## Exact PDF of an ARMA process - Efficient method using Levinson algorithm

The exact PDF of an ARMA process can be written as in (9.6) where is the symmetric Toeplitz covariance matrix of an ARMA process formed from (9.15). Computing , and even just storing it, for large is inefficient. Using the Levinson algorithm, an efficient implementation of the exact PDF may be obtained.

When is large, the computing the term is prohibitive and when is very large, even storing is inefficient. The problem is solved efficiently using the Levinson algorithm. Let x be an N-by-1 vector and let be the symmetric N-by-N Toeplitz matrix formed from the N-by-1 ACF vector r. The function

   [y,ldetR] = toepsolve(r,x,N);

is the same as
   R = toeplitz(r(1:N));
y = R \ x;
ldetR = log(det(R));

We then have

The execution time of software/toepsolve.m is much faster (see Figure 9.3) and includes as a by-product the determinant of R. As predicted by theory, the standard approach is order , while Levinson is .
The exact PDF can be therefore computed by
   [y,ldetR] = toepsolve(r,x,N);
lpx = -N/2*log(2*pi) -.5*ldetR -.5*x'*y;


Baggenstoss 2017-05-19