bayesclasses-general

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

[Michael S. <ms...@21...>]

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...

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? 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. The documentation doesn't make it
completely clear what all the pieces that I'm specifying are. Is there
any help anyone could offer?

Thanks,
Mike Simon

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

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

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?

> 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?

The Covariance_filter implements the EKF in a fairly standard form. It should 
be usable for any non-linear problem that can be solved by an EKF.
You need to turn the maths of you model into 3 things:
	f(x) : This should be easy once you have discrete time state equation of the 
system!
	Fx : Is the Jacobian of f(x)
	GqG' : This is often written as the symmetric matrix Q and is the covariance 
of the addative noise in you discrete state equations. Generally it is 
possible (an physically more meaningful) to write this a coupling matrix G 
and an noise variance q. If this is not possible then you can numerically 
obtain G and q from Q with the following:

        // Equivalent de-correlated form as GqG'
	Float rcond = Bayesian_filter_matrix::UdUfactor (Qtemp, Q);
	rclimit.check_PSD(rcond, "decorrelating Q not PSD");
	Bayesian_filter_matrix::UdUseperate (G, q, Qtemp);

Hope this helps you in the right direction,
	Michael

-- 
___________________________________
Michael Stevens Systems Engineering

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

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

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

[Michael S. <ms...@21...>]

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 Covariance_filter implements the EKF in a fairly standard form. It should 
> be usable for any non-linear problem that can be solved by an EKF.
> You need to turn the maths of you model into 3 things:
> 	f(x) : This should be easy once you have discrete time state equation of the 
> system!

I suppose this is where my biggest problem was when the question was 
asked, but after some rooting through the code, I am under the 
impression you are supposed to replace the virtual f function with a 
function that takes x, computes f(x), computes the Jacobian for that 
time step and puts it in Fx, and returns the new value. Is that correct?

> 	Fx : Is the Jacobian of f(x)
> 	GqG' : This is often written as the symmetric matrix Q and is the covariance 
> of the addative noise in you discrete state equations. Generally it is 
> possible (an physically more meaningful) to write this a coupling matrix G 
> and an noise variance q. If this is not possible then you can numerically 
> obtain G and q from Q with the following:
> 
>         // Equivalent de-correlated form as GqG'
> 	Float rcond = Bayesian_filter_matrix::UdUfactor (Qtemp, Q);
> 	rclimit.check_PSD(rcond, "decorrelating Q not PSD");
> 	Bayesian_filter_matrix::UdUseperate (G, q, Qtemp);

The two above things are simply variables to be set, yes?

> 
> Hope this helps you in the right direction,
> 	Michael

Yes, this has been helpful so far, thanks.

Michael S.

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
___________________________________