car_tracking

probnum.problems.zoo.filtsmooth.car_tracking(measurement_variance=0.5, process_diffusion=1.0, model_ordint=1, timespan=(0.0, 20.0), step=0.2, initrv=None, forward_implementation='classic', backward_implementation='classic')

Filtering/smoothing setup for a simple car-tracking scenario.

A discrete, linear, time-invariant Gaussian state space model for car-tracking, based on Example 3.6 in Särkkä, 2013. 1 Let \(X = (\dot{x}_1, \dot{x}_2, \ddot{x}_1, \ddot{x}_2)\). Then the state space model has the following discretized formulation

\[\begin{split}X(t_{n}) &= \begin{pmatrix} 1 & 0 & \Delta t& 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} X(t_{n-1}) + q_n \\ y_{n} &= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix} X(t_{n}) + r_n\end{split}\]

where \(q_n \sim \mathcal{N}(0, Q)\) and \(r_n \sim \mathcal{N}(0, R)\) for process noise covariance matrix \(Q\) and measurement noise covariance matrix \(R\).

Parameters

• measurement_variance (Real) – Marginal measurement variance.

• process_diffusion (Real) – Diffusion constant for the dynamics.

• model_ordint (Integral) – Order of integration for the dynamics model. Defaults to one, which corresponds to a Wiener velocity model.

• timespan (Tuple[Real, Real]) – \(t_0\) and \(t_{\max}\) of the time grid.

• step (Real) – Step size of the time grid.

• initrv (Optional[RandomVariable]) – Initial random variable.

Returns

• regression_problem – RegressionProblem object with time points and noisy observations.

• statespace_components – Dictionary containing

  • dynamics model

  • measurement model

  • initial random variable

References

1

Särkkä, Simo. Bayesian Filtering and Smoothing. Cambridge University Press, 2013.