Switching Kalman Filter

The switching Kalman filtering (SKF) method is a variant of the Kalman filter. In its generalised form, it is often attributed to Kevin P. Murphy,K. P. Murphy, "Switching Kalman Filters", Compaq Cambridge Research Lab Tech. Report 98-10, 1998Kalman Filtering and Neural Networks. Edited by Simon Haykin. but related switching state-space models have been in use.

Applications

Applications of the switching Kalman filter include: Brain–computer interfaces and neural decoding, real-time decoding for continuous neural-prosthetic control,Wu, Wei, Michael J. Black, David Bryant Mumford, Yun Gao, Elie Bienenstock, and John P. Donoghue. 2004. Modelling and decoding motor cortical activity using a switching Kalman filter. IEEE Transactions on Biomedical Engineering 51(6): 933-942. and sensorimotor learning in humans.Heald JB, Ingram JN, Flanagan JR, Wolpert DM. Multiple motor memories
are learned to control different points on a tool. ''Nature Human Behaviour''. 2, 300–311, (2018).
It also has application in econometrics, signal processing, tracking, computer vision, etc. It is an alternative to the Kalman filter when the system's state has a discrete component. The additional error when using a Kalman filter instead of a Switching Kalman filter may be quantified in terms of the switching system's parameters. For example, when an industrial plant has "multiple discrete modes of behaviour, each of which having a linear (Gaussian) dynamics".Zoubin Ghahramani, Geoffrey E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996.

Model

There are several variants of SKF discussed in.

Special case

In the simpler case, switching state-space models are defined based on a switching variable which evolves independent of the hidden variable. The probabilistic model of such variant of SKF is as the following:

his section is badly written: It does not explain the notation used below.
: $\begin & \Pr(\) \\ = & \Pr(S_1)\prod_^T \Pr(S_t \mid S_) \times \prod_^M \Pr(X_1^) \prod_^T \Pr(X_t^\mid X_^) \times \prod_^T \Pr(Y_t\mid X_t^,\ldots,X_t^,S_t). \end$

The hidden variables include not only the continuous $X$, but also a discrete *switch* (or switching) variable $S_t$. The dynamics of the switch variable are defined by the term $\Pr(S_t \mid S_)$. The probability model of $X$ and $Y$ can depend on $S_t$.

The switch variable can take its values from a set $S_t\in\$. This changes the joint distribution $(X_t,Y_t)$ which is a separate multivariate Gaussian distribution in case of each value of $S_t$.

General case

In more generalised variants, the switch variable affects the dynamics of $X_t$, e.g. through $\Pr(X_t\mid X_, S_t)$.Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Artech House, Boston, MA.Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1–22.
The filtering and smoothing procedure for general cases is discussed in.

References

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Control theory
Nonlinear filters
Linear filters
Signal estimation
Stochastic differential equations
Robot control
Markov models