IntegratedWienerTransition

class probnum.randprocs.markov.integrator.IntegratedWienerTransition(num_derivatives, wiener_process_dimension, forward_implementation='classic', backward_implementation='classic')

Bases: probnum.randprocs.markov.integrator.IntegratorTransition, probnum.randprocs.markov.continuous.LTISDE

Integrated Brownian motion in \(d\) dimensions.

Attributes Summary

equivalent_discretisation_preconditioned

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_transform_base_measure_realization_list_backward(…)

Transform samples from a base measure into joint backward samples from a list of random variables.

jointly_transform_base_measure_realization_list_forward(…)

Transform samples from a base measure into joint backward samples from a list of random variables.

proj2coord(coord)

Projection matrix to \(i\) th coordinates.

smooth_list(rv_list, locations, _diffusion_list)

Apply smoothing to a list of random variables, according to the present transition.

Attributes Documentation

equivalent_discretisation_preconditioned

Discretised IN THE PRECONDITIONED SPACE.

The preconditioned state transition is the flipped Pascal matrix. The preconditioned process noise covariance is the flipped Hilbert matrix. The shift is always zero.

Reference: https://arxiv.org/abs/2012.10106

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 provided

  • gain – 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 provided

  • gain – 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_transform_base_measure_realization_list_backward(base_measure_realizations, t, rv_list, _diffusion_list, _previous_posterior=None)

Transform samples from a base measure into joint backward samples from a list of random variables.

Parameters
  • base_measure_realizations (ndarray) – Base measure realizations (usually samples from a standard Normal distribution). These are transformed into joint realizations of the random variable list.

  • rv_list (_RandomVariableList) – List of random variables to be jointly sampled from.

  • t (Real) – Locations of the random variables in the list. Assumed to be sorted.

  • _diffusion_list (ndarray) – List of diffusions that correspond to the intervals in the locations. If locations=(t0, …, tN), then _diffusion_list=(d1, …, dN), i.e. it contains one element less.

  • _previous_posterior – Previous posterior. Used for iterative posterior linearisation.

Returns

Jointly transformed realizations.

Return type

np.ndarray

jointly_transform_base_measure_realization_list_forward(base_measure_realizations, t, initrv, _diffusion_list, _previous_posterior=None)

Transform samples from a base measure into joint backward samples from a list of random variables.

Parameters
  • base_measure_realizations (ndarray) – Base measure realizations (usually samples from a standard Normal distribution). These are transformed into joint realizations of the random variable list.

  • initrv (RandomVariable) – Initial random variable.

  • t (Real) – Locations of the random variables in the list. Assumed to be sorted.

  • _diffusion_list (ndarray) – List of diffusions that correspond to the intervals in the locations. If locations=(t0, …, tN), then _diffusion_list=(d1, …, dN), i.e. it contains one element less.

  • _previous_posterior – Previous posterior. Used for iterative posterior linearisation.

Returns

Jointly transformed realizations.

Return type

np.ndarray

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, _diffusion_list, _previous_posterior=None)

Apply smoothing to a list of random variables, according to the present transition.

Parameters
  • rv_list (_randomvariablelist._RandomVariableList) – List of random variables to be smoothed.

  • locations – Locations \(t\) of the random variables in the time-domain. Used for continuous-time transitions.

  • _diffusion_list – List of diffusions that correspond to the intervals in the locations. If locations=(t0, …, tN), then _diffusion_list=(d1, …, dN), i.e. it contains one element less.

  • _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._RandomVariableList