LTISDE¶

class probnum.filtsmooth.statespace.LTISDE(driftmat, forcevec, dispmat)[source]¶

Bases: probnum.filtsmooth.statespace.sde.LinearSDE

Linear time-invariant continuous Markov models of the form.

\[d x(t) = [G x(t) + v] d t + L d w(t).\]

In the language of dynamic models, x(t) : state process G : drift matrix v : force term/vector L : dispersion matrix. w(t) : Wiener process with unit diffusion.

Parameters

driftmat (ndarray) – This is F. It is the drift matrix of the SDE.
forcevec (ndarray) – This is U. It is the force vector of the SDE.
dispmat (ndarray) – This is L. It is the dispersion matrix of the SDE.

Attributes Summary

dimension

Spatial dimension (utility attribute).

Methods Summary

`discretise`(step)	Returns a discrete transition model (i.e.
`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, **kwargs)	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¶: Spatial dimension (utility attribute).

Methods Documentation

discretise(step)[source]¶

Returns a discrete transition model (i.e. mild solution to SDE) using matrix fraction decomposition.

That is, matrices A(h) and Q(h) and vector s(h) such that the transition is

\[x | x_\text{old} \sim \mathcal{N}(A(h) x_\text{old} + s(h), Q(h)) ,\]

which is the transition of the mild solution to the LTI SDE.

transition_realization(real, start, stop, **kwargs)[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 \(t + \Delta t\).
step – Intermediate step-size. Optional, default is None.
linearise_at – For approximate transitions , for instance ContinuousEKFComponent, this argument overloads the state at which the Jacobian is computed.

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.