LTISDEModel¶

class probnum.filtsmooth.LTISDEModel(driftmatrix, force, dispmatrix, diffmatrix)[source]¶

Bases: probnum.filtsmooth.LinearSDEModel

Linear time-invariant continuous Markov models of the form dx = [F x(t) + u] dt + L dBt. In the language of dynamic models, x(t) : state process F : drift matrix u : forcing term L : dispersion matrix. Bt : Brownian motion with constant diffusion matrix Q.

Parameters:

driftmatrix (np.ndarray, shape=(n, n)) – This is F. It is the drift matrix of the SDE.
force (np.ndarray, shape=(n,)) – This is U. It is the force vector of the SDE.
dispmatrix (np.ndarray, shape(n, s)) – This is L. It is the dispersion matrix of the SDE.
diffmatrix (np.ndarray, shape=(s, s)) – This is the diffusion matrix Q of the Brownian motion driving the SDE.

Notes

It assumes Gaussian initial conditions (otherwise it is no Gauss-Markov process).

Attributes Summary

`diffusionmatrix`	Evaluates Q.
`dimension`	Spatial dimension (utility attribute).
`dispersionmatrix`
`driftmatrix`
`force`

Methods Summary

`__call__`(arr_or_rv, RandomVariable], start, …)	Transition a random variable or a realization of one.
`dispersion`(time, state, **kwargs)	Evaluates l(t, x(t)) = L(t).
`drift`(time, state, **kwargs)	Evaluates f(t, x(t)) = F(t) x(t) + u(t).
`jacobian`(time, state, **kwargs)	maps t -> F(t)
`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

diffusionmatrix¶: Evaluates Q.

dimension¶: Spatial dimension (utility attribute).

dispersionmatrix¶

driftmatrix¶

force¶

Methods Documentation

__call__(arr_or_rv: Union[numpy.ndarray, RandomVariable], start: float = None, stop: float = None, **kwargs) -> ('RandomVariable', typing.Dict)¶

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().

dispersion(time, state, **kwargs)¶: Evaluates l(t, x(t)) = L(t).

drift(time, state, **kwargs)¶: Evaluates f(t, x(t)) = F(t) x(t) + u(t).

jacobian(time, state, **kwargs)¶: maps t -> F(t)

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 \(t + \Delta t\).

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.