Re: [Bayes++] Bayes++ w/EKF and a non-linear equation: How to model?

[Michael S. <ma...@mi...>]

On Tuesday, 12. December 2006 22:41, Michael Simon wrote:
> Michael Stevens wrote:
> > On Monday, 11. December 2006 17:24, Michael Simon wrote:
> >> Hello, everyone.
> >>
> >> I am attempting to use the Covariance_filter as an EKF for a non-linear
> >> model. However, I do not understand which type of model class to use,
> >> and how to specify the data for the models, even after looking at the
> >> Bayesian Filtering overview. I'm pretty sure I need to use a
> >> linrz_predict_model (or something like that) but beyond that...
> >
> > 'Linrz_predict_model' is definately what you want.
> >
> >> The main problem is that the elements of the f function I want to use
> >> for the model are exponential. It's non-linear, with noise, but no
> >> control (it's essentially 'the object I'm trying to predict is walking
> >> around randomly, but within the laws of physics') How do you specify
> >> that?
> >
> > No-control is fine. Control inputs just make the implementation more
> > complex! I don't understand how you model can be 'exponential' when the
> > physics are of something moving around at random. Or do you mean that you
> > have exponentially correlated noise in you model?
>
> Actually, after sketching the algorithms out by hand, I don't understand
> it either. ;)  I do want a model that would be sufficient to track
> something that might move at random, but the formula I had down clearly
> wouldn't give it.
>
> >> I've been looking at the Welch and Bishop introduction and trying
> >> to compare back to Bayes++, but with no avail. I'm fairly sure I could
> >> use the libraries if my model was linear, but I don't quite see how to
> >> put the non-linearity in there.
> >
> > I not sure of where you are stubling. Would it be possible to post the
> > equations that model your system?
>
> As I mentioned above, I was using the wrong formulas. At the time I was
> reading a report about EKF that purported to be using unconstrained
> Brownian Motion for modeling and trying to reproduce their results, but
> the formulas they gave don't produce anything approaching that, at
> least, not how I read them. (One example: x_k = exp(-1/4(x_(k-1) + 1.5
> (deltax_(k-1)) from
> http://page.mi.fu-berlin.de/~zaldivar/files/tr-b-05-12.pdf )

The paper looks to be totaly bogus to me! The first 13 pages are just a 
standard derivation of the Kalman filter then can be found in many books. The 
dynamic system model at the end of page 16 which you quote from is just 
nonsense. I'm not sure where they go it from.

I think Welch and Bishop should have the equations you need. Otherwise take a 
look in:
Estimation with Applications to Tracking and Navigation: Theory Algorithms and 
Software
Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan
2001 John Wiley & Sons, Inc.
ISBNs: 0-471-41655-X (Hardback) 0-471-22127-9 (Electronic)

Start with a "Continuous White Noise Acceleration Model".

My own PV example in Bayes++ is a extension of this. It implements the IOU 
(see reference in code) in 1D. This is a little better as it bounds the 
growth of velocity uncertainty.

Sorry I can't wrie much more at the moment. I will be checking my emails over 
the Christmas holidays and will attempt to help out if you have more 
questions.

All the best,
	Michaelesclasses-general

-- 
___________________________________
Michael Stevens Systems Engineering

34128 Kassel, Germany
Phone/Fax: +49 561 5218038

Navigation Systems, Estimation  and
                 Bayesian Filtering
    http://bayesclasses.sf.net
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