Transition¶
- class probnum.randprocs.markov.Transition(input_dim, output_dim)¶
Bases:
abc.ABC
Interface for Markov transitions in discrete and continuous time.
This framework describes transition probabilities
\[p(\mathcal{G}_t[x(t)] \,|\, x(t))\]for some operator \(\mathcal{G}: \mathbb{R}^d \rightarrow \mathbb{R}^m\), which are used to describe the evolution of Markov processes
\[p(x(t+\Delta t) \,|\, x(t))\]both in discrete time (Markov chains) and in continuous time (Markov processes). In continuous time, Markov processes are modelled as the solution of a stochastic differential equation (SDE)
\[d x(t) = f(t, x(t)) d t + d w(t)\]driven by a Wiener process \(w\). In discrete time, Markov chain are described by a transformation
\[x({t + \Delta t}) \,|\, x(t) \sim p(x({t + \Delta t}) \,|\, x(t)).\]Sometimes, these can be equivalent. For example, linear, time-invariant SDEs have a mild solution that can be written as a discrete transition. In ProbNum, we also use discrete-time transition objects to describe observation models,
\[z_k \,|\, x(t_k) \sim p(z_k \,|\, x(t_k))\]for some \(k=0,...,K\). All three building blocks are used heavily in filtering and smoothing, as well as solving ODEs.
See also
SDE
Markov-processes in continuous time.
NonlinearGaussian
Markov-chains and general discrete-time transitions (likelihoods).
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.
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.
Methods Documentation
- abstract backward_realization(realization_obtained, rv, rv_forwarded=None, gain=None, t=None, dt=None, _diffusion=1.0, _linearise_at=None)[source]¶
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.
- abstract backward_rv(rv_obtained, rv, rv_forwarded=None, gain=None, t=None, dt=None, _diffusion=1.0, _linearise_at=None)[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.
- abstract forward_realization(realization, t, dt=None, compute_gain=False, _diffusion=1.0, _linearise_at=None)[source]¶
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).
- abstract forward_rv(rv, t, dt=None, compute_gain=False, _diffusion=1.0, _linearise_at=None)[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)[source]¶
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)[source]¶
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
- smooth_list(rv_list, locations, _diffusion_list, _previous_posterior=None)[source]¶
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