IBM¶
-
class
probnum.statespace.
IBM
(ordint, spatialdim, forward_implementation='classic', backward_implementation='classic')[source]¶ Bases:
probnum.statespace.integrator.Integrator
,probnum.statespace.sde.LTISDE
Integrated Brownian motion in \(d\) dimensions.
Attributes Summary
Discretised IN THE PRECONDITIONED SPACE.
Methods Summary
backward_realization
(realization_obtained, rv)Backward-pass of a realisation of a state, according to the transition.
backward_rv
(rv_obtained, rv[, rv_forwarded, …])Backward-pass of a state, according to the transition.
discretise
(dt)Equivalent discretisation of the process.
forward_realization
(realization, t[, dt, …])Forward-pass of a realization of a state, according to the transition.
forward_rv
(rv, t[, dt, compute_gain, _diffusion])Forward-pass of a state, according to the transition.
jointly_sample_list_backward
(rv_list, locations)Jointly sample from a list of random variables, according to the present transition.
proj2coord
(coord)Projection matrix to \(i\) th coordinates.
smooth_list
(rv_list, locations[, …])Apply smoothing to a list of random variables, according to the present transition.
Attributes Documentation
-
equivalent_discretisation_preconditioned
¶ Discretised IN THE PRECONDITIONED SPACE.
Methods Documentation
-
backward_realization
(realization_obtained, rv, rv_forwarded=None, gain=None, t=None, dt=None, _diffusion=1.0, **kwargs)¶ Backward-pass of a realisation of a state, according to the transition. In other words, return a description of
\[p(x(t) \,|\, {\mathcal{G}_t(x(t)) = \xi})\]for an observed realization \(\xi\) of \({\mathcal{G}_t}(x(t))\). For example, this function is called in a Kalman update step.
- Parameters
realization_obtained – Observed realization \(\xi\) as an array.
rv – “Current” distribution \(p(x(t))\) as a RandomVariable.
rv_forwarded – “Forwarded” distribution (think: \(p(\mathcal{G}_t(x(t)) \,|\, x(t))\)) as a RandomVariable. Optional. If provided (in conjunction with gain), computation might be more efficient, because most backward passes require the solution of a forward pass. If rv_forwarded is not provided,
forward_rv()
might be called internally (depending on the object) which is skipped if rv_forwarded has been providedgain – Expected gain. Optional. If provided (in conjunction with rv_forwarded), some additional computations may be avoided (depending on the object).
t – Current time point.
dt – Increment \(\Delta t\). Ignored for discrete-time transitions.
_diffusion – Special diffusion of the driving stochastic process, which is used internally.
_linearise_at – Specific point of linearisation for approximate forward passes (think: extended Kalman filtering). Used internally for iterated filtering and smoothing.
- Returns
RandomVariable – New state, after applying the backward-pass.
Dict – Information about the backward-pass.
-
backward_rv
(rv_obtained, rv, rv_forwarded=None, gain=None, t=None, dt=None, _diffusion=1.0, **kwargs)[source]¶ Backward-pass of a state, according to the transition. In other words, return a description of
\[p(x(t) \,|\, z_{\mathcal{G}_t}) = \int p(x(t) \,|\, z_{\mathcal{G}_t}, \mathcal{G}_t(x(t))) p(\mathcal{G}_t(x(t)) \,|\, z_{\mathcal{G}_t})) d \mathcal{G}_t(x(t)),\]for observations \(z_{\mathcal{G}_t}\) of \({\mathcal{G}_t}(x(t))\). For example, this function is called in a Rauch-Tung-Striebel smoothing step, which computes a Gaussian distribution
\[p(x(t) \,|\, z_{\leq t+\Delta t}) = \int p(x(t) \,|\, z_{\leq t+\Delta t}, x(t+\Delta t)) p(x(t+\Delta t) \,|\, z_{\leq t+\Delta t})) d x(t+\Delta t),\]from filtering distribution \(p(x(t) \,|\, z_{\leq t})\) and smoothing distribution \(p(x(t+\Delta t) \,|\, z_{\leq t+\Delta t})\), where \(z_{\leq t + \Delta t}\) contains both \(z_{\leq t}\) and \(z_{t + \Delta t}\).
