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

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:^[9]^{[clarification needed]}

{\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 $\Pr(X_{t}\mid X_{t-1},S_{t})$ .^[7]^[6] The filtering and smoothing procedure for general cases is discussed in ^[1].

References

1 2 3 4 Murphy, K. P. (1998). Switching Kalman Filters (PDF) (Technical report). Compaq Cambridge Research Lab. 98-10.
↑ Murphy., K. (2002). Dynamic Bayesian Networks: Representation, Inference and Learning (PDF) (PhD thesis). University of California, Berkeley, Computer Science Division.
↑ Haykin, Simon (ed.). Kalman Filtering and Neural Networks. doi:10.1002/0471221546. ISBN 0-471-22154-6.
↑ Wu, Wei; Black, Michael J.; Mumford, David Bryant; Gao, Yun; Bienenstock, Elie; Donoghue, John P. (2004). "Modelling and decoding motor cortical activity using a switching Kalman filter" (PDF). IEEE Transactions on Biomedical Engineering. 51 (6): 933–942. doi:10.1109/TBME.2004.826666.
↑ Heald, JB; Ingram, JN; Flanagan, JR; Wolpert, DM (2018). "Multiple motor memories are learned to control different points on a tool". Nature Human Behaviour. 2: 300–311. doi:10.1038/s41562-018-0324-5. PMC 5935225. PMID 29736420.
1 2 Kim, C.-J. (1994). "Dynamic linear models with Markov-switching". J. Econometrics. 60: 1–22. doi:10.1016/0304-4076(94)90036-1.
1 2 Bar-Shalom, Y.; Li, X.-R. (1993). Estimation and Tracking. Boston, MA: Artech House. ISBN 978-0-89006-643-0.
↑ 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.
1 2 Ghahramani, Zoubin; Hinton, Geoffrey E. "Variational Learning for Switching State-Space Models" (PDF). Neural Computation. 12 (4): 963–996. doi:10.1162/089976600300015619. PMID 10770834.

[Murphy-SKF-1] 1 2 3 4 Murphy, K. P. (1998). Switching Kalman Filters (PDF) (Technical report). Compaq Cambridge Research Lab. 98-10.

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

[SKF-Haykin-book-3] Haykin, Simon (ed.). Kalman Filtering and Neural Networks. doi:10.1002/0471221546. ISBN 0-471-22154-6.

[Wu-Mumford-etal-4] Wu, Wei; Black, Michael J.; Mumford, David Bryant; Gao, Yun; Bienenstock, Elie; Donoghue, John P. (2004). "Modelling and decoding motor cortical activity using a switching Kalman filter" (PDF). IEEE Transactions on Biomedical Engineering. 51 (6): 933–942. doi:10.1109/TBME.2004.826666.

[HeaIngFlaWol-5] Heald, JB; Ingram, JN; Flanagan, JR; Wolpert, DM (2018). "Multiple motor memories are learned to control different points on a tool". Nature Human Behaviour. 2: 300–311. doi:10.1038/s41562-018-0324-5. PMC 5935225. PMID 29736420.

[Kim-Econometrics-6] 1 2 Kim, C.-J. (1994). "Dynamic linear models with Markov-switching". J. Econometrics. 60: 1–22. doi:10.1016/0304-4076(94)90036-1.

[BarShalom-Tracking-7] 1 2 Bar-Shalom, Y.; Li, X.-R. (1993). Estimation and Tracking. Boston, MA: Artech House. ISBN 978-0-89006-643-0.

[8] 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-9] 1 2 Ghahramani, Zoubin; Hinton, Geoffrey E. "Variational Learning for Switching State-Space Models" (PDF). Neural Computation. 12 (4): 963–996. doi:10.1162/089976600300015619. PMID 10770834.

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