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 (
Union
[float
,Real
,floating
]) – Marginal measurement variance.process_diffusion (
Union
[float
,Real
,floating
]) – Diffusion constant for the dynamics.num_prior_derivatives (
Union
[int
,Integral
,integer
]) – Order of integration for the dynamics model. Defaults to one, which corresponds to a Wiener velocity model.timespan (
Tuple
[Union
[float
,Real
,floating
],Union
[float
,Real
,floating
]]) – \(t_0\) and \(t_{\max}\) of the time grid.step (
Union
[float
,Real
,floating
]) – 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_problem –
TimeSeriesRegressionProblem
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.