"""
State Space Representation and Kalman Filter, Smoother
Author: Chad Fulton
License: Simplified-BSD
"""
import numpy as np
from types import SimpleNamespace
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, int}
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)
def _clone_kwargs(self, endog, **kwargs):
# See Representation._clone_kwargs for docstring
kwargs = super(KalmanSmoother, self)._clone_kwargs(endog, **kwargs)
# Get defaults for options
kwargs.setdefault('smoother_output', self.smoother_output)
kwargs.setdefault('smooth_method', self.smooth_method)
return 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
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 : int, 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])
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 : int, 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
def smooth(self, smoother_output=None, smooth_method=None, results=None,
run_filter=True, prefix=None, complex_step=False,
update_representation=True, update_filter=True,
update_smoother=True, **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 : str
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)
if update_representation:
results.update_representation(self)
if update_filter:
results.update_filter(kfilter)
else:
# (even if we don't update all filter results, still need to
# update this)
results.nobs_diffuse = kfilter.nobs_diffuse
# Run the smoother
if smoother_output is None:
smoother_output = self.smoother_output
smoother = self._smooth(smoother_output, results=results, **kwargs)
# Update the results
if update_smoother:
results.update_smoother(smoother)
return results
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 : ndarray
The observation vector.
design : ndarray
The design matrix, :math:`Z`.
obs_intercept : ndarray
The intercept for the observation equation, :math:`d`.
obs_cov : ndarray
The covariance matrix for the observation equation :math:`H`.
transition : ndarray
The transition matrix, :math:`T`.
state_intercept : ndarray
The intercept for the transition equation, :math:`c`.
selection : ndarray
The selection matrix, :math:`R`.
state_cov : ndarray
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 : ndarray
The filtered state vector at each time period.
filtered_state_cov : ndarray
The filtered state covariance matrix at each time period.
predicted_state : ndarray
The predicted state vector at each time period.
predicted_state_cov : ndarray
The predicted state covariance matrix at each time period.
kalman_gain : ndarray
The Kalman gain at each time period.
forecasts : ndarray
The one-step-ahead forecasts of observations at each time period.
forecasts_error : ndarray
The forecast errors at each time period.
forecasts_error_cov : ndarray
The forecast error covariance matrices at each time period.
loglikelihood : ndarray
The loglikelihood values at each time period.
collapsed_forecasts : ndarray
If filtering using collapsed observations, stores the one-step-ahead
forecasts of collapsed observations at each time period.
collapsed_forecasts_error : ndarray
If filtering using collapsed observations, stores the one-step-ahead
forecast errors of collapsed observations at each time period.
collapsed_forecasts_error_cov : ndarray
If filtering using collapsed observations, stores the one-step-ahead
forecast error covariance matrices of collapsed observations at each
time period.
standardized_forecast_error : ndarray
The standardized forecast errors
smoother_output : int
Bitmask representing the generated Kalman smoothing output
scaled_smoothed_estimator : ndarray
The scaled smoothed estimator at each time period.
scaled_smoothed_estimator_cov : ndarray
The scaled smoothed estimator covariance matrices at each time period.
smoothing_error : ndarray
The smoothing error covariance matrices at each time period.
smoothed_state : ndarray
The smoothed state at each time period.
smoothed_state_cov : ndarray
The smoothed state covariance matrices at each time period.
smoothed_state_autocov : ndarray
The smoothed state lago-one autocovariance matrices at each time
period: :math:`Cov(\alpha_{t+1}, \alpha_t)`.
smoothed_measurement_disturbance : ndarray
The smoothed measurement at each time period.
smoothed_state_disturbance : ndarray
The smoothed state at each time period.
smoothed_measurement_disturbance_cov : ndarray
The smoothed measurement disturbance covariance matrices at each time
period.
smoothed_state_disturbance_cov : ndarray
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', 'innovations_transition'
]
_smoother_options = KalmanSmoother.smoother_outputs
_attributes = FilterResults._model_attributes + _smoother_attributes
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 : bool, 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
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)
self.innovations_transition = (
np.array(smoother.innovations_transition, copy=True))
# 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
# Cache
self.__smoothed_state_autocovariance = {}
def _smoothed_state_autocovariance(self, shift, start, end,
extend_kwargs=None):
"""
Compute "forward" autocovariances, Cov(t, t+j)
Parameters
----------
shift : int
The number of period to shift forwards when computing the
autocovariance. This has the opposite sign as `lag` from the
`smoothed_state_autocovariance` method.
start : int, optional
The start of the interval (inclusive) of autocovariances to compute
and return.
end : int, optional
The end of the interval (exclusive) autocovariances to compute and
return. Note that since it is an exclusive endpoint, the returned
autocovariances do not include the value at this index.
extend_kwargs : dict, optional
Keyword arguments containing updated state space system matrices
for handling out-of-sample autocovariance computations in
time-varying state space models.
