# 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')[source]

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
Returns

• regression_problemRegressionProblem 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.