This toolbox supports filtering, smoothing and parameter estimation (using EM) for Linear Dynamical Systems.
A Linear Dynamical System is a partially observed stochastic process with linear dynamics and linear observations, both subject to Gaussian noise. It can be defined as follows, where X(t) is the hidden state at time t, and Y(t) is the observation.
x(t+1) = F*x(t) + w(t), w ~ N(0, Q), x(0) ~ N(X(0), V(0))y(t) = H*x(t) + v(t), v ~ N(0, R)
The Kalman filter is an algorithm for performing filtering on this model, i.e., computing P(X(t) | Y(1), ..., Y(t)).
The Rauch-Tung-Striebel (RTS) algorithm performs fixed-interval offline smoothing, i.e., computing P(X(t) | Y(1), ..., Y(T)), for t <= T.
[ x1(t) ] = [1 0 1 0] [ x1(t-1) ] + [ wx1 ][ x2(t) ] [0 1 0 1] [ x2(t-1) ] [ wx2 ][ dx1(t) ] [0 0 1 0] [ dx1(t-1) ] [ wdx1 ][ dx2(t) ] [0 0 0 1] [ dx2(t-1) ] [ wdx2 ]We assume we only observe the position of the particle.
[ y1(t) ] = [1 0 0 0] [ x1(t) ] + [ vx1 ][ y2(t) ] [0 1 0 0] [ x2(t) ] [ vx2 ][ dx1(t) ][ dx2(t) ]Suppose we start out at position (10,10) moving to the right with velocity (1,0). We sampled a random trajectory of length 15. Below we show the filtered and smoothed trajectories.
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For systems with non-Gaussian noise, I recommend Particle filtering (PF), which is a popular sequential Monte Carlo technique. See also this discussion on pros/cons of particle filters. and the following tutorial: M. Arulampalam, S. Maskell, N. Gordon, T. Clapp, IEEE Transactions on Signal Processing, Volume 50, Number 2, February 2002, pp 174-189 (pdf cached here The EKF can be used as a proposal distribution for a PF. This method is better than either one alone. The Unscented Particle Filter, by R van der Merwe, A Doucet, JFG de Freitas and E Wan, May 2000. Matlab software for the UPF is also available. Gatsby reading group on nonlinear dynamical systemsOther packages for Kalman filtering and state-space models
Recommended reading
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