Kalman filter and cousins, recursive estimation, predictive state models, Data assimilation. 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.

## Awaiting filing

Recently enjoyed:
Sahani Pathiraja’s state filter does something cool, in attempting to identify
process *model* noise - a conditional nonparametric density of process errors, that may be used to come up with some neat process models.
I’m not convinced about her use of
kernel density estimators, since these scale badly precisely when you need them most, in high dimension; but any nonparametric density estimator would, I assume, work.

## 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 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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- 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.
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- Sark07
- Sarkka, S. (2007) On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.
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- 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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- 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.