Transition¶

class
probnum.filtsmooth.
Transition
¶ Bases:
abc.ABC
Markov transition rules in discrete or continuous time.
In continuous time, this is a Markov process and described by a stochastic differential equation (SDE)
\[d x_t = f(t, x_t) d t + d w_t\]driven by a Wiener process \(w\). In discrete time, it is defined by a transformation
\[x_{t + \Delta t} \sim p(x_{t + \Delta t}  x_t).\]Sometimes, these can be equivalent. For example: mild solutions to linear, timeinvariant SDEs have an equivalent, discretised form that can be written as a transformation.
See also
ContinuousModel
 Continuously indexed transitions (SDEs)
DiscreteModel
 Discretely indexed transitions (transformations)
Attributes Summary
dimension
Dimension of the transition model. Methods Summary
__call__
(arr_or_rv, RandomVariable], start, …)Transition a random variable or a realization of one. transition_realization
(real, start, stop, …)Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\). transition_rv
(rv, start, stop, **kwargs)Transition a random variable from time \(t\) to time \(t+\Delta t\). Attributes Documentation

dimension
¶ Dimension of the transition model.
Methods Documentation

__call__
(arr_or_rv: Union[numpy.ndarray, RandomVariable], start: float = None, stop: float = None, **kwargs) > ('RandomVariable', typing.Dict)[source]¶ Transition a random variable or a realization of one.
The input is either interpreted as a random variable or as a realization. Accordingly, the respective methods are called:
transition_realization()
ortransition_rv()
.

transition_realization
(real: numpy.ndarray, start: float, stop: float, **kwargs) > ('RandomVariable', typing.Dict)[source]¶ Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\).
For random variable \(x_t\), it returns the random variable defined by
\[x_{t + \Delta t} \sim p(x_{t + \Delta t}  x_t = r) .\]This is different to
transition_rv()
which computes the parametrization of \(x_{t + \Delta t}\) based on the parametrization of \(x_t\).Nb: Think of transition as a verb, i.e. this method “transitions” a realization of a random variable.
Parameters:  real – Realization of the random variable.
 start – Starting point \(t\).
 stop – End point \(t + \Delta t\).
Returns:  RandomVariable – Random variable, describing the state at time \(t + \Delta t\) based on realization at time \(t\).
 dict – Additional information in form of a dictionary, for instance the crosscovariance in the prediction step, access to which is useful in smoothing.
See also
transition_rv()
 Apply transition to a random variable.

transition_rv
(rv: probnum.random_variables.RandomVariable, start: float, stop: float, **kwargs) > ('RandomVariable', typing.Dict)[source]¶ Transition a random variable from time \(t\) to time \(t+\Delta t\).
For random variable \(x_t\), it returns the random variable defined by
\[x_{t + \Delta t} \sim p(x_{t + \Delta t}  x_t) .\]This returns a random variable where the parametrization depends on the paramtrization of \(x_t\). This is different to
transition_rv()
which computes the parametrization of \(x_{t + \Delta t}\) based on a realization of \(x_t\).Nb: Think of transition as a verb, i.e. this method “transitions” a random variable.
Parameters:  rv – Realization of the random variable.
 start – Starting point \(t\).
 stop – End point \(t + \Delta t\).
Returns:  RandomVariable – Random variable, describing the state at time \(t + \Delta t\) based on realization at time \(t\).
 dict – Additional information in form of a dictionary, for instance the crosscovariance in the prediction step, access to which is useful in smoothing.
See also
transition_realization()
 Apply transition to a realization of a random variable.