LTISDE¶
-
class
probnum.filtsmooth.
LTISDE
(driftmatrix, forcevec, dispmatrix)[source]¶ Bases:
probnum.filtsmooth.statespace.LinearSDE
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.
forcevec (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.
Notes
It assumes Gaussian initial conditions (otherwise it is no Gauss-Markov process).
Attributes Summary
Spatial dimension (utility attribute).
Methods Summary
__call__
(arr_or_rv[, start, stop])Transition a random variable or a realization of one.
discretise
(step)Returns a discrete transition model (i.e.
drift
(time, state, **kwargs)jacobian
(time, state, **kwargs)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
¶ Spatial dimension (utility attribute).
-
dispersionmatrix
¶
-
driftmatrix
¶
-
forcevec
¶
Methods Documentation
-
__call__
(arr_or_rv, start=None, stop=None, **kwargs)¶ 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()
ortransition_rv()
.- Return type
(‘RandomVariable’, typing.Dict)
-
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.
-
drift
(time, state, **kwargs)¶
-
jacobian
(time, state, **kwargs)¶
-
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.
See also
transition_rv()
Apply transition to a random variable.
-
transition_rv
(rv, start, stop, **kwargs)[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 \(t + \Delta t\).
- 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.
See also
transition_realization()
Apply transition to a realization of a random variable.