I wanted to compare the performance of a standard Kalman Filter and an Information Filter (propagation of precisions instead of covariances) for a sensor network scenario, i.e. when one has way more measurements per time step than states in the state space model.
When the measurement vector y_k is affected by white Gaussian noise, both the Kalman Filter and Information Filters' performance is comparable, therefore I wanted to look into correlated measurement noise (not correlated in time though, i.e. no colored noise, but correlation between the sensor readings at each time step).
I was looking around, but could find any interesting application or papers discussing that. Can anyone point me to some interesting work?
Thanks!
Thanks a lot, that is an interesting example!
I additionally found an application in sensor networks, where the measurement noise is dependent on the distance between the sensors.
I found a nice model in this paper: Sensor Selection for Estimation
with Correlated Measurement Noise
http://arxiv.org/pdf/1508.03690.pdf
Equation (43)
Hello. Your question is a little unclear. The main cases of noise correlation in standard Kalman filtering are: correlated measurement noise, correlation between measurement and process noise. I think you can also formulate the problem forcorrelation between the initial state and the other noise sequences also. The reference is Jazwinski (Stochastic processes and filtering theory).
If you do not have coloured noise (i.e. time correlation), then there is no problem. You have a non diagonal measurement covariance, but you can still apply regular KF theory.
As far as the information filter is concerned, I am not aware of a formulation that allows for correlated noise. I think that you generally have to whiten the noise by transforming the state space or measurement variables to a system in which the whiteness assumptions hold, then derive a new Kalman filter for this system. In the transformed system, you can have an information filter. This is the way I would go about it anyhow.
BTW, the information filter and the standard Kalman filter are perfectly equivalent when the standard assumptions hold (whiteness, controllability, observability).
Thanks for your answer, I'll have a look into "whitening" the noise.
I guess one has to find a new basis (eigenbasis) and there the covariance matrix is diagonal. This involves transforming the observation vector at each time step. In a sensor network scenario this sounds computationally expensive.
"As far as the information filter is concerned, I am not aware of a formulation that allows for correlated noise."
As long as the covariance matrix of the measurement noise is invertible there is no problem in implementing an information filter. Singular covariance matrices would mean perfect measurements in a specific subspace which would not really make sense.
"BTW, the information filter and the standard Kalman filter are perfectly equivalent when the standard assumptions hold (whiteness, controllability, observability)."
Mathematically yes, numerically (both in terms of stability and FLOPS) no, since the filters involve different algebraic manipulations of the mean and covariance matrices (or precisions in the case of the information filter).
Hello.
"As long as the covariance matrix of the measurement noise is invertible there is no problem in implementing an information filter."
I think we are talking about different things. The noise covariances Q and R in the system description for a Kalman filter can naturally be fully populated, and are assumed to be positive definite. This is not what I call correlated noise however - I mean E[v(k) v(k-1)^T] not equal to 0 or similar for the measurement noise w(k). This assumption is what the KF is based on. If it doesn't hold, the KF loses its optimality.
When the usual KF assumptions are satisfied, the information filter is algebraically equivalent (via the matrix inversion lemma). It may not be equivalent in finite precision arithmetic.But this does not seem to be what your original question was about.
True,
"I mean E[v(k) v(k-1)^T] not equal to 0 or similar for the measurement noise w(k)."
I guess I should have added to my question that I was looking into spatially and not temporally correlated noise. Regarding output noise R this of course only makes numerically a difference if the measurement vector (at a given time) is large, ideally larger than the state vector dimension.
The question arose, since in this paper (ITA 2016)
http://www.isiweb.ee.ethz.ch/papers/docu/aloe-lubr-hama-fewa-noza-2016.pdf,
among other things, I suggested a Kalman smoother which is dual to the Modified-Bryson-Frazier. Now I wanted to compare the numerical stability of the two smoothers.
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