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][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.]
- [math]\displaystyle{ \begin{align} & \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{align} }[/math]
The hidden variables include not only the continuous [math]\displaystyle{ X }[/math], but also a discrete *switch* (or switching) variable [math]\displaystyle{ S_t }[/math]. The dynamics of the switch variable are defined by the term [math]\displaystyle{ \Pr(S_t \mid S_{t-1}) }[/math]. The probability model of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] can depend on [math]\displaystyle{ S_t }[/math].
The switch variable can take its values from a set [math]\displaystyle{ S_t\in\{1,2,\ldots,M\} }[/math]. This changes the joint distribution [math]\displaystyle{ (X_t,Y_t) }[/math] which is a separate multivariate Gaussian distribution in case of each value of [math]\displaystyle{ S_t }[/math].
General case
In more generalised variants,[1] the switch variable affects the dynamics of [math]\displaystyle{ X_t }[/math], e.g. through [math]\displaystyle{ \Pr(X_t\mid X_{t-1}, S_t) }[/math].[8][7] The filtering and smoothing procedure for general cases is discussed in.[1]
References
- ↑ 1.0 1.1 1.2 1.3 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).
- ↑ 7.0 7.1 Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1–22.
- ↑ 8.0 8.1 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. doi:10.1109/LSP.2021.3116504. Bibcode: 2021ISPL...28.2008K. https://ieeexplore.ieee.org/document/9552473.
- ↑ 10.0 10.1 Zoubin Ghahramani, Geoffrey E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996.
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