"""
State Space Representation and Kalman Filter, Smoother
Author: Chad Fulton
License: Simplified-BSD
"""
from __future__ import division, absolute_import, print_function
import numpy as np
from statsmodels.tsa.statespace.representation import OptionWrapper
from statsmodels.tsa.statespace.kalman_filter import (KalmanFilter,
FilterResults)
from statsmodels.tsa.statespace.tools import (
reorder_missing_matrix, reorder_missing_vector, copy_index_matrix)
from statsmodels.tsa.statespace import tools
SMOOTHER_STATE = 0x01 # Durbin and Koopman (2012), Chapter 4.4.2
SMOOTHER_STATE_COV = 0x02 # ibid., Chapter 4.4.3
SMOOTHER_DISTURBANCE = 0x04 # ibid., Chapter 4.5
SMOOTHER_DISTURBANCE_COV = 0x08 # ibid., Chapter 4.5
SMOOTHER_STATE_AUTOCOV = 0x10 # ibid., Chapter 4.7
SMOOTHER_ALL = (
SMOOTHER_STATE | SMOOTHER_STATE_COV | SMOOTHER_DISTURBANCE |
SMOOTHER_DISTURBANCE_COV | SMOOTHER_STATE_AUTOCOV
)
SMOOTH_CONVENTIONAL = 0x01
SMOOTH_CLASSICAL = 0x02
SMOOTH_ALTERNATIVE = 0x04
SMOOTH_UNIVARIATE = 0x08
[docs]class KalmanSmoother(KalmanFilter):
r"""
State space representation of a time series process, with Kalman filter
and smoother.
Parameters
----------
k_endog : array_like or integer
The observed time-series process :math:`y` if array like or the
number of variables in the process if an integer.
k_states : int
The dimension of the unobserved state process.
k_posdef : int, optional
The dimension of a guaranteed positive definite covariance matrix
describing the shocks in the measurement equation. Must be less than
or equal to `k_states`. Default is `k_states`.
results_class : class, optional
Default results class to use to save filtering output. Default is
`SmootherResults`. If specified, class must extend from
`SmootherResults`.
**kwargs
Keyword arguments may be used to provide default values for state space
matrices, for Kalman filtering options, or for Kalman smoothing
options. See `Representation` for more details.
"""
smoother_outputs = [
'smoother_state', 'smoother_state_cov', 'smoother_state_autocov',
'smoother_disturbance', 'smoother_disturbance_cov', 'smoother_all',
]
smoother_state = OptionWrapper('smoother_output', SMOOTHER_STATE)
smoother_state_cov = OptionWrapper('smoother_output', SMOOTHER_STATE_COV)
smoother_disturbance = (
OptionWrapper('smoother_output', SMOOTHER_DISTURBANCE)
)
smoother_disturbance_cov = (
OptionWrapper('smoother_output', SMOOTHER_DISTURBANCE_COV)
)
smoother_state_autocov = (
OptionWrapper('smoother_output', SMOOTHER_STATE_AUTOCOV)
)
smoother_all = OptionWrapper('smoother_output', SMOOTHER_ALL)
smooth_methods = [
'smooth_conventional', 'smooth_alternative', 'smooth_classical'
]
smooth_conventional = OptionWrapper('smooth_method', SMOOTH_CONVENTIONAL)
"""
(bool) Flag for conventional (Durbin and Koopman, 2012) Kalman smoothing.
"""
smooth_alternative = OptionWrapper('smooth_method', SMOOTH_ALTERNATIVE)
"""
(bool) Flag for alternative (modified Bryson-Frazier) smoothing.
"""
smooth_classical = OptionWrapper('smooth_method', SMOOTH_CLASSICAL)
"""
(bool) Flag for classical (see e.g. Anderson and Moore, 1979) smoothing.
"""
smooth_univariate = OptionWrapper('smooth_method', SMOOTH_UNIVARIATE)
"""
(bool) Flag for univariate smoothing (uses modified Bryson-Frazier timing).
