DiscreteGaussianModel

class probnum.filtsmooth.DiscreteGaussianModel(dynafct, diffmatfct, jacfct=None)

Bases: probnum.filtsmooth.DiscreteModel

Discrete Gaussian transition models of the form

\[x_{i+1} \sim \mathcal{N}(g(t_i, x_i), S(t_i))\]

for some (potentially non-linear) dynamics \(g\) and diffusion matrix \(S\).

Parameters:
  • dynafct (callable) – Dynamics function \(g=g(t, x)\). Signature: dynafct(t, x).
  • diffmatfct (callable) – Diffusion matrix function \(S=S(t)\). Signature: diffmatfct(t).
  • jacfct (callable, optional.) – Jacobian of the dynamics function \(g\), \(Jg=Jg(t, x)\). Signature: jacfct(t, x).

See also

DiscreteModel, DiscreteGaussianLinearModel

Attributes Summary

dimension Dimension of the transition model.

Methods Summary

__call__(arr_or_rv, RandomVariable], start, …) Transition a random variable or a realization of one.
diffusionmatrix(time, **kwargs) Compute diffusion matrix \(S=S(t)\) at time \(t\).
dynamics(time, state, **kwargs) Compute dynamics \(g=g(t, x)\) at time \(t\) and state \(x\).
jacobian(time, state, **kwargs) Compute diffusion matrix \(S=S(t)\) at time \(t\).
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]) Transition a random variable from time \(t\) to time \(t+\Delta t\).

Attributes Documentation

dimension

Dimension of the transition model.

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().

diffusionmatrix(time, **kwargs)[source]

Compute diffusion matrix \(S=S(t)\) at time \(t\).

Parameters:time (float) – Time \(t\).
Returns:Diffusion matrix \(S=S(t)\).
Return type:np.ndarray
dynamics(time, state, **kwargs)[source]

Compute dynamics \(g=g(t, x)\) at time \(t\) and state \(x\).

Parameters:
  • time (float) – Time \(t\).
  • state (array_like) – State \(x\). For instance, realization of a random variable.
Returns:

Evaluation of \(g=g(t, x)\).
Return type:

np.ndarray
jacobian(time, state, **kwargs)[source]

Compute diffusion matrix \(S=S(t)\) at time \(t\).

Parameters:
  • time (float) – Time \(t\).
  • state (array_like) – State \(x\). For instance, realization of a random variable.
Raises:

NotImplementedError – If the Jacobian is not implemented. This is the case if jacfct() is not specified at initialization.
Returns:

Evaluation of the Jacobian \(J g=Jg(t, x)\).
Return type:

np.ndarray
transition_realization(real, start, stop=None)[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=None, **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.