# NAG C Library Function Document

## 1Purpose

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

## 2Specification

 #include #include
 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)

## 3Description

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 ${y}_{t}$:
 $rxyl=∑t=1 n-lxt-x-yt+l-y- nsxsy , l=0,1,…,L$
where
 $x-=∑t= 1nxtn$
 $sx2=∑t=1n xt-x- 2n$
and similarly for $y$.
The ratio of standard deviations ${s}_{y}/{s}_{x}$ is also returned, and a portmanteau statistic is calculated:
 $stat=n∑l=1Lrxy l 2.$
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 ${\chi }^{2}$-distribution with $L$ degrees of freedom. Values of stat in the upper tail of this distribution provide evidence against the null hypothesis.
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

## 5Arguments

1:    $\mathbf{x}\left[{\mathbf{nxy}}\right]$const doubleInput
On entry: the $n$ values of the $x$ series.
2:    $\mathbf{y}\left[{\mathbf{nxy}}\right]$const doubleInput
On entry: the $n$ values of the $y$ series.
3:    $\mathbf{nxy}$IntegerInput
On entry: $n$, the length of the time series.
Constraint: ${\mathbf{nxy}}\ge 2$.
4:    $\mathbf{nl}$IntegerInput
On entry: $L$, the maximum lag for calculating cross-correlations.
Constraint: $1\le {\mathbf{nl}}<{\mathbf{nxy}}$.
5:    $\mathbf{s}$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:    $\mathbf{r0}$double *Output
On exit: the cross-correlation between the $x$ and $y$ series at lag zero.
7:    $\mathbf{r}\left[{\mathbf{nl}}\right]$doubleOutput
On exit: ${\mathbf{r}}\left[\mathit{l}-1\right]$ contains the cross-correlations between the $x$ and $y$ series at lags $L$, ${r}_{xy}\left(\mathit{l}\right)$, for $\mathit{l}=1,2,\dots ,L$.
8:    $\mathbf{stat}$double *Output
On exit: the statistic for testing for absence of cross-correlation.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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 $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nl}}\ge 1$.
On entry, ${\mathbf{nxy}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nxy}}>1$.
NE_INT_2
On entry, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nxy}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{nl}}<{\mathbf{nxy}}$.
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
One or both of the $x$ and $y$ series have zero variance.

## 7Accuracy

All computations are believed to be stable.

## 8Parallelism 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.

## 9Further Comments

If $n<100$, or $L<10\mathrm{log}\left(n\right)$ then the autocorrelations are calculated directly and the time taken by nag_tsa_cross_corr (g13bcc) is approximately proportional to $nL$, otherwise the autocorrelations are calculated by utilizing fast Fourier transforms (FFTs) and the time taken is approximately proportional to $n\mathrm{log}\left(n\right)$. If FFTs are used then nag_tsa_cross_corr (g13bcc) internally allocates approximately $6n$ real elements.

## 10Example

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.

### 10.1Program Text

Program Text (g13bcce.c)

### 10.2Program Data

Program Data (g13bcce.d)

### 10.3Program Results

Program Results (g13bcce.r)