"""
# Default smoother options
smoother_output = SMOOTHER_ALL
smooth_method = 0
def __init__(self, k_endog, k_states, k_posdef=None, results_class=None,
kalman_smoother_classes=None, **kwargs):
# Set the default results class
if results_class is None:
results_class = SmootherResults
super(KalmanSmoother, self).__init__(
k_endog, k_states, k_posdef, results_class=results_class, **kwargs
)
# Options
self.prefix_kalman_smoother_map = (
kalman_smoother_classes
if kalman_smoother_classes is not None
else tools.prefix_kalman_smoother_map.copy())
# Setup the underlying Kalman smoother storage
self._kalman_smoothers = {}
# Set the smoother options
self.set_smoother_output(**kwargs)
self.set_smooth_method(**kwargs)
@property
def _kalman_smoother(self):
prefix = self.prefix
if prefix in self._kalman_smoothers:
return self._kalman_smoothers[prefix]
return None
def _initialize_smoother(self, smoother_output=None, smooth_method=None,
prefix=None, **kwargs):
if smoother_output is None:
smoother_output = self.smoother_output
if smooth_method is None:
smooth_method = self.smooth_method
# Make sure we have the required Kalman filter
prefix, dtype, create_filter, create_statespace = (
self._initialize_filter(prefix, **kwargs)
)
# Determine if we need to (re-)create the smoother
# (definitely need to recreate if we recreated the filter)
create_smoother = (create_filter or
prefix not in self._kalman_smoothers)
if not create_smoother:
kalman_smoother = self._kalman_smoothers[prefix]
create_smoother = (kalman_smoother.kfilter is not
self._kalman_filters[prefix])
# If the dtype-specific _kalman_smoother does not exist (or if we
# need to re-create it), create it
if create_smoother:
# Setup the smoother
cls = self.prefix_kalman_smoother_map[prefix]
self._kalman_smoothers[prefix] = cls(
self._statespaces[prefix], self._kalman_filters[prefix],
smoother_output, smooth_method
)
# Otherwise, update the smoother parameters
else:
self._kalman_smoothers[prefix].set_smoother_output(
smoother_output, False)
self._kalman_smoothers[prefix].set_smooth_method(smooth_method)
return prefix, dtype, create_smoother, create_filter, create_statespace
[docs] def set_smoother_output(self, smoother_output=None, **kwargs):
"""
Set the smoother output
The smoother can produce several types of results. The smoother output
variable controls which are calculated and returned.
Parameters
----------
smoother_output : integer, optional
Bitmask value to set the smoother output to. See notes for details.
**kwargs
Keyword arguments may be used to influence the smoother output by
setting individual boolean flags. See notes for details.
Notes
-----
The smoother output is defined by a collection of boolean flags, and
is internally stored as a bitmask. The methods available are:
SMOOTHER_STATE = 0x01
Calculate and return the smoothed states.
SMOOTHER_STATE_COV = 0x02
Calculate and return the smoothed state covariance matrices.
SMOOTHER_STATE_AUTOCOV = 0x10
Calculate and return the smoothed state lag-one autocovariance
matrices.
SMOOTHER_DISTURBANCE = 0x04
Calculate and return the smoothed state and observation
disturbances.
SMOOTHER_DISTURBANCE_COV = 0x08
Calculate and return the covariance matrices for the smoothed state
and observation disturbances.
SMOOTHER_ALL
Calculate and return all results.
If the bitmask is set directly via the `smoother_output` argument, then
the full method must be provided.
If keyword arguments are used to set individual boolean flags, then
the lowercase of the method must be used as an argument name, and the
value is the desired value of the boolean flag (True or False).
Note that the smoother output may also be specified by directly
modifying the class attributes which are defined similarly to the
keyword arguments.
The default smoother output is SMOOTHER_ALL.
If performance is a concern, only those results which are needed should
be specified as any results that are not specified will not be
calculated. For example, if the smoother output is set to only include
SMOOTHER_STATE, the smoother operates much more quickly than if all
output is required.
Examples
--------
>>> import statsmodels.tsa.statespace.kalman_smoother as ks
>>> mod = ks.KalmanSmoother(1,1)
>>> mod.smoother_output
15
>>> mod.set_smoother_output(smoother_output=0)
>>> mod.smoother_state = True
>>> mod.smoother_output
1
>>> mod.smoother_state
True
"""
if smoother_output is not None:
self.smoother_output = smoother_output
for name in KalmanSmoother.smoother_outputs:
if name in kwargs:
setattr(self, name, kwargs[name])
[docs] def set_smooth_method(self, smooth_method=None, **kwargs):
r"""
Set the smoothing method
The smoothing method can be used to override the Kalman smoother
approach used. By default, the Kalman smoother used depends on the
Kalman filter method.
Parameters
----------
smooth_method : integer, optional
Bitmask value to set the filter method to. See notes for details.
**kwargs
Keyword arguments may be used to influence the filter method by
setting individual boolean flags. See notes for details.
Notes
-----
The smoothing method is defined by a collection of boolean flags, and
is internally stored as a bitmask. The methods available are:
SMOOTH_CONVENTIONAL = 0x01
Default Kalman smoother, as presented in Durbin and Koopman, 2012
chapter 4.
SMOOTH_CLASSICAL = 0x02
Classical Kalman smoother, as presented in Anderson and Moore, 1979
or Durbin and Koopman, 2012 chapter 4.6.1.
SMOOTH_ALTERNATIVE = 0x04
Modified Bryson-Frazier Kalman smoother method; this is identical
to the conventional method of Durbin and Koopman, 2012, except that
an additional intermediate step is included.
SMOOTH_UNIVARIATE = 0x08
Univariate Kalman smoother, as presented in Durbin and Koopman,
2012 chapter 6, except with modified Bryson-Frazier timing.
Practically speaking, these methods should all produce the same output
but different computational implications, numerical stability
implications, or internal timing assumptions.
Note that only the first method is available if using a Scipy version
older than 0.16.
If the bitmask is set directly via the `smooth_method` argument, then
the full method must be provided.
If keyword arguments are used to set individual boolean flags, then
the lowercase of the method must be used as an argument name, and the
value is the desired value of the boolean flag (True or False).
Note that the filter method may also be specified by directly modifying
the class attributes which are defined similarly to the keyword
arguments.
The default filtering method is SMOOTH_CONVENTIONAL.
Examples
--------
>>> mod = sm.tsa.statespace.SARIMAX(range(10))
>>> mod.smooth_method
1
>>> mod.filter_conventional
True
>>> mod.filter_univariate = True
>>> mod.smooth_method
17
>>> mod.set_smooth_method(filter_univariate=False,
filter_collapsed=True)
>>> mod.smooth_method
33
>>> mod.set_smooth_method(smooth_method=1)
>>> mod.filter_conventional
True
>>> mod.filter_univariate
False
>>> mod.filter_collapsed
False
>>> mod.filter_univariate = True
>>> mod.smooth_method
17
"""
if smooth_method is not None:
self.smooth_method = smooth_method
for name in KalmanSmoother.smooth_methods:
if name in kwargs:
setattr(self, name, kwargs[name])
def _smooth(self, smoother_output=None, smooth_method=None, prefix=None,
complex_step=False, results=None, **kwargs):
# Initialize the smoother
prefix, dtype, create_smoother, create_filter, create_statespace = (
self._initialize_smoother(
smoother_output, smooth_method, prefix=prefix, **kwargs
))
# Check that the filter and statespace weren't just recreated
if create_filter or create_statespace:
raise ValueError('Passed settings forced re-creation of the'
' Kalman filter. Please run `_filter` before'
' running `_smooth`.')
# Get the appropriate smoother
smoother = self._kalman_smoothers[prefix]
# Run the smoother
smoother()
return smoother
[docs] def smooth(self, smoother_output=None, smooth_method=None, results=None,
run_filter=True, prefix=None, complex_step=False,
**kwargs):
"""
Apply the Kalman smoother to the statespace model.
Parameters
----------
smoother_output : int, optional
Determines which Kalman smoother output calculate. Default is all
(including state, disturbances, and all covariances).
results : class or object, optional
If a class, then that class is instantiated and returned with the
result of both filtering and smoothing.
If an object, then that object is updated with the smoothing data.
If None, then a SmootherResults object is returned with both
filtering and smoothing results.
run_filter : bool, optional
Whether or not to run the Kalman filter prior to smoothing. Default
is True.
prefix : string
The prefix of the datatype. Usually only used internally.
Returns
-------
SmootherResults object
"""
# Run the filter
kfilter = self._filter(**kwargs)
# Create the results object
results = self.results_class(self)
results.update_representation(self)
results.update_filter(kfilter)
# Run the smoother
if smoother_output is None:
smoother_output = self.smoother_output
smoother = self._smooth(smoother_output, results=results, **kwargs)
# Update the results
results.update_smoother(smoother)
return results
[docs]class SmootherResults(FilterResults):
r"""
Results from applying the Kalman smoother and/or filter to a state space
model.
Parameters
----------
model : Representation
A Statespace representation
Attributes
----------
nobs : int
Number of observations.
k_endog : int
The dimension of the observation series.
k_states : int
The dimension of the unobserved state process.
k_posdef : int
The dimension of a guaranteed positive definite covariance matrix
describing the shocks in the measurement equation.
dtype : dtype
Datatype of representation matrices
prefix : str
BLAS prefix of representation matrices
shapes : dictionary of name:tuple
A dictionary recording the shapes of each of the representation
matrices as tuples.
endog : array
The observation vector.
design : array
The design matrix, :math:`Z`.
obs_intercept : array
The intercept for the observation equation, :math:`d`.
obs_cov : array
The covariance matrix for the observation equation :math:`H`.
transition : array
The transition matrix, :math:`T`.
state_intercept : array
The intercept for the transition equation, :math:`c`.
selection : array
The selection matrix, :math:`R`.
state_cov : array
The covariance matrix for the state equation :math:`Q`.
missing : array of bool
An array of the same size as `endog`, filled with boolean values that
are True if the corresponding entry in `endog` is NaN and False
otherwise.
nmissing : array of int
An array of size `nobs`, where the ith entry is the number (between 0
and k_endog) of NaNs in the ith row of the `endog` array.
time_invariant : bool
Whether or not the representation matrices are time-invariant
initialization : str
Kalman filter initialization method.
initial_state : array_like
The state vector used to initialize the Kalamn filter.
initial_state_cov : array_like
The state covariance matrix used to initialize the Kalamn filter.
filter_method : int
Bitmask representing the Kalman filtering method
inversion_method : int
Bitmask representing the method used to invert the forecast error
covariance matrix.
stability_method : int
Bitmask representing the methods used to promote numerical stability in
the Kalman filter recursions.
conserve_memory : int
Bitmask representing the selected memory conservation method.
tolerance : float
The tolerance at which the Kalman filter determines convergence to
steady-state.
loglikelihood_burn : int
The number of initial periods during which the loglikelihood is not
recorded.
converged : bool
Whether or not the Kalman filter converged.
period_converged : int
The time period in which the Kalman filter converged.
filtered_state : array
The filtered state vector at each time period.
filtered_state_cov : array
The filtered state covariance matrix at each time period.
predicted_state : array
The predicted state vector at each time period.
predicted_state_cov : array
The predicted state covariance matrix at each time period.
kalman_gain : array
The Kalman gain at each time period.
forecasts : array
The one-step-ahead forecasts of observations at each time period.
forecasts_error : array
The forecast errors at each time period.
forecasts_error_cov : array
The forecast error covariance matrices at each time period.
loglikelihood : array
The loglikelihood values at each time period.
collapsed_forecasts : array
If filtering using collapsed observations, stores the one-step-ahead
forecasts of collapsed observations at each time period.
collapsed_forecasts_error : array
If filtering using collapsed observations, stores the one-step-ahead
forecast errors of collapsed observations at each time period.
collapsed_forecasts_error_cov : array
If filtering using collapsed observations, stores the one-step-ahead
forecast error covariance matrices of collapsed observations at each
time period.
standardized_forecast_error : array
The standardized forecast errors
smoother_output : int
Bitmask representing the generated Kalman smoothing output
scaled_smoothed_estimator : array
The scaled smoothed estimator at each time period.
scaled_smoothed_estimator_cov : array
The scaled smoothed estimator covariance matrices at each time period.
smoothing_error : array
The smoothing error covariance matrices at each time period.
smoothed_state : array
The smoothed state at each time period.
smoothed_state_cov : array
The smoothed state covariance matrices at each time period.
smoothed_state_autocov : array
The smoothed state lago-one autocovariance matrices at each time
period: :math:`Cov(\alpha_{t+1}, \alpha_t)`.
smoothed_measurement_disturbance : array
The smoothed measurement at each time period.
smoothed_state_disturbance : array
The smoothed state at each time period.
smoothed_measurement_disturbance_cov : array
The smoothed measurement disturbance covariance matrices at each time
period.
smoothed_state_disturbance_cov : array
The smoothed state disturbance covariance matrices at each time period.
"""
_smoother_attributes = [
'smoother_output', 'scaled_smoothed_estimator',
'scaled_smoothed_estimator_cov', 'smoothing_error',
'smoothed_state', 'smoothed_state_cov', 'smoothed_state_autocov',
'smoothed_measurement_disturbance', 'smoothed_state_disturbance',
'smoothed_measurement_disturbance_cov',
'smoothed_state_disturbance_cov'
]
_smoother_options = KalmanSmoother.smoother_outputs
_attributes = FilterResults._model_attributes + _smoother_attributes
[docs] def update_representation(self, model, only_options=False):
"""
Update the results to match a given model
Parameters
----------
model : Representation
The model object from which to take the updated values.
only_options : boolean, optional
If set to true, only the smoother and filter options are updated,
and the state space representation is not updated. Default is
False.
Notes
-----
This method is rarely required except for internal usage.
"""
super(SmootherResults, self).update_representation(model, only_options)
# Save the options as boolean variables
for name in self._smoother_options:
setattr(self, name, getattr(model, name, None))
# Initialize holders for smoothed forecasts
self._smoothed_forecasts = None
self._smoothed_forecasts_error = None
self._smoothed_forecasts_error_cov = None
[docs] def update_smoother(self, smoother):
"""
Update the smoother results
Parameters
----------
smoother : KalmanSmoother
The model object from which to take the updated values.
Notes
-----
This method is rarely required except for internal usage.
"""
# Copy the appropriate output
attributes = []
# Since update_representation will already have been called, we can
# use the boolean options smoother_* and know they match the smoother
# itself
if self.smoother_state or self.smoother_disturbance:
attributes.append('scaled_smoothed_estimator')
if self.smoother_state_cov or self.smoother_disturbance_cov:
attributes.append('scaled_smoothed_estimator_cov')
if self.smoother_state:
attributes.append('smoothed_state')
if self.smoother_state_cov:
attributes.append('smoothed_state_cov')
if self.smoother_state_autocov:
attributes.append('smoothed_state_autocov')
if self.smoother_disturbance:
attributes += [
'smoothing_error',
'smoothed_measurement_disturbance',
'smoothed_state_disturbance'
]
if self.smoother_disturbance_cov:
attributes += [
'smoothed_measurement_disturbance_cov',
'smoothed_state_disturbance_cov'
]
has_missing = np.sum(self.nmissing) > 0
for name in self._smoother_attributes:
if name == 'smoother_output':
pass
elif name in attributes:
if name in ['smoothing_error',
'smoothed_measurement_disturbance']:
vector = getattr(smoother, name, None)
if vector is not None and has_missing:
vector = np.array(reorder_missing_vector(
vector, self.missing, prefix=self.prefix))
else:
vector = np.array(vector, copy=True)
setattr(self, name, vector)
elif name == 'smoothed_measurement_disturbance_cov':
matrix = getattr(smoother, name, None)
if matrix is not None and has_missing:
matrix = reorder_missing_matrix(
matrix, self.missing, reorder_rows=True,
reorder_cols=True, prefix=self.prefix)
# In the missing data case, we want to set the missing
# components equal to their unconditional distribution
copy_index_matrix(
self.obs_cov, matrix, self.missing,
index_rows=True, index_cols=True, inplace=True,
prefix=self.prefix)
else:
matrix = np.array(matrix, copy=True)
setattr(self, name, matrix)
else:
setattr(self, name,
np.array(getattr(smoother, name, None), copy=True))
else:
setattr(self, name, None)
# Diffuse objects
self.scaled_smoothed_diffuse_estimator = None
self.scaled_smoothed_diffuse1_estimator_cov = None
self.scaled_smoothed_diffuse2_estimator_cov = None
if self.nobs_diffuse > 0:
self.scaled_smoothed_diffuse_estimator = np.array(
smoother.scaled_smoothed_diffuse_estimator, copy=True)
self.scaled_smoothed_diffuse1_estimator_cov = np.array(
smoother.scaled_smoothed_diffuse1_estimator_cov, copy=True)
self.scaled_smoothed_diffuse2_estimator_cov = np.array(
smoother.scaled_smoothed_diffuse2_estimator_cov, copy=True)
# Adjustments
# For r_t (and similarly for N_t), what was calculated was
# r_T, ..., r_{-1}. We only want r_0, ..., r_T
# so exclude the appropriate element so that the time index is
# consistent with the other returned output
# r_t stored such that scaled_smoothed_estimator[0] == r_{-1}
start = 1
end = None
if 'scaled_smoothed_estimator' in attributes:
self.scaled_smoothed_estimator = (
self.scaled_smoothed_estimator[:, start:end]
)
if 'scaled_smoothed_estimator_cov' in attributes:
self.scaled_smoothed_estimator_cov = (
self.scaled_smoothed_estimator_cov[:, :, start:end]
)
# Clear the smoothed forecasts
self._smoothed_forecasts = None
self._smoothed_forecasts_error = None
self._smoothed_forecasts_error_cov = None
# Note: if we concentrated out the scale, need to adjust the
# loglikelihood values and all of the covariance matrices and the
# values that depend on the covariance matrices
if self.filter_concentrated and self.model._scale is None:
self.smoothed_state_cov *= self.scale
self.smoothed_state_autocov *= self.scale
self.smoothed_state_disturbance_cov *= self.scale
self.smoothed_measurement_disturbance_cov *= self.scale
self.scaled_smoothed_estimator /= self.scale
self.scaled_smoothed_estimator_cov /= self.scale
self.smoothing_error /= self.scale
def _get_smoothed_forecasts(self):
if self._smoothed_forecasts is None:
# Initialize empty arrays
self._smoothed_forecasts = np.zeros(self.forecasts.shape,
dtype=self.dtype)
self._smoothed_forecasts_error = (
np.zeros(self.forecasts_error.shape, dtype=self.dtype)
)
self._smoothed_forecasts_error_cov = (
np.zeros(self.forecasts_error_cov.shape, dtype=self.dtype)
)
for t in range(self.nobs):
design_t = 0 if self.design.shape[2] == 1 else t
obs_cov_t = 0 if self.obs_cov.shape[2] == 1 else t
obs_intercept_t = 0 if self.obs_intercept.shape[1] == 1 else t
mask = ~self.missing[:, t].astype(bool)
# We can recover forecasts
self._smoothed_forecasts[:, t] = np.dot(
self.design[:, :, design_t], self.smoothed_state[:, t]
) + self.obs_intercept[:, obs_intercept_t]
if self.nmissing[t] > 0:
self._smoothed_forecasts_error[:, t] = np.nan
self._smoothed_forecasts_error[mask, t] = (
self.endog[mask, t] - self._smoothed_forecasts[mask, t]
)
self._smoothed_forecasts_error_cov[:, :, t] = np.dot(
np.dot(self.design[:, :, design_t],
self.smoothed_state_cov[:, :, t]),
self.design[:, :, design_t].T
) + self.obs_cov[:, :, obs_cov_t]
return (
self._smoothed_forecasts,
self._smoothed_forecasts_error,
self._smoothed_forecasts_error_cov
)
@property
def smoothed_forecasts(self):
return self._get_smoothed_forecasts()[0]
@property
def smoothed_forecasts_error(self):
return self._get_smoothed_forecasts()[1]
@property
def smoothed_forecasts_error_cov(self):
return self._get_smoothed_forecasts()[2]
