phc

Scilab Function
Last update : 25/04/2007

phc - Markovian representation

### Calling Sequence

  `[H,F,G]=phc(hk,d,r)`

### Parameters

- `hk` : hankel matrix
- `d` : dimension of the observation
- `r` : desired dimension of the state vector for the approximated model
- `H, F, G` : relevant matrices of the Markovian model

### Description

Function which computes the matrices `H, F, G` of a Markovian representation by the principal hankel component approximation method, from the hankel matrix built from the covariance sequence of a stochastic process.

### Examples

    //
    //This example may usefully be compared with the results from
    //the 'levin' macro (see the corresponding help and example)
    //
    //We consider the process defined by two sinusoids (1Hz and 2 Hz)
    //in additive Gaussian noise (this is the observation);
    //the simulated process is sampled at 10 Hz.
    //
    t=0:.1:100;rand('normal');
    y=sin(2*%pi*t)+sin(2*%pi*2*t);y=y+rand(y);plot(t,y)
    //
    //covariance of y
    //
    nlag=128;
    c=corr(y,nlag);
    //
    //hankel matrix from the covariance sequence
    //(we can choose to take more information from covariance
    //by taking greater n and m; try it to compare the results !
    //
    n=20;m=20;
    h=hank(n,m,c);
    //
    //compute the Markov representation (mh,mf,mg)
    //We just take here a state dimension equal to 4 :
    //this is the rather difficult problem of estimating the order !
    //Try varying ns !
    //(the observation dimension is here equal to one)
    ns=4;
    [mh,mf,mg]=phc(h,1,ns);
    //
    //verify that the spectrum of mf contains the
    //frequency spectrum of the observed process y
    //(remember that y is sampled -in our example
    //at 10Hz (T=0.1s) so that we need
    //to retrieve the original frequencies through the log
    //and correct scaling by the frequency sampling)
    //
    s=spec(mf);s=log(s);
    s=s/2/%pi/.1;
    //
    //now we get the estimated spectrum
    imag(s),
    //

### See Also

`levin` ,