# LinearSDE¶

class probnum.randprocs.markov.continuous.LinearSDE(state_dimension, wiener_process_dimension, drift_matrix_function, force_vector_function, dispersion_matrix_function, mde_atol=1e-06, mde_rtol=1e-06, mde_solver='RK45', forward_implementation='classic')

Bases: SDE

Linear stochastic differential equation (SDE),

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

For Gaussian initial conditions, this solution is a Gaussian process.

Parameters
• drift_matrix_function (Callable[[Union[float, Real, floating]], ndarray]) – This is G = G(t). The evaluations of this function are called the drift matrix of the SDE. Returns np.ndarray with shape=(n, n)

• force_vector_function (Callable[[Union[float, Real, floating]], ndarray]) – This is v = v(t). Evaluations of this function are called the force(vector) of the SDE. Returns np.ndarray with shape=(n,)

• dispersion_matrix_function (Callable[[Union[float, Real, floating]], ndarray]) – This is L = L(t). Evaluations of this function are called the dispersion(matrix) of the SDE. Returns np.ndarray with shape=(n, s)

• mde_atol (Optional[Union[float, Real, floating]]) – Absolute tolerance passed to the solver of the moment differential equations (MDEs). Optional. Default is 1e-6.

• mde_rtol (Optional[Union[float, Real, floating]]) – Relative tolerance passed to the solver of the moment differential equations (MDEs). Optional. Default is 1e-6.

• mde_solver (Optional[str]) – Method that is chosen in scipy.integrate.solve_ivp. Any string that is compatible with solve_ivp(..., method=mde_solve,...) works here. Usual candidates are [RK45, LSODA, Radau, BDF, RK23, DOP853]. Optional. Default is LSODA.

• forward_implementation (Optional[str]) – Implementation style for forward transitions.

• state_dimension (Union[int, Integral, integer]) –

• wiener_process_dimension (Union[int, Integral, integer]) –

Attributes Summary

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. drift_function(t, x) drift_jacobian(t, x) forward_realization(realization, t[, dt, ...]) Forward-pass of a realization of a state, according to the transition. forward_rv(rv, t[, dt, _compute_gain, ...]) Forward-pass of a state, according to the transition. Transform samples from a base measure into joint backward samples from a list of random variables. 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

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)[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.

dispersion_function(t, x)
dispersion_matrix_function(t)[source]
drift_function(t, x)
drift_jacobian(t, x)
drift_matrix_function(t)[source]
force_vector_function(t)[source]
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 (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