StatTimeSerAnalysis.DurbinLevinson Method

StatTimeSerAnalysis

procedure DurbinLevinson(const gamma: TVec; const Phi: TVec; out Sigma2: Double; const NumEvals: Integer; const PhiVar: TVec);

The Durbin-Levinson algorithm.

#NameDescription
1gammaDefines covariances for Durbin-Levinson algorithm.
2PhiReturns phi[n,1]...phi[n,n] coefficients.
3NumEvalsDefines number of iterations of the Durbin-Levinson algorithm.
4Sigma2Returns estimated variance.
5PhiVarIf not nil, it returns phi coefficients variances.

Result: stored in self (calling object)

Remarks:

Uses the Durbin-Levinson algorithm to calculate phi[n,1]...phi[n,n] coefficients. Coefficients are calculated resursively from the following relations:

ϕnn=[γ(n)−∑j=1n−1ϕn−1,jγ(n−j)]vn−1−1,[ϕn,1⋮ϕn,n−1]=[ϕn−1,1⋮ϕn−1,n−1]−ϕnn[ϕn−1,n−1⋮ϕn−1,1]vn=vn−1[1−ϕnn2]\begin{aligned} \phi_{nn}&= \left[ \gamma(n)-\sum_{j=1}^{n-1} \phi_{n-1,j} \gamma(n-j)\right] v_{n-1}^ {-1} \quad , \\ \left[ \begin{array}{c}\phi_{n,1} \\ \vdots \\ \phi_{n,n-1} \\ \end{array}\right] &= \left[ \begin{array}{c}\phi_{n-1,1} \\ \vdots \\ \phi_{n-1,n-1}\\ \end{array}\right] - \phi_{nn} \left[ \begin{array}{c}\phi_{n-1,n-1} \\ \vdots \\ \phi_{n-1,1} \\ \end{array}\right] \\ v_n &= v_{n-1}[1-\phi_{nn}^2] \end{aligned}

where phi(1,1) = gamma(1)/gamma(0) and v(0)=gamma(0).

See Also: StatTimeSerAnalysis.Innovations