: the $n$ values of the $y$ series.
3: $nxy$ – IntegerInput: On entry: $n$ , the length of the time series.

Constraint: $nxy \geq 2$ .
4: $nl$ – IntegerInput: On entry: $L$ , the maximum lag for calculating cross-correlations.

Constraint: $1 \leq nl < nxy$ .
5: $s$ – double *Output: On exit: the ratio of the standard deviation of the $y$ series to the standard deviation of the $x$ series, $s_{y} / s_{x}$ .
6: $r0$ – double *Output: On exit: the cross-correlation between the $x$ and $y$ series at lag zero.
7: $r [nl]$ – doubleOutput: On exit: $r [l - 1]$ contains the cross-correlations between the $x$ and $y$ series at lags $L$ , $r_{x y} (l)$ , for $l = 1, 2, \dots, L$ .
8: $stat$ – double *Output: On exit: the statistic for testing for absence of cross-correlation.
9: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 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, $nl = 〈value〉$ .
Constraint: $nl \geq 1$ .

On entry, $nxy = 〈value〉$ .
Constraint: $nxy > 1$ .
NE_INT_2: On entry, $nl \geq nxy$ : $nl = 〈value〉$ and $nxy = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
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: One or both of the $x$ and $y$ series have zero variance.

7 Accuracy

All computations are believed to be stable.

8 Parallelism and Performance

nag_tsa_cross_corr (g13bcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_tsa_cross_corr (g13bcc) 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

n < 100

, or

L < 10 \log (n)

then the autocorrelations are calculated directly and the time taken by nag_tsa_cross_corr (g13bcc) is approximately proportional to

n L

, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to

n \log (n)

. If FFTs are used then nag_tsa_cross_corr (g13bcc) internally allocates approximately

6 n

real elements.

10 Example

This example reads two time series of length

20

. It calculates and prints the cross-correlations up to lag

15

for the first series leading the second series and then for the second series leading the first series.

NAG Library Function Documentnag_tsa_cross_corr (g13bcc)