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.TH phc 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
phc - Markovian representation
.SH CALLING SEQUENCE
.nf
[H,F,G]=phc(hk,d,r)
.fi
.SH PARAMETERS
.TP 14
hk
: hankel matrix
.TP
d
: dimension of the observation
.TP
r
: desired dimension of the state vector for the approximated model
.TP
H, F, G
: relevant matrices of the Markovian model
.SH DESCRIPTION
Function which computes the matrices \fVH, F, G\fR of a Markovian
representation by the principal hankel
component approximation method, from the hankel matrix built
from the covariance sequence of a stochastic process.
.SH EXAMPLE
.nf
//
//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),
//
.fi
.SH SEE ALSO
levin
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