DiscreteGaussian

class probnum.filtsmooth.statespace.DiscreteGaussian(dynamicsfun, diffmatfun, jacobfun=None)

Bases: probnum.filtsmooth.statespace.transition.Transition

Random variable transitions with additive Gaussian noise.

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

for some (potentially non-linear) dynamics \(g\) and diffusion matrix \(S\). This is used for, but not restricted to time-series.

Parameters

• dynamicsfun (Callable[[Real, ndarray], ndarray]) – Dynamics function \(g=g(t, x)\). Signature: dynafct(t, x).

• diffmatfun (Callable[[Real], ndarray]) – Diffusion matrix function \(S=S(t)\). Signature: diffmatfct(t).

• jacobfun (Optional[Callable[[Real, ndarray], ndarray]]) – 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

transition_realization(real, start, **kwargs)	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, **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

dimension

Dimension of the transition model.

Not all transition models have a unique dimension. Some turn a state (x, y) into a scalar z and it is not clear whether the dimension should be 2 or 1.

Return type

int

Methods Documentation

transition_realization(real, start, **kwargs)

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.

See also

transition_rv()

Apply transition to a random variable.

transition_realization_preconditioned(real, start, stop=None, step=None, linearise_at=None)

Applies the transition, assuming that the state is already preconditioned.

This is useful for numerically stable implementation of Kalman smoothing steps and Kalman updates.

Return type

(probnum.random_variables.RandomVariable, typing.Dict)

transition_rv(rv, start, **kwargs)

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 \(t + \Delta t\).

• 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.

See also

transition_realization()

Apply transition to a realization of a random variable.

transition_rv_preconditioned(rv, start, stop=None, step=None, linearise_at=None)

Applies the transition, assuming that the state is already preconditioned.

This is useful for numerically stable implementation of Kalman smoothing steps and Kalman updates.

Return type

(probnum.random_variables.RandomVariable, typing.Dict)