Matern¶
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class
probnum.filtsmooth.statespace.
Matern
(ordint, spatialdim, lengthscale, diffconst)[source]¶ Bases:
probnum.filtsmooth.statespace.integrator.Integrator
,probnum.filtsmooth.statespace.sde.LTISDE
Matern process in \(d\) dimensions.
Attributes Summary
Spatial dimension (utility attribute).
Methods Summary
discretise
(step)Returns a discrete transition model (i.e.
proj2coord
(coord)Projection matrix to \(i\) th coordinates.
transition_realization
(real, start, stop, …)Transition a realization of a random variable from time \(t\) to time \(t+\Delta t\).
Applies the transition, assuming that the state is already preconditioned.
transition_rv
(rv, start, stop, **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
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dimension
¶ Spatial dimension (utility attribute).
Methods Documentation
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discretise
(step)¶ 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.
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proj2coord
(coord)¶ Projection matrix to \(i\) th coordinates.
Computes the matrix
\[H_i = \left[ I_d \otimes e_i \right] P^{-1},\]where \(e_i\) is the \(i\) th unit vector, that projects to the \(i\) th coordinate of a vector. If the ODE is multidimensional, it projects to each of the \(i\) th coordinates of each ODE dimension.
- Parameters
coord (int) – Coordinate index \(i\) which to project to. Expected to be in range \(0 \leq i \leq q + 1\).
- Returns
Projection matrix \(H_i\).
- Return type
np.ndarray, shape=(d, d*(q+1))
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transition_realization
(real, start, stop, **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.
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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
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transition_rv
(rv, start, stop, **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.
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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
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