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From: RRogers <rerogers@plaidheron.com>
Newsgroups: sci.stat.math,comp.soft-sys.matlab,sci.engr.control
Subject: Re: Kalman filtering with multiplicative noise
Date: Mon, 21 Jul 2008 07:22:38 -0700 (PDT)
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On Jul 20, 10:51 pm, d...@myallit.com wrote:
> I'm trying to implement a Kalman filter in MATLAB that will use two
> types of measurements: volume and in/out flow rate. For the flow rate,
> the measurement error is additive Gaussian, but for the volume the
> measurement error is expressed as a percentage of the volume, so that
> the volume measurement is less accurate when its value is higher. I
> think the measurement model should therefore be:
>
> Flow rate measurement model:
> z1 = x1 + v1 where v1 ~ N(0,e1)
>
> Volume measurement model:
> z2 = x2*v2 where v2 ~ N(1,e2)
>
> I assumed the volume filtering should be done in the log domain to
> make the noise additive but how do I deal with a noise mean of one
> when the Kalman filter assumes a mean of zero? And how can I have a
> Kalman filter using both the measurements if one is in the log domain
> and the other one isn't?
>
> I am also dealing with a system where measurements will usually be
> missing (they are arriving sequentially) and at an uneven sampling
> rate, any other pointers on these too would be appreciated.

A couple of comments:
1) Unless v1,v2 have covariance then they are independent and you can
deal with them separately.  It doesn't seem to make any sense to
combine the estimation of z1,z2 in one Kalman filter.
2) In the case of equation two, typically v2 would be restricted to be
>0.  I would expect v2=lognormal(1,mu) so that ln(v2)=N(0,mu); or see
comment 4.
3) Given the above then defining nz2=log(z2), nx2=log(x2), nv2=log(v2)
would yield a perfectly normal distribution; sorry for the pun:)
4) Even is v2 wasn't truly log-normal you could try to use the first
and second moments as a description of the real ln(v2) distribution
and see how adequate the description works.

IMHO: all of engineering mathematics is descriptive and the only
question is how adequate the descriptions are.

Ray