DiscreteGaussian¶

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

Bases: probnum.filtsmooth.statespace.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) – Dynamics function \(g=g(t, x)\). Signature: dynafct(t, x).
diffmatfun (callable) – Diffusion matrix function \(S=S(t)\). Signature: diffmatfct(t).
jacobfun (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[, start, stop])	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.

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

__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() or transition_rv().

Return type: (‘RandomVariable’, typing.Dict)

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, **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.