naginterfaces.library.tsa.multi_​xcorr

naginterfaces.library.tsa.multi_xcorr(x, y, nl)

multi_xcorr calculates cross-correlations between two time series.

For full information please refer to the NAG Library document for g13bc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g13/g13bcf.html

Parameters

xfloat, array-like, shape $\left(\text{nxy}\right)$

The $n$ values of the $x$ series.

yfloat, array-like, shape $\left(\text{nxy}\right)$

The $n$ values of the $y$ series.

nlint

$L$, the maximum lag for calculating cross-correlations.

Returns

sfloat

The ratio of the standard deviation of the $y$ series to the standard deviation of the $x$ series, $s_y/s_x$.

r0float

The cross-correlation between the $x$ and $y$ series at lag zero.

rfloat, ndarray, shape $\left(nl\right)$

$r[\text{l}-1]$ contains the cross-correlations between the $x$ and $y$ series at lags $L$, $r_{xy}\left(\text{l}\right)$, for $\text{l}=1,2,\dots ,L$.

statfloat

The statistic for testing for absence of cross-correlation.

Raises

NagValueError

(errno $1$)

On entry, $\text{nxy}=⟨value⟩$.

Constraint: $\text{nxy}>1$.

(errno $1$)

On entry, $nl=⟨value⟩$ and $\text{nxy}=⟨value⟩$.

Constraint: $1\le nl<\text{nxy}$.

(errno $1$)

On entry, $nl=⟨value⟩$.

Constraint: $nl\ge 1$.

(errno $2$)

One or both of the $x$ and $y$ series have zero variance.

Notes

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 $y_t$:

$$r_{xy}\left(l\right)=\frac{{\sum }_{t=1}^{n-l}(x_t-\overline{x})(y_{t+l}-\overline{y})}{n{s}_x{s}_y}\text{,}l=0,1,\dots ,L$$

where

$$\overline{x}=\frac{{\sum }_{t=1}^n{x}_t}{n}$$

$$s_x^2=\frac{{\sum }_{t=1}^n{(x_t-\overline{x})}^2}{n}$$

and similarly for $y$.

The ratio of standard deviations $s_y/s_x$ is also returned, and a portmanteau statistic is calculated:

$$stat=n\sum _{l=1}^L{r}_{xy}{\left(l\right)}^2\text{.}$$

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.

References

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