[Bayes++] Re: bayes++

[Nicola B. <nic...@gm...>]

Hi Baskar,

On Monday 26 Sep 2005 01:20, Baskar Jayaraman wrote:
> Hi Nicola, I saw your following question on the bayes++ ML. I am also a
> little bit confused about modeling Q on my problem in bayes++ and would
> like to see if you can help me. I am using linear_predict model to model
> the following problem:
>  x1(k+1) =3D x1(k) + u1(k)-u2(k)+w1(k)
> x2(k+1) =3D x2(k) + u2(k)-u3(k)+w2(k)
> y(k) =3D x1(k) + x2(k) + v(k)
>  w1,w2,v are noises. u1, u2 are control actions. u3 is a disturbance on x=
2.
> y is the observation. u1 u2 and u3 have noises which I am modeling as bei=
ng
> included in w1 and w2. This causes a difficulty because u2 is there in bo=
th
> x1 and x2 equations and hence would make w1 and w2 correlated and hence t=
he
> covariance matrix would have off-diagonal elements. The filter is to
> estimate x1 and x2. My question is how to specify G and q of bayes++. I
> know from the examples that G is a 2x2 matrix and q is a 2x1 vector. But I
> am lost because the code calls q as covariance. I would appreciate any
> light you can throw on modeling this in bayes++.

Actually that comment is incorrect because the covariance is a matrix indee=
d.
I think the following answer, that I got from Michael a few weeks ago, may=
=20
help to understand how to use 'q' and 'G':

=2E..
G and q together represent the process (predict) noise. q is the noise=20
variance (a vector) and G is the noise coupling.
In this case the process model is=20
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0x(k+1) =3D f(x(k)) + G.q(k)
where q(k) in Gaussian white noise with variance q

This leads to a Kalman filter covariance update for the linear case
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0X(k+1) =3D F.X(k).F' + G.q.=
G'
This is equivalent to
=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0X(k+1) =3D F.X(k).F' + Q
where Q =3D G.q.G'

The are a couple of reasons for expressing the process noise in this way.
a) For factorised filters (such as the UD_scheme) it is in the perfect form
b) The same noise is often additive to more then one element of the state. =
In=20
this case the size of q is less then x and G provides a physically easily=20
interpreted of how the elements of q effect x.
=2E..

However, instead of including the control noise in w(k), I'd rather extend =
the=20
state vector trying to estimate such a noise. For example, a simple system=
=20
like this:

x(k+1) =3D x(k) + u(k) + n(k) + w(k)
y(k+1) =3D x(k) + v(k)

where n(k) is the noise of the control u(k), could be extended as follows:

x1(k+1) =3D x1(k) + x2(k) + u(k) + w(k)
x2(k+1) =3D n(k)
y(k+1) =3D x1(k) + v(k)

I'm posting this on the bayes++ mailing list, so maybe someone with more=20
experience than me can give you some suggestion.
Cheers,
Nicola

>  Thanks for your help.
>  Baskar
>  Your question on ML:
> -------------------------------
> "I'm using a "Linear_predict_model" (but this is related also to other
> models...) for implementing an EKF and there's something I cannot
> understand.
> In the "bayesFlt.hpp" is reported that the vector "q" is the covariance of
> the state noise w(k)... but isn't such a covariance a symmetric matrix
> (normally called "Q")? So, if I wanna represent my covariance Q, how shou=
ld
> I
> do?"

=2D-=20
=2D-----------------------------------------
Nicola Bellotto
University of Essex
Department of Computer Science
Wivenhoe Park
Colchester CO4 3SQ
United Kingdom

Room: 1N1.2.8
Tel. +44 (0)1206 874094
E-Mail: nb...@es...
URL: http://privatewww.essex.ac.uk/~nbello
=2D-----------------------------------------