LinearSDE¶
-
class
probnum.filtsmooth.
LinearSDE
(driftmatrixfun, forcevecfun, dispmatrixfun)¶ Bases:
probnum.filtsmooth.statespace.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
driftmatrixfun (callable, signature=(t, **kwargs)) – This is F = F(t). The evaluations of this function are called the drift(matrix) of the SDE. Returns np.ndarray with shape=(n, n)
forcevecfun (callable, signature=(t, **kwargs)) – This is u = u(t). Evaluations of this function are called the force(vector) of the SDE. Returns np.ndarray with shape=(n,)
dispmatrixfun (callable, signature=(t, **kwargs)) – This is L = L(t). Evaluations of this function are called the dispersion(matrix) of the SDE. Returns np.ndarray with shape=(n, s)
Notes
If initial conditions are Gaussian, the solution is a 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.
dispersionmatrix
(time, **kwargs)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, step, **kwargs)Transition a random variable from time \(t\) to time \(t+\Delta t\).
Attributes Documentation
-
dimension
¶ Spatial dimension (utility attribute).
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)
-
dispersionmatrix
(time, **kwargs)¶
-
drift
(time, state, **kwargs)¶
-
jacobian
(time, state, **kwargs)¶
-
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 \(t + \Delta t\).
- 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, step, **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.