LTISDE

class probnum.randprocs.markov.continuous.LTISDE(drift_matrix, force_vector, dispersion_matrix, forward_implementation='classic', backward_implementation='classic')

Bases: LinearSDE

Linear time-invariant continuous Markov models of the form.

\[d x(t) = [G x(t) + v] d t + L d w(t).\]

In the language of dynamic models,

x(t) : state process G : drift matrix v : force term/vector L : dispersion matrix. w(t) : Wiener process with unit diffusion.

Parameters

• drift_matrix (ndarray) – This is F. It is the drift matrix of the SDE.

• force_vector (ndarray) – This is U. It is the force vector of the SDE.

• dispersion_matrix (ndarray) – This is L. It is the dispersion matrix of the SDE.

Attributes Summary

backward_implementation
dispersion_matrix
drift_matrix
force_vector
forward_implementation
mde_atol
mde_rtol
mde_solver
state_dimension
wiener_process_dimension

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)	Return a discrete transition model (i.e.
dispersion_function(t, x)
dispersion_matrix_function(t)
drift_function(t, x)
drift_jacobian(t, x)
drift_matrix_function(t)
force_vector_function(t)
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.
smooth_list(rv_list, locations, _diffusion_list)	Apply smoothing to a list of random variables, according to the present transition.

Attributes Documentation

backward_implementation

dispersion_matrix

drift_matrix

force_vector

forward_implementation

mde_atol

mde_rtol

mde_solver

state_dimension

wiener_process_dimension

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)

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)

Return 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.

dispersion_function(t, x)

dispersion_matrix_function(t)

drift_function(t, x)

drift_jacobian(t, x)

drift_matrix_function(t)

force_vector_function(t)

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)

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 (Union[float, Real, floating]) – 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 (Union[float, Real, floating]) – 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

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 (randvars._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

randvars._RandomVariableList