Path: utzoo!yunexus!ists!jarvis.csri.toronto.edu!mailrus!cornell!uw-beaver!ubc-cs!alberta!calgary!!smit
From: smit@.ucalgary.ca (Theo Smit)
Newsgroups: comp.dsp
Subject: Re: AR and MA questions
Message-ID: <2074@cs-spool.calgary.UUCP>
Date: 14 Nov 89 17:11:22 GMT
Article-I.D.: cs-spool.2074
References: <1389@mrsvr.UUCP>
Sender: news@calgary.UUCP
Reply-To: smit@enel.ucalgary.ca (Theo Smit)
Organization: U. of Calgary, Calgary, Alberta, Canada
Lines: 57

In article <1389@mrsvr.UUCP> kohli@gemed.ge.com (Mr. Bad Judgment) writes:
>
>I've been working a little bit with some ARMA stuff
>and have run into some questions which I can't
>generalize an answer to from my experience:
>
>1. Are the AR "parameters" the poles of the waveform?
>	If so, does this mean that an AR parameter of magnitude
>	> 1 indicates an unstable system (assuming no zero to
>	counteract it)?

Yes. Common ways to 'create' a stable system include shrinking all of the
pole radii by some factor (alpha). Then the AR polynomial becomes:

                             -1        2     -2        3     -3
	A(z) = 1 + alpha a1 z   + alpha  a2 z   + alpha  a3 z   + ...

alpha will be slightly less than 1.0. (0.99, or 0.999; it doesn't take much).
Mostly this is used to counter arithmetic errors; if your system is unstable
the poles will likely be well outside the unit circle and you have no hope in
heck of doing much about it.

>2. (And the MA parameters *are* the zeroes?)

Right again. In general you can set up an ARMA process as separable AR and
MA processes, i.e.

	        B(z)
	H(z) = ------
	        A(z)

where B(z) is the MA process, and A(z) is the AR process.

>3. Is the noise process usually neglected when
>	using the AR and MA parameters to model the
>	time series?  If not, is a Gaussian model
>	used?  If the noise process is calculated
>	using the calculated ARMA parameters and
>	the original sampled data, doesn't that mean
>	that your model will be identical to the
>	sampled data?

As long as there is a noise process, the process described by the model cannot
be identical to the sample process. At the 'ideal' model order (whatever that
is), the noise process will (should, anyway) be perfectly white, i.e. totally
uncorrelated, since the model has predicted all there is to predict.
>
>Thanks for your thoughts!

No trouble!

>
>Jim Kohli
>GE Medical Systems
>ge.crd.com!gemed!hal!kohli

Theo Smit (smit@enel.ucalgary.ca)