naginterfaces.library.linsys.real_​toeplitz_​yule

naginterfaces.library.linsys.real_toeplitz_yule(t, wantp, wantv)[source]

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://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f04/f04fef.html

Parameters
tfloat, array-like, shape

must contain the value of the diagonal elements of , and the remaining elements of must contain the elements of the vector .

wantpbool

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

wantvbool

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

Returns
xfloat, ndarray, shape

The solution vector .

pfloat, ndarray, shape

With as , the th element of contains the partial (auto)correlation coefficient, or reflection coefficient, for the th step. (See Further Comments and submodule tsa.) If is , is not referenced. Note that in any case, .

vfloat, ndarray, shape

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

vlastfloat

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

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

Principal minor is not positive definite. Value of the reflection coefficient is .

Notes

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

real_toeplitz_yule solves the equations

where is the symmetric positive definite Toeplitz matrix

and is the vector

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