naginterfaces.library.tsa.multi_​autocorr_​part

naginterfaces.library.tsa.multi_autocorr_part(c0, c, nk)

multi_autocorr_part calculates the multivariate partial autocorrelation function of a multivariate time series.

For full information please refer to the NAG Library document for g13db

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13dbf.html

Parameters

c0float, array-like, shape $(\text{ns},\text{ns})$

Contains the zero lag cross-covariances between the $\text{ns}$ series as returned by multi_corrmat_cross(). ($c0$ is assumed to be symmetric, upper triangle only is used.)

cfloat, array-like, shape $(\text{ns},\text{ns},\text{nl})$

Contains the cross-covariances at lags $1$ to $\text{nl}$. $c[i-1,j-1,k-1]$ must contain the cross-covariance, $c_{ijk}$, of series $i$ and series $j$ at lag $k$. Series $j$ leads series $i$.

nkint

The number of lags to which partial auto-correlations are to be calculated.

Returns

pfloat, ndarray, shape $\left(nvp\right)$

The multiple squared partial autocorrelations from lags $1$ to $nvp$; that is, $p[l-1]$ contains $p_{\text{l}}^2$, for $\text{l}=1,2,\dots ,nvp$. For lags $nvp+1$ to $nk$ the elements of $p$ are set to zero.

v0float

The lag zero prediction error variance (equal to the determinant of $c0$).

vfloat, ndarray, shape $\left(nvp\right)$

The prediction error variance ratios from lags $1$ to $nvp$; that is, $v[\text{l}-1]$ contains $v_{\text{l}}$, for $\text{l}=1,2,\dots ,nvp$. For lags $nvp+1$ to $nk$ the elements of $v$ are set to zero.

dfloat, ndarray, shape $(\text{ns},\text{ns},nk)$

The prediction error variance matrices at lags $1$ to $nvp$.

Element $(i,j,\text{k})$ of $d$ contains the prediction error covariance of series $i$ and series $j$ at lag $\text{k}$, for $\text{k}=1,2,\dots ,nvp$.

Series $j$ leads series $i$; that is, the $(i,j)$th element of $D_k$.

For lags $nvp+1$ to $nk$ the elements of $d$ are set to zero.

dbfloat, ndarray, shape $(\text{ns},\text{ns})$

The backward prediction error variance matrix at lag $nvp$.

$db[i-1,j-1]$ contains the backward prediction error covariance of series $i$ and series $j$; that is, the $(i,j)$th element of the $G_k$, where $k=nvp$.

wfloat, ndarray, shape $(\text{ns},\text{ns},nk)$

The prediction coefficient matrices at lags $1$ to $nvp$.

$w[i-1,j-1,\text{l}-1]$ contains the $j$th prediction coefficient of series $i$ at lag $\text{l}$; that is, the $(i,j)$th element of ${\Phi }_{k\text{l}}$, where $k=nvp$, for $\text{l}=1,2,\dots ,nvp$.

For lags $nvp+1$ to $nk$ the elements of $w$ are set to zero.

wbfloat, ndarray, shape $(\text{ns},\text{ns},nk)$

The backward prediction coefficient matrices at lags $1$ to $nvp$.

$wb[i-1,j-1,\text{l}-1]$ contains the $j$th backward prediction coefficient of series $i$ at lag $\text{l}$; that is, the $(i,j)$th element of ${\Psi }_{k\text{l}}$, where $k=nvp$, for $\text{l}=1,2,\dots ,nvp$.

For lags $nvp+1$ to $nk$ the elements of $wb$ are set to zero.

nvpint

The maximum lag, $L$, for which calculation of $p$, $v$, $d$, $db$, $w$ and $wb$ was successful. If the function completes successfully $nvp$ will equal $nk$.

Raises

NagValueError

(errno $1$)

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

Constraint: $nk\le \text{nl}$.

(errno $1$)

On entry, $nk=⟨value⟩$.

Constraint: $nk\ge 1$.

(errno $1$)

On entry, $\text{nl}=⟨value⟩$.

Constraint: $\text{nl}\ge 1$.

(errno $1$)

On entry, $\text{ns}=⟨value⟩$ and $\text{ldc0}=⟨value⟩$.

Constraint: $\text{ns}\le \text{ldc0}$.

(errno $1$)

On entry, $\text{ns}=⟨value⟩$.

Constraint: $\text{ns}\ge 1$.

(errno $2$)

$c0$ is not positive definite.

Warns

NagAlgorithmicWarning

(errno $3$)

For $nvp=⟨value⟩$, at lag $k=nvp+1$, $D_k$ was found not to be positive definite.

Notes

The input is a set of lagged autocovariance matrices $C_0,C_1,C_2,\dots ,C_m$. These will generally be sample values such as are obtained from a multivariate time series using multi_corrmat_cross().

The main calculation is the recursive determination of the coefficients in the finite lag (forward) prediction equation

$$x_t={\Phi }_{l,1}{x}_{t-1}+\cdots +{\Phi }_{l,l}{x}_{t-l}+e_{l,t}$$

and the associated backward prediction equation

$$x_{t-l-1}={\Psi }_{l,1}{x}_{t-l}+\cdots +{\Psi }_{l,l}{x}_{t-1}+f_{l,t}$$

together with the covariance matrices $D_l$ of $e_{l,t}$ and $G_l$ of $f_{l,t}$.

The recursive cycle, by which the order of the prediction equation is extended from $l$ to $l+1$, is to calculate

$$M_{l+1}=C_{l+1}^T-{\Phi }_{l,1}{C}_l^T-\cdots -{\Phi }_{l,l}{C}_1^T$$

then ${\Phi }_{l+1,l+1}=M_{l+1}{D}_l^{-1}$, ${\Psi }_{l+1,l+1}=M_{l+1}^T{G}_l^{-1}$

from which

$${\Phi }_{l+1,j}={\Phi }_{l,j}-{\Phi }_{l+1,l+1}{\Psi }_{l,l+1-j}\text{,}j=1,2,\dots ,l$$

and

$${\Psi }_{l+1,j}={\Psi }_{l,j}-{\Psi }_{l+1,l+1}{\Phi }_{l,l+1-j}\text{,}j=1,2,\dots ,l\text{.}$$

Finally, $D_{l+1}=D_l-M_{l+1}{\Phi }_{l+1,l+1}^T$ and $G_{l+1}=G_l-M_{l+1}^T{\Psi }_{l+1,l+1}^T$.

(Here $T$ denotes the transpose of a matrix.)

The cycle is initialized by taking (for $l=0$)

$$D_0=G_0=C_0\text{.}$$

In the step from $l=0$ to $1$, the above equations contain redundant terms and simplify. Thus (1) becomes $M_1=C_1^T$ and neither (2) and (3) are needed.

Quantities useful in assessing the effectiveness of the prediction equation are generalized variance ratios

$$v_l=det\left(D_l\right)/det\left(C_0\right)\text{,}l=1,2,\dots$$

and multiple squared partial autocorrelations

$$p_l^2=1-v_l/v_{l-1}\text{.}$$

References

Akaike, H, 1971, Autoregressive model fitting for control, Ann. Inst. Statist. Math. (23), 163–180

Whittle, P, 1963, On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix, Biometrika (50), 129–134