lattn

recursive solution of normal equations

Syntax

[la,lb]=lattn(n,p,cov)

Arguments

n

maximum order of the filter

p

fixed dimension of the MA part. If p= -1, the algorithm reduces to the classical Levinson recursions.

cov

matrix containing the Rk's (d*d matrices for a d-dimensional process).It must be given the following way

$\begin{eqnarray} \begin{pmatrix} R_0\\R_1\\R_2\\ \vdots \\R_{nlags} \end{pmatrix} \end{eqnarray}$

la

list-type variable, giving the successively calculated polynomials (degree 1 to degree n),with coefficients Ak

Description

solves recursively on n (p being fixed) the following system (normal equations), i.e. identifies the AR part (poles) of a vector ARMA(n,p) process,

$\begin{eqnarray} \begin{pmatrix} I&-A_1&\cdots&-A_n \end{pmatrix} \ast \begin{pmatrix} R_{p+1}&R_{p+2}&\cdots&R_{p+n} \\ R_p&R_{p+1}&\cdots&R_{p+n-1} \\ R_{p+n-1}&R_p&\cdots&R_{p+n-2} \\ \vdots&\vdots&\cdots&\vdots \\ R_{p+1-n}&R_{p+2-n}&\cdots&R_p \end{pmatrix} = 0 \end{eqnarray}$

where {Rk;k=1,nlag} is the sequence of empirical covariances.

Example

//Generate the process
t1=0:0.1:100;
y1=sin(2*%pi*t1)+sin(2*%pi*2*t1);
y1=y1+rand(y1,"normal");

//Covariance of y1
nlag=128;
c1=corr(y1,nlag);
c1=c1';

//Compute the filter with maximum order=15 and p=1
n=15;
[la1,sig1]=lattn(n,1,c1);

//Compare result of poles with p=-1 and with levin function
[la2,sig2]=lattn(n,-1,c1);
for i=1:n
  s2=roots(la2(i));
  s2=log(s2)/2/%pi/.1; //estimated poles
  s2=gsort(imag(s2));
  s2=s2(1:i/2);
end;
[la3,sig3]=levin(n,c1);
for i=1:n
  s3=roots(la3(i));
  s3=log(s3)/2/%pi/.1; //estimated poles
  s3=gsort(imag(s3));
  s3=s3(1:i/2);
end;

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lattn

Syntax

Arguments

Description

Example

See also