Transition¶

class probnum.filtsmooth.statespace.Transition[source]¶

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.

See also

ContinuousModel: Continuously indexed transitions (SDEs)
DiscreteModel: Discretely indexed transitions (transformations)

Attributes Summary

dimension

Dimension of the transition model.

Methods Summary

`transition_realization`(real, start[, stop, …])	Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\).
`transition_realization_preconditioned`(real, …)	Applies the transition, assuming that the state is already preconditioned.
`transition_rv`(rv, start[, stop, step, …])	Transition a random variable from time \(t\) to time \(t+\Delta t\).
`transition_rv_preconditioned`(rv, start[, …])	Applies the transition, assuming that the state is already preconditioned.

Attributes Documentation

dimension¶

Dimension of the transition model.

Not all transition models have a unique dimension. Some turn a state (x, y) into a scalar z and it is not clear whether the dimension should be 2 or 1.

Return type: int

Methods Documentation

abstract transition_realization(real, start, stop=None, step=None, linearise_at=None)[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 (ndarray) – Realization of the random variable.
start (float) – Starting point \(t\).
stop (Optional[float]) – End point \(t + \Delta t\).
step (Optional[float]) – Intermediate step-size. Optional, default is None.
linearise_at (Optional[RandomVariable]) – For approximate transitions , for instance ContinuousEKFComponent, this argument overloads the state at which the Jacobian is computed.

Return type

(probnum.random_variables.RandomVariable, typing.Dict)

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 cross-covariance in the prediction step, access to which is useful in smoothing.