- Parameters
rv_obtained – “Incoming” distribution (think: \(p(x(t+\Delta t) \,|\, z_{\leq t+\Delta t})\)) as a RandomVariable.
rv – “Current” distribution (think: \(p(x(t) \,|\, z_{\leq t})\)) as a RandomVariable.
rv_forwarded – “Forwarded” distribution (think: \(p(x(t+\Delta t) \,|\, z_{\leq t})\)) as a RandomVariable. Optional. If provided (in conjunction with gain), computation might be more efficient, because most backward passes require the solution of a forward pass. If rv_forwarded is not provided,
forward_rv()
might be called internally (depending on the object) which is skipped if rv_forwarded has been providedgain – Expected gain from “observing states at time \(t+\Delta t\) from time \(t\)). Optional. If provided (in conjunction with rv_forwarded), some additional computations may be avoided (depending on the object).
t – Current time point.
dt – Increment \(\Delta t\). Ignored for discrete-time transitions.
_diffusion – Special diffusion of the driving stochastic process, which is used internally.
_linearise_at – Specific point of linearisation for approximate forward passes (think: extended Kalman filtering). Used internally for iterated filtering and smoothing.
- Returns
RandomVariable – New state, after applying the backward-pass.
Dict – Information about the backward-pass.
-
discretise
(dt)[source]¶ Equivalent discretisation of the process.
Overwrites matrix-fraction decomposition in the super-class. Only present for user’s convenience and to maintain a clean interface. Not used for forward_rv, etc..
-
forward_realization
(realization, t, dt=None, compute_gain=False, _diffusion=1.0, **kwargs)¶ Forward-pass of a realization of a state, according to the transition. In other words, return a description of
\[p(\mathcal{G}_t[x(t)] \,|\, x(t)=\xi),\]for some realization \(\xi\).
- Parameters
realization – Realization \(\xi\) of the random variable \(x(t)\) that describes the current state.
t – Current time point.
dt – Increment \(\Delta t\). Ignored for discrete-time transitions.
compute_gain – Flag that indicates whether the expected gain of the forward transition shall be computed. This is important if the forward-pass is computed as part of a forward-backward pass, as it is for instance the case in a Kalman update.
_diffusion – Special diffusion of the driving stochastic process, which is used internally.
_linearise_at – Specific point of linearisation for approximate forward passes (think: extended Kalman filtering). Used internally for iterated filtering and smoothing.
- Returns
RandomVariable – New state, after applying the forward-pass.
Dict – Information about the forward pass. Can for instance contain a gain key, if compute_gain was set to True (and if the transition supports this functionality).
-
forward_rv
(rv, t, dt=None, compute_gain=False, _diffusion=1.0, **kwargs)[source]¶ Forward-pass of a state, according to the transition. In other words, return a description of
\[p(\mathcal{G}_t[x(t)] \,|\, x(t)),\]or, if we take a message passing perspective,
\[p(\mathcal{G}_t[x(t)] \,|\, x(t), z_{\leq t}),\]for past observations \(z_{\leq t}\). (This perspective will be more interesting in light of
backward_rv()
).- Parameters
rv – Random variable that describes the current state.
t – Current time point.
dt – Increment \(\Delta t\). Ignored for discrete-time transitions.
compute_gain – Flag that indicates whether the expected gain of the forward transition shall be computed. This is important if the forward-pass is computed as part of a forward-backward pass, as it is for instance the case in a Kalman update.
_diffusion – Special diffusion of the driving stochastic process, which is used internally.
_linearise_at – Specific point of linearisation for approximate forward passes (think: extended Kalman filtering). Used internally for iterated filtering and smoothing.
- Returns
RandomVariable – New state, after applying the forward-pass.
Dict – Information about the forward pass. Can for instance contain a gain key, if compute_gain was set to True (and if the transition supports this functionality).
-
jointly_sample_list_backward
(rv_list, locations, _previous_posterior=None)¶ Jointly sample from a list of random variables, according to the present transition.
An explanation of the algorithm can be found in, for instance, this link 1.
References
- Parameters
rv_list (_RandomVariableList) – List of random variables to be sampled from (jointly).
locations – Locations \(t\) of the random variables in the time-domain. Used for continuous-time transitions.
_previous_posterior – Specify a previous posterior to improve linearisation in approximate backward passes. Used in iterated smoothing based on posterior linearisation.
- Returns
List of smoothed random variables.
- Return type
_RandomVariableList
-
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))
-
smooth_list
(rv_list, locations, _previous_posterior=None)¶ Apply smoothing to a list of random variables, according to the present transition.
- Parameters
rv_list (_RandomVariableList) – List of random variables to be smoothed.
locations – Locations \(t\) of the random variables in the time-domain. Used for continuous-time transitions.
_previous_posterior – Specify a previous posterior to improve linearisation in approximate backward passes. Used in iterated smoothing based on posterior linearisation.
- Returns
List of smoothed random variables.
- Return type
_RandomVariableList
-