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I have been experimenting with using an augmented state space in which I store / memorise previous states as new variables at the bottom of the state vector when performing discrete Kalman filtering.

I can then approximate a high-order discrete model by passing the lagged terms down through the state vector after each time step.

My initial results seem to work well, so I was wondering why the Markov assumption is so firmly stated in the filtering literature. Besides the massive increase in dimensionality of the state, are there any possible side effects I should look out for, or reasons why this approach is not generalisable?

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  • $\begingroup$ Your question seems rather broad & open-ended to me. That isn't generally what the site is for. Be aware that people may object & may vote to close as too broad. You may want to edit this to make it narrower & more concrete. $\endgroup$ – gung - Reinstate Monica♦ Aug 2 '17 at 17:39
  • $\begingroup$ I disagree I think this is actually a fairly well posed. $\endgroup$ – jds Aug 3 '17 at 0:44
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So you mention storing "observations" but I think you should instead be thinking of augmenting the state space by "storing the previous state". Either way, if I am correct in what you are thinking, this concept is used all the time.

For example its the standard way of working with higher order ARMA models (for example an AR(2)) as well as a number of other situations.

For example, Take a look at Example 1 of Eric Zivot's Notes on State Space Models (found using a quick google search for "AR model and state-space"). Or from this post on the Signal Processing Stack Exchage.

Assuming you have a linear gaussian model as you explicitly mentioned the Kalman filter, there not a problem, it is just an alternative representation of an identical model.

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  • $\begingroup$ Thanks for your answer and the reference. I'll edit to clarify that I do mean storing previous states. Out of interest, you mention use of a linear Gaussian model, but in nonlinear Kalman filtering we could do the same thing couldn't we? We wouldn't have a full state-space model necessarily, but we'd have a state vector in which we could similarly store previous states. $\endgroup$ – wil_j_wil Aug 3 '17 at 9:25
  • $\begingroup$ Yes. You can do this in a non-linear filter as well. $\endgroup$ – jds Aug 4 '17 at 22:04

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