naginterfaces.library.linsys.real_​toeplitz_​yule

naginterfaces.library.linsys.real_toeplitz_yule(t, wantp, wantv)

real_toeplitz_yule solves the Yule–Walker equations for a real symmetric positive definite Toeplitz system.

For full information please refer to the NAG Library document for f04fe

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04fef.html

Parameters

tfloat, array-like, shape $(n+1)$

$t\left[0\right]$ must contain the value ${\tau }_0$ of the diagonal elements of $T$, and the remaining $\text{n}$ elements of $t$ must contain the elements of the vector $t$.

wantpbool

Must be set to $True$ if the partial (auto)correlation coefficients are required, and must be set to $False$ otherwise.

wantvbool

Must be set to $True$ if the mean square prediction errors are required, and must be set to $False$ otherwise.

Returns

xfloat, ndarray, shape $\left(n\right)$

The solution vector $x$.

pfloat, ndarray, shape $(:)$

With $wantp$ as $True$, the $i$th element of $p$ contains the partial (auto)correlation coefficient, or reflection coefficient, $p_i$ for the $i$th step. (See Further Comments and submodule tsa.) If $wantp$ is $False$, $p$ is not referenced. Note that in any case, $x_n=p_n$.

vfloat, ndarray, shape $(:)$

With $wantv$ as $True$, the $i$th element of $v$ contains the mean square prediction error, or predictor error variance ratio, $v_i$, for the $i$th step. (See Further Comments and submodule tsa.) If $wantv$ is $False$, $v$ is not referenced.

vlastfloat

The value of $v_n$, the mean square prediction error for the final step.

Raises

NagValueError

(errno $-1$)

On entry, $t\left[0\right]=⟨value⟩$.

Constraint: $t\left[0\right]>0.0$.

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Principal minor $⟨value⟩$ is not positive definite. Value of the reflection coefficient is $⟨value⟩$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

real_toeplitz_yule solves the equations

$$Tx=-t\text{,}$$

where $T$ is the $n\times n$ symmetric positive definite Toeplitz matrix

$$\begin{array}{c}T=\left(\begin{array}{ccccc}{\tau }_0& {\tau }_1& {\tau }_2& \dots & {\tau }_{n-1}\\ {\tau }_1& {\tau }_0& {\tau }_1& \dots & {\tau }_{n-2}\\ {\tau }_2& {\tau }_1& {\tau }_0& \dots & {\tau }_{n-3}\\ .& .& .& & .\\ {\tau }_{n-1}& {\tau }_{n-2}& {\tau }_{n-3}& \dots & {\tau }_0\end{array}\right)\end{array}$$

and $t$ is the vector

$$t^T=({\tau }_1,{\tau }_2\dots {\tau }_n)\text{.}$$

The function uses the method of Durbin (see Durbin (1960) and Golub and Van Loan (1996)). Optionally the mean square prediction errors and/or the partial correlation coefficients for each step can be returned.

References

Bunch, J R, 1985, Stability of methods for solving Toeplitz systems of equations, SIAM J. Sci. Statist. Comput. (6), 349–364

Bunch, J R, 1987, The weak and strong stability of algorithms in numerical linear algebra, Linear Algebra Appl. (88/89), 49–66

Cybenko, G, 1980, The numerical stability of the Levinson–Durbin algorithm for Toeplitz systems of equations, SIAM J. Sci. Statist. Comput. (1), 303–319

Durbin, J, 1960, The fitting of time series models, Rev. Inst. Internat. Stat. (28), 233

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore