"""Gaussian filtering and smoothing."""
from typing import Iterable, Optional, Union
import numpy as np
from probnum import problems, statespace
from .._bayesfiltsmooth import BayesFiltSmooth
from .._timeseriesposterior import TimeSeriesPosterior
from ._extendedkalman import DiscreteEKFComponent
from ._kalmanposterior import FilteringPosterior, SmoothingPosterior
from ._stoppingcriterion import StoppingCriterion
from ._unscentedkalman import DiscreteUKFComponent
# Measurement models for a Kalman filter can be all sorts of things:
KalmanSingleMeasurementModelType = Union[
statespace.DiscreteLinearGaussian,
DiscreteEKFComponent,
DiscreteUKFComponent,
]
KalmanMeasurementModelArgType = Union[
KalmanSingleMeasurementModelType, Iterable[KalmanSingleMeasurementModelType]
]
[docs]class Kalman(BayesFiltSmooth):
"""Gaussian filtering and smoothing, i.e. Kalman-like filters and smoothers.
Parameters
----------
prior_process
Prior Gauss-Markov process. Usually a :class:`MarkovProcess` with a :class:`Normal` initial random variable,
and an LTISDE transition or an Integrator transition, but LinearSDE, ContinuousEKFComponent,
or ContinuousUKFComponent are also valid. Describes a random process in :math:`K` dimensions.
If the transition is an integrator, `K=spatialdim*(ordint+1)` for some spatialdim and ordint.
"""
[docs] def iterated_filtsmooth(
self,
regression_problem: problems.TimeSeriesRegressionProblem,
init_posterior: Optional[SmoothingPosterior] = None,
stopcrit: Optional[StoppingCriterion] = None,
):
"""Compute an iterated smoothing estimate with repeated posterior linearisation.
If the extended Kalman filter is used, this yields the IEKS. In
any case, the result is an approximation to the maximum-a-
posteriori estimate.
Parameters
----------
regression_problem :
Regression problem.
init_posterior
Initial posterior to linearize at. If not specified, linearizes
at the prediction random variable.
stopcrit: StoppingCriterion
A stopping criterion for iterated filtering.
Returns
-------
SmoothingPosterior
See Also
--------
TimeSeriesRegressionProblem: a regression problem data class
"""
smoothing_post = init_posterior
info_dicts = None
for smoothing_post, info_dicts in self.iterated_filtsmooth_posterior_generator(
regression_problem, init_posterior, stopcrit
):
pass
return smoothing_post, info_dicts
[docs] def iterated_filtsmooth_posterior_generator(
self,
regression_problem: problems.TimeSeriesRegressionProblem,
init_posterior: Optional[SmoothingPosterior] = None,
stopcrit: Optional[StoppingCriterion] = None,
):
"""Compute iterated smoothing estimates with repeated posterior linearisation.
If the extended Kalman filter is used, this yields the IEKS. In
any case, the result is an approximation to the maximum-a-
posteriori estimate.
Parameters
----------
regression_problem :
Regression problem.
init_posterior
Initial posterior to linearize at. Defaults to computing a (non-iterated)
smoothing posterior, which amounts to linearizing at the prediction
random variable.
stopcrit: StoppingCriterion
A stopping criterion for iterated filtering.
Yields
------
SmoothingPosterior
info_dicts
list of dictionaries containing filtering information
See Also
--------
TimeSeriesRegressionProblem: a regression problem data class
"""
if stopcrit is None:
stopcrit = StoppingCriterion()
if init_posterior is None:
# Initialise iterated smoother
new_posterior, info_dicts = self.filtsmooth(
regression_problem,
_previous_posterior=None,
)
else:
new_posterior = init_posterior
info_dicts = []
yield new_posterior, info_dicts
new_mean = new_posterior.states.mean
old_mean = np.inf * np.ones(new_mean.shape)
while not stopcrit.terminate(error=new_mean - old_mean, reference=new_mean):
old_posterior = new_posterior
new_posterior, info_dicts = self.filtsmooth(
regression_problem,
_previous_posterior=old_posterior,
)
yield new_posterior, info_dicts
new_mean = new_posterior.states.mean
old_mean = old_posterior.states.mean
[docs] def filtsmooth(
self,
regression_problem: problems.TimeSeriesRegressionProblem,
_previous_posterior: Optional[TimeSeriesPosterior] = None,
):
"""Apply Gaussian filtering and smoothing to a data set.
Parameters
----------
regression_problem :
Regression problem.
_previous_posterior: KalmanPosterior
If specified, approximate Gaussian filtering and smoothing linearises at this, prescribed posterior.
This is used for iterated filtering and smoothing. For standard filtering, this can be ignored.
Returns
-------
KalmanPosterior
Posterior distribution of the filtered output
info_dicts
list of dictionaries containing filtering information
See Also
--------
TimeSeriesRegressionProblem: a regression problem data class
"""
filter_result = self.filter(
regression_problem,
_previous_posterior=_previous_posterior,
)
filter_posterior, info_dicts = filter_result
smooth_posterior = self.smooth(filter_posterior)
return smooth_posterior, info_dicts
[docs] def filter(
self,
regression_problem: problems.TimeSeriesRegressionProblem,
_previous_posterior: Optional[TimeSeriesPosterior] = None,
):
"""Apply Gaussian filtering (no smoothing!) to a data set.
Parameters
----------
regression_problem :
Regression problem.
_previous_posterior: KalmanPosterior
If specified, approximate Gaussian filtering and smoothing linearises at this, prescribed posterior.
This is used for iterated filtering and smoothing. For standard filtering, this can be ignored.
Returns
-------
KalmanPosterior
Posterior distribution of the filtered output
info_dicts
list of dictionaries containing filtering information
See Also
--------
TimeSeriesRegressionProblem: a regression problem data class
"""
posterior = FilteringPosterior(transition=self.prior_process.transition)
info_dicts = []
for t, rv, info in self.filtered_states_generator(
regression_problem, _previous_posterior
):
posterior.append(location=t, state=rv)
info_dicts.append(info)
return posterior, info_dicts
[docs] def filtered_states_generator(
self,
regression_problem: problems.TimeSeriesRegressionProblem,
_previous_posterior: Optional[TimeSeriesPosterior] = None,
):
"""Apply Gaussian filtering (no smoothing!) to a data set.
Parameters
----------
regression_problem :
Regression problem.
_previous_posterior: KalmanPosterior
If specified, approximate Gaussian filtering and smoothing linearises at this, prescribed posterior.
This is used for iterated filtering and smoothing. For standard filtering, this can be ignored.
Yields
------
filtrv
Random variable returned from prediction and update of the Kalman filter.
info_dict
Dictionary containing filtering information
See Also
--------
TimeSeriesRegressionProblem: a regression problem data class
"""
# It is not clear at the moment how to implement this cleanly.
if not np.all(np.diff(regression_problem.locations) > 0):
raise ValueError(
"Gaussian filtering expects sorted, non-repeating time points."
)
# Initialise
t_old = self.prior_process.initarg
curr_rv = self.prior_process.initrv
# Iterate over data and measurement models
for t, data, measmod in regression_problem:
dt = t - t_old
info_dict = {}
# Predict if there is a time-increment
if dt > 0:
linearise_predict_at = (
None if _previous_posterior is None else _previous_posterior(t_old)
)
output = self.prior_process.transition.forward_rv(
curr_rv, t, dt=dt, _linearise_at=linearise_predict_at
)
curr_rv, info_dict["predict_info"] = output
# Update (even if there is no increment)
linearise_update_at = (
None if _previous_posterior is None else _previous_posterior(t)
)
curr_rv, info_dict["update_info"] = measmod.backward_realization(
realization_obtained=data, rv=curr_rv, _linearise_at=linearise_update_at
)
yield t, curr_rv, info_dict
t_old = t
[docs] def smooth(self, filter_posterior, _previous_posterior=None):
"""Apply Gaussian smoothing to the filtering outcome (i.e. a KalmanPosterior).
Parameters
----------
filter_posterior : KalmanPosterior
Posterior distribution obtained after filtering
Returns
-------
KalmanPosterior
Posterior distribution of the smoothed output
"""
diffusion_list = np.ones_like(filter_posterior.locations[1:])
rv_list = self.prior_process.transition.smooth_list(
filter_posterior.states,
filter_posterior.locations,
_diffusion_list=diffusion_list,
)
return SmoothingPosterior(
filtering_posterior=filter_posterior,
transition=self.prior_process.transition,
locations=filter_posterior.locations,
states=rv_list,
)