Switching Kalman filter

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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,^[1]^[2]^[3]^[4] 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,^[5] and sensorimotor learning in humans.^[6] It also has application in econometrics,^[7] signal processing, tracking,^[8] 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.^[9] For example, when an industrial plant has "multiple discrete modes of behaviour, each of which having a linear (Gaussian) dynamics".^[10]

Model

There are several variants of SKF discussed in.^[1]

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:^[10]

[This section is badly written: It does not explain the notation used below.]

{\begin{aligned}&\Pr(\{S_{t},X_{t}^{(1)},\ldots ,X_{t}^{(M)},Y_{t}\})\\={}&\Pr(S_{1})\prod _{t=2}^{T}\Pr(S_{t}\mid S_{t-1})\times \prod _{m=1}^{M}\Pr(X_{1}^{(m)})\prod _{t=2}^{T}\Pr(X_{t}^{(m)}\mid X_{t-1}^{(m)})\times \prod _{t=1}^{T}\Pr(Y_{t}\mid X_{t}^{(1)},\ldots ,X_{t}^{(M)},S_{t}).\end{aligned}}

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 $\Pr(S_{t}\mid S_{t-1})$ . The probability model of $X$ and $Y$ can depend on $S_{t}$ .

The switch variable can take its values from a set $S_{t}\in \{1,2,\ldots ,M\}$ . 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,^[1] the switch variable affects the dynamics of $X_{t}$ , e.g. through $\Pr(X_{t}\mid X_{t-1},S_{t})$ .^[8]^[7] The filtering and smoothing procedure for general cases is discussed in.^[1]

References

^ ^a ^b ^c ^d K. P. Murphy, "Switching Kalman Filters", Compaq Cambridge Research Lab Tech. Report 98-10, 1998
^ K. Murphy. Switching Kalman filters. Technical report, U. C. Berkeley, 1998.
^ K. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, University of California, Berkeley, Computer Science Division, 2002.
^ Kalman Filtering and Neural Networks. Edited by Simon Haykin. ISBN 0-471-22154-6
^ 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. doi:10.1109/TBME.2004.826666
^ 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).
^ ^a ^b Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1–22.
^ ^a ^b Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Artech House, Boston, MA.
^ Karimi, Parisa (2021). "Quantification of mismatch error in randomly switching linear state-space models". IEEE Signal Processing Letters. 28: 2008–2012. arXiv:2012.04542. Bibcode:2021ISPL...28.2008K. doi:10.1109/LSP.2021.3116504. S2CID 227745283.
^ ^a ^b Zoubin Ghahramani, Geoffrey E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996.

[Murphy-SKF-1] K. P. Murphy, "Switching Kalman Filters", Compaq Cambridge Research Lab Tech. Report 98-10, 1998

[2] K. Murphy. Switching Kalman filters. Technical report, U. C. Berkeley, 1998.

[3] K. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, University of California, Berkeley, Computer Science Division, 2002.

[SKF-Haykin-book-4] Kalman Filtering and Neural Networks. Edited by Simon Haykin. ISBN 0-471-22154-6

[Wu-Mumford-etal-5] 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. doi:10.1109/TBME.2004.826666

[HeaIngFlaWol-6] 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).

[Kim-Econometrics-7] Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1–22.

[BarShalom-Tracking-8] Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Artech House, Boston, MA.

[9] Karimi, Parisa (2021). "Quantification of mismatch error in randomly switching linear state-space models". IEEE Signal Processing Letters. 28: 2008–2012. arXiv:2012.04542. Bibcode:2021ISPL...28.2008K. doi:10.1109/LSP.2021.3116504. S2CID 227745283.

[Ghahramani-Hinton-Variational-10] Zoubin Ghahramani, Geoffrey E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996.

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