LinearSDE¶

class probnum.filtsmooth.statespace.LinearSDE(driftmatfun, forcevecfun, dispmatfun)[source]¶

Bases: probnum.filtsmooth.statespace.sde.SDE

Linear stochastic differential equation (SDE),

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

For Gaussian initial conditions, this solution is a Gaussian process.

Parameters

driftmatfun (Callable[[Real], ndarray]) – This is G = G(t). The evaluations of this function are called the driftmatrix of the SDE. Returns np.ndarray with shape=(n, n)
forcevecfun (Callable[[Real], ndarray]) – This is v = v(t). Evaluations of this function are called the force(vector) of the SDE. Returns np.ndarray with shape=(n,)
dispmatfun (Callable[[Real], ndarray]) – This is L = L(t). Evaluations of this function are called the dispersion(matrix) of the SDE. Returns np.ndarray with shape=(n, s)

Attributes Summary

Spatial dimension (utility attribute).

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, **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

Methods Documentation

transition_realization(real, start, stop, step, **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.