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.]

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

The hidden variables include not only the continuous ${\displaystyle X}$, but also a discrete *switch* (or switching) variable ${\displaystyle S_{t}}$. The dynamics of the switch variable are defined by the term ${\displaystyle \Pr(S_{t}\mid S_{t-1})}$. The probability model of ${\displaystyle X}$ and ${\displaystyle Y}$ can depend on ${\displaystyle S_{t}}$.

The switch variable can take its values from a set ${\displaystyle S_{t}\in \{1,2,\ldots ,M\}}$. This changes the joint distribution ${\displaystyle (X_{t},Y_{t})}$ which is a separate multivariate Gaussian distribution in case of each value of ${\displaystyle S_{t}}$.

General case

In more generalised variants,^[1] the switch variable affects the dynamics of ${\displaystyle X_{t}}$, e.g. through ${\displaystyle \Pr(X_{t}\mid X_{t-1},S_{t})}$.^[8][7] The filtering and smoothing procedure for general cases is discussed in.^[1]