naginterfaces.library.tsa.multi_​corrmat_​cross

naginterfaces.library.tsa.multi_corrmat_cross(matrix, m, w)

multi_corrmat_cross calculates the sample cross-correlation (or cross-covariance) matrices of a multivariate time series.

For full information please refer to the NAG Library document for g13dm

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13dmf.html

Parameters

matrixstr, length 1

Indicates whether the cross-covariance or cross-correlation matrices are to be computed.

$matrix=\text{'V'}$

The cross-covariance matrices are computed.

$matrix=\text{'R'}$

The cross-correlation matrices are computed.

mint

$m$, 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 $n/4$.

wfloat, array-like, shape $(k,n)$

$w[\text{i}-1,\text{t}-1]$ must contain the observation $w_{\text{i}\text{t}}$, for $\text{t}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,k$.

Returns

wmeanfloat, ndarray, shape $\left(k\right)$

The means, ${\overline{w}}_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

r0float, ndarray, shape $(k,k)$

If $i\ne j$, then $r0[i-1,j-1]$ contains an estimate of the $(i,j)$th element of the cross-correlation (or cross-covariance) matrix at lag zero, ${\hat{R}}_{ij}\left(0\right)$; if $i=j$, then if $matrix=\text{'V'}$, $r0[i-1,i-1]$ contains the variance of the $i$th series, ${\hat{C}}_{ii}\left(0\right)$, and if $matrix=\text{'R'}$, $r0[i-1,i-1]$ contains the standard deviation of the $i$th series, $\sqrt{{\hat{C}}_{ii}\left(0\right)}$.

If $errno$ = 2 and $matrix=\text{'R'}$, then on exit all the elements in $r0$ whose computation involves the zero variance are set to zero.

rfloat, ndarray, shape $(k,k,m)$

$r[\text{i}-1,\text{j}-1,\text{l}-1]$ contains an estimate of the ($\text{i},\text{j}$)th element of the cross-correlation (or cross-covariance) at lag $\text{l}$, ${\hat{R}}_{\text{i}\text{j}}\left(\text{l}\right)$, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,k$, for $\text{l}=1,2,\dots ,m$.

If $errno$ = 2 and $matrix=\text{'R'}$, then on exit all the elements in $r$ whose computation involves the zero variance are set to zero.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m\ge 1$ and $m<n$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $1$)

On entry, $matrix=⟨value⟩$.

Constraint: $matrix=\text{'V'}$ or $\text{'R'}$.

Warns

NagAlgorithmicWarning

(errno $2$)

On entry, at least one of the series is such that all its elements are practically identical giving zero (or near zero) variance.

Notes

Let $W_t={(w_{1t},w_{2t},\dots ,w_{kt})}^T$, 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 $k\times k$ matrix $\hat{C}\left(l\right)$, whose ($i,j$)th element is given by

$${\hat{C}}_{ij}\left(l\right)=\frac{1}{n}\sum _{t=l+1}^n(w_{i(t-l)}-{\overline{w}}_i)(w_{jt}-{\overline{w}}_j)\text{,}l=0,1,2,\dots ,m\text{,}i=1,2,\dots ,k\text{and}j=1,2,\dots ,k\text{,}$$

where ${\overline{w}}_i$ and ${\overline{w}}_j$ 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 $k\times k$ matrix $\hat{R}\left(l\right)$, whose $(i,j)$th element is given by

$${\hat{R}}_{ij}\left(l\right)=\frac{{\hat{C}}_{ij}\left(l\right)}{\sqrt{{\hat{C}}_{ii}\left(0\right){\hat{C}}_{jj}\left(0\right)}}\text{,}l=0,1,2,\dots ,m\text{,}i=1,2,\dots ,k\text{and}j=1,2,\dots ,k\text{.}$$

The number of lags, $m$, is usually taken to be at most $n/4$.

If $W_t$ follows a vector moving average model of order $q$, then it can be shown that the theoretical cross-correlation matrices $\left(R\left(l\right)\right)$ are zero beyond lag $q$. In order to help spot a possible cut-off point, the elements of $\hat{R}\left(l\right)$ 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.

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