# car_tracking¶

probnum.problems.zoo.filtsmooth.car_tracking(rng, measurement_variance=0.5, process_diffusion=1.0, num_prior_derivatives=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
• rng (Generator) – Random number generator.

• measurement_variance (FloatLike) – Marginal measurement variance.

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

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

• timespan (Tuple[<sphinx.util.inspect.TypeAliasForwardRef object at 0x7f92c4222c10>, <sphinx.util.inspect.TypeAliasForwardRef object at 0x7f92c4222940>]) – $$t_0$$ and $$t_{\max}$$ of the time grid.

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

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

• forward_implementation (str) – Implementation of the forward transitions inside prior and measurement model. Optional. Default is classic. For improved numerical stability, use sqrt.

• backward_implementation (str) – Implementation of the backward transitions inside prior and measurement model. Optional. Default is classic. For improved numerical stability, use sqrt.

Returns

• regression_problemTimeSeriesRegressionProblem object with time points and noisy observations.

• info – Dictionary containing additional information like the prior process.

References

1

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