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Fitting AR model to stohastic signal

Started by TI September 3, 2003
Hi
I have a sample of a Stohastic signal with very slowly variable mean,
and would like to find AR model for it perhaps using Burg parameter
estimation model (explained in book by Kay).
I would like to try diffent model order until I get the Power spectral
density of white  noise driven AR model's output, to look like the PSD
of my sample.
Is the aforementioned  possible?(any other ideas?)
Also can I expect AR model's ouput(to white noise) in time to resemble
my Stohastic sample provided PSDs 'match'?

Thanks
In <18a34598.0309030805.59e9cb9e@posting.google.com> anglomont@yahoo.com (TI)
writes:

>Hi >I have a sample of a Stohastic signal with very slowly variable mean, >and would like to find AR model for it perhaps using Burg parameter >estimation model (explained in book by Kay). >I would like to try diffent model order until I get the Power spectral >density of white noise driven AR model's output, to look like the PSD >of my sample. >Is the aforementioned possible?(any other ideas?) >Also can I expect AR model's ouput(to white noise) in time to resemble >my Stohastic sample provided PSDs 'match'?
>Thanks
You need the Signal Processing Toolbox. Use arburg(), looping over model order, to get AR coefficients. Then use the AR model to predict the value of the signal from previous values of the signal, and look at the difference between the actual and predicted values. This difference should be white noise at the correct model order. The power spectrum of your AR model's output to white noise input should match the power spectrum of the sample; there's no reason why the output of the model should match the sample. After all, there are an infinity of different white noise samples which have the same statistical characteristics. By the way, you should look at a discussion of the criteria used to determine an appropriate model order. -- David B. Chorlian Neurodynamics Lab SUNY/HSCB chorlian@spot.cns.hscbklyn.edu davidc@panix.com
anglomont@yahoo.com (TI) wrote in message
news:<18a34598.0309030805.59e9cb9e@posting.google.com>...
> Hi > I have a sample of a Stohastic signal with very slowly variable mean, > and would like to find AR model for it perhaps using Burg parameter > estimation model (explained in book by Kay).
Kay has written several books. I assume you mean "Modern Spectral Estimation" (1988).
> I would like to try diffent model order until I get the Power spectral > density of white noise driven AR model's output, to look like the PSD > of my sample. > Is the aforementioned possible?(any other ideas?)
Yes, this is possible. First, be aware that Burg's method and the regular AR models are the same. Burg used a different mathematical starting point but found the same prediction coefficients that the "usual" AR method, based on the Yule-Walker equations, estimates. One could say that Burg showed that the AR coefficients in a certain sense are optimal, in that they maximize the entropy of the residual signal. It's easy to implement a basic AR estimator yourself, but you need to read up on the Yule-Walker equations that describe why the basic AR model works. You should also read about the Levinson recursion, which is the practical way to estimate the AR model from observed data. The Levinson recursion is described as a flowchart in Kay's book, but it isn't very clear how it works. If you have access to a good library, try to get Therrien: "Discrete Random Signals and Statistical Signal Processing" Prentice-Hall 1992 (it's hard to find, but a reprint can be bought at www.amazon.com). Therrien shows the Levinson recursion in a way that lets you implement it from scratch in five minutes, using matlab (his table 8.2). Last, take some time to find out about order estimators. The Akaike Information Criterion (AIC) and the Rissanen/Schwartz Minimum Description Length (MDL) are mathematical functions that help you to estimate the order from the data. Put simply, order estimators are tests that tell you when to stop the Levinson recursion.
> Also can I expect AR model's ouput(to white noise) in time to resemble > my Stohastic sample provided PSDs 'match'?
Well... maybe. If your data are generated by a process that is truly AR, you will be able to generate a sequence that has the same statistical properties as your input data. If your data are not entirely AR, then you may experience problems with getting the model to fit. Rune
TI wrote:

> Hi > I have a sample of a Stohastic signal with very slowly variable mean, > and would like to find AR model for it perhaps using Burg parameter > estimation model (explained in book by Kay). > I would like to try diffent model order until I get the Power spectral > density of white noise driven AR model's output, to look like the PSD > of my sample. > Is the aforementioned possible?(any other ideas?) > Also can I expect AR model's ouput(to white noise) in time to resemble > my Stohastic sample provided PSDs 'match'? > > Thanks
The algorithms from Kay's book are implemented in my TSA-toolbox for Matlab/Octave. http://www.dpmi.tu-graz.ac.at/~schloegl/matlab/tsa/ There are hardly any good tests for checking whether a PSD corresponds to whith noise or not. Instead, various criteria for model order selection are also implemented in the TSA-tb. I recommend to use TSADEMO and replace the sample data with your own data. You can resemble your orginal data only if you know the "innovation process"; otherwise you get a different realization of the stochastic process. The following commands should do the job: [AR,RC,PE]=lattice(Y(:).',p); % estimate AR paramters using BURG, Y must be a row vector X = filter(ar2poly(AR),1,Y); % gives the residual X; X is similar to the innovation process Y1 = filter(1,ar2poly(AR),X); % resembles the original process. Alois