# 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, time-invariant SDEs have an equivalent, discretised form that can be written as a transformation.

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() or transition_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 $$t + \Delta t$$. 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 cross-covariance in the prediction step, access to which is useful in smoothing.

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 $$t + \Delta t$$. 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 cross-covariance in the prediction step, access to which is useful in smoothing.

transition_realization()