Switching Kalman filter

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Short description: Type of mathematical 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]

Pr⁡({St,Xt(1),…,Xt(M),Yt})=Pr⁡(S1)∏t=2TPr⁡(St∣St−1)×∏m=1MPr⁡(X1(m))∏t=2TPr⁡(Xt(m)∣Xt−1(m))×∏t=1TPr⁡(Yt∣Xt(1),…,Xt(M),St).

The hidden variables include not only the continuous X, but also a discrete switch (or switching) variable St. The dynamics of the switch variable are defined by the term Pr⁡(St∣St−1). The probability model of X and Y can depend on St.

The switch variable can take its values from a set St∈{1,2,…,M}. This changes the joint distribution (Xt,Yt) which is a separate multivariate Gaussian distribution in case of each value of St.

General case

In more generalised variants,[1] the switch variable affects the dynamics of Xt, e.g. through Pr⁡(Xt∣Xt−1,St).[7][6] The filtering and smoothing procedure for general cases is discussed in [1].

References

  1. ↑ 1.0 1.1 1.2 1.3 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.
  3. ↑ Haykin, Simon, ed. Kalman Filtering and Neural Networks. doi:10.1002/0471221546. ISBN 0-471-22154-6. 
  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". IEEE Transactions on Biomedical Engineering 51 (6): 933-942. doi:10.1109/TBME.2004.826666. https://sites.stat.columbia.edu/liam/teaching/neurostat-spr12/papers/EM/switching-kalman-prosthetic.pdf. 
  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. PMID 29736420. 
  6. ↑ 6.0 6.1 Kim, C.-J. (1994). "Dynamic linear models with Markov-switching". J. Econometrics 60: 1–22. doi:10.1016/0304-4076(94)90036-1. https://www.researchgate.net/publication/223036114_Dynamic_Linear_Model_with_Markov_Switching. 
  7. ↑ 7.0 7.1 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. doi:10.1109/LSP.2021.3116504. Bibcode: 2021ISPL...28.2008K. 
  9. ↑ 9.0 9.1 Ghahramani, Zoubin; Hinton, Geoffrey E.. "Variational Learning for Switching State-Space Models". Neural Computation 12 (4): 963–996. doi:10.1162/089976600300015619. PMID 10770834. https://www.cs.toronto.edu/~fritz/absps/switch.pdf. 



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