Kalman filter and cousins, recursive estimation, predictive state models. A particular sub-field of signal processing for models with hidden state.

In statistics terms, the state filters are a kind of online-updating hierarchical model for sequential observations of a dynamical system where the random state is unobserved, but you can get an optimal estimate of it based on incoming measurements and known parameters.

A unifying feature of all these is by assuming a sparse influence graph between observations and dynamics, that you can estimate behaviour using effiicent message passing.

## Linear systems

In Kalman filters *per se* you are usually concerned with multivariate real vector signals representing different axes of some telemetry data problem.

The classic Kalman filter (Kalm60) assumes a linear model with Gaussian noise, although it might work with not-quite Gaussian, not-quite linear models if you prod it. You can extend this flavour to somewhat more general dynamics.

If you are doing telemetry then you probably know a prori that your model is not linear in this case, and extensions are advisable.

## Non-linear dynamical systems

Cute exercise: you can derive the analytic Kalman filter for any noise and process dynamics of with Bayesian conjugate, and this leads to filters of nonlinear behaviour. Multivariate distributions are a bit of a mess for non-Gaussians, though, and a beta-Kalman filter feels a little contrived.

Upshot is, the non-linear extensions don’t usually rely on non-Gaussian conjugate distributions and anlytic forms, but rather do some Gaussian/linear approximation, or use randomised methods such as particle filters.

## Other interesting state filters

TODO: learn if the “Predictive state representation” of WoJS05 and SiJR04 is in fact a state filter? Looks similar to something I’ve derived… Or is this better classified as state space reconstruction?

Note that state filters can also do approximate gaussian process regresssion, apparently.

## State filter inference

How about learning the parameters of the model generating your states? See linear system identification, general system identification, .

## Implementations

## Refs

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- Robe11
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- RoPl07
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- RoSD13
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- RoSP11
- Robertson, A., Stark, A. M., & Plumbley, M. D.(2011) Real-Time Visual Beat Tracking Using a Comb Filter Matrix.
- RoRu09
- Rodriguez, A., & Ruiz, E. (2009) Bootstrap prediction intervals in state–space models.
*Journal of Time Series Analysis*, 30(2), 167–178. DOI. - RuSW05
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- Sark07
- Sarkka, S. (2007) On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.
*IEEE Transactions on Automatic Control*, 52(9), 1631–1641. DOI. - SiJR04
- Singh, S., James, M. R., & Rudary, M. R.(2004) Predictive State Representations: A New Theory for Modeling Dynamical Systems. In Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence (pp. 512–519). Arlington, Virginia, United States: AUAI Press
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*Journal of the American Statistical Association*, 1–31. DOI. - WoJS05
- Wolfe, B., James, M. R., & Singh, S. (2005) Learning Predictive State Representations in Dynamical Systems Without Reset. In Proceedings of the 22Nd International Conference on Machine Learning (pp. 980–987). New York, NY, USA: ACM DOI.