"""
if extend_kwargs is None:
extend_kwargs = {}
# Size of returned array in the time dimension
n = end - start
# Get number of post-sample periods we need to create an extended
# model to compute
if shift == 0:
max_insample = self.nobs - shift
else:
max_insample = self.nobs - shift + 1
n_postsample = max(0, end - max_insample)
# Get full in-sample arrays
if shift != 0:
L = self.innovations_transition
P = self.predicted_state_cov
N = self.scaled_smoothed_estimator_cov
else:
acov = self.smoothed_state_cov
# If applicable, append out-of-sample arrays
if n_postsample > 0:
# Note: we need 1 less than the number of post
endog = np.zeros((n_postsample, self.k_endog)) * np.nan
mod = self.model.extend(endog, start=self.nobs, **extend_kwargs)
mod.initialize_known(self.predicted_state[..., self.nobs],
self.predicted_state_cov[..., self.nobs])
res = mod.smooth()
if shift != 0:
start_insample = max(0, start)
L = np.concatenate((L[..., start_insample:],
res.innovations_transition), axis=2)
P = np.concatenate((P[..., start_insample:],
res.predicted_state_cov[..., 1:]),
axis=2)
N = np.concatenate((N[..., start_insample:],
res.scaled_smoothed_estimator_cov),
axis=2)
end -= start_insample
start -= start_insample
else:
acov = np.concatenate((acov, res.predicted_state_cov), axis=2)
if shift != 0:
# Subset to appropriate start, end
start_insample = max(0, start)
LT = L[..., start_insample:end + shift - 1].T
P = P[..., start_insample:end + shift].T
N = N[..., start_insample:end + shift - 1].T
# Intermediate computations
tmpLT = np.eye(self.k_states)[None, :, :]
length = P.shape[0] - shift # this is the required length of LT
for i in range(1, shift + 1):
tmpLT = LT[shift - i:length + shift - i] @ tmpLT
eye = np.eye(self.k_states)[None, ...]
# Compute the autocovariance
acov = np.zeros((n, self.k_states, self.k_states))
acov[:start_insample - start] = np.nan
acov[start_insample - start:] = (
P[:-shift] @ tmpLT @ (eye - N[shift - 1:] @ P[shift:]))
else:
acov = acov.T[start:end]
return acov
def smoothed_state_autocovariance(self, lag=1, t=None, start=None,
end=None, extend_kwargs=None):
r"""
Compute state vector autocovariances, conditional on the full dataset
Computes:
.. math::
Cov(\alpha_t - \hat \alpha_t, \alpha_{t - j} - \hat \alpha_{t - j})
where the `lag` argument gives the value for :math:`j`. Thus when
the `lag` argument is positive, the autocovariance is between the
current and previous periods, while if `lag` is negative the
autocovariance is between the current and future periods.
Parameters
----------
lag : int, optional
The number of period to shift when computing the autocovariance.
Default is 1.
t : int, optional
A specific period for which to compute and return the
autocovariance. Cannot be used in combination with `start` or
`end`. See the Returns section for details on how this
parameter affects what is what is returned.
start : int, optional
The start of the interval (inclusive) of autocovariances to compute
and return. Cannot be used in combination with the `t` argument.
See the Returns section for details on how this parameter affects
what is what is returned. Default is 0.
end : int, optional
The end of the interval (exclusive) autocovariances to compute and
return. Note that since it is an exclusive endpoint, the returned
autocovariances do not include the value at this index. Cannot be
used in combination with the `t` argument. See the Returns section
for details on how this parameter affects what is what is returned
and what the default value is.
extend_kwargs : dict, optional
Keyword arguments containing updated state space system matrices
for handling out-of-sample autocovariance computations in
time-varying state space models.
Returns
-------
acov : ndarray
Array of autocovariance matrices. If the argument `t` is not
provided, then it is shaped `(k_states, k_states, n)`, while if `t`
given then the third axis is dropped and the array is shaped
`(k_states, k_states)`.
The output under the default case differs somewhat based on the
state space model and the sign of the lag. To see how these cases
differ, denote the output at each time point as Cov(t, t-j). Then:
- If `lag > 0` (and the model is either time-varying or
time-invariant), then the returned array is shaped `(*, *, nobs)`
and each entry [:, :, t] contains Cov(t, t-j). However, the model
does not have enough information to compute autocovariances in
the pre-sample period, so that we cannot compute Cov(1, 1-lag),
Cov(2, 2-lag), ..., Cov(lag, 0). Thus the first `lag` entries
have all values set to NaN.
- If the model is time-invariant and `lag < -1` or if `lag` is
0 or -1, and the model is either time-invariant or time-varying,
then the returned array is shaped `(*, *, nobs)` and each
entry [:, :, t] contains Cov(t, t+j). Moreover, all entries are
available (i.e. there are no NaNs).
- If the model is time-varying and `lag < -1` and `extend_kwargs`
is not provided, then the returned array is shaped
`(*, *, nobs - lag + 1)`.
- However, if the model is time-varying and `lag < -1`, then
`extend_kwargs` can be provided with `lag - 1` additional
matrices so that the returned array is shaped `(*, *, nobs)` as
usual.
More generally, the dimension of the last axis will be
`start - end`.
Notes
-----
This method computes:
.. math::
Cov(\alpha_t - \hat \alpha_t, \alpha_{t - j} - \hat \alpha_{t - j})
where the `lag` argument determines the autocovariance order :math:`j`,
and `lag` is an integer (positive, zero, or negative). This method
cannot compute values associated with time points prior to the sample,
and so it returns a matrix of NaN values for these time points.
For example, if `start=0` and `lag=2`, then assuming the output is
assigned to the variable `acov`, we will have `acov[..., 0]` and
`acov[..., 1]` as matrices filled with NaN values.
Based only on the "current" results object (i.e. the Kalman smoother
applied to the sample), there is not enough information to compute
Cov(t, t+j) for the last `lag - 1` observations of the sample. However,
the values can be computed for these time points using the transition
equation of the state space representation, and so for time-invariant
state space models we do compute these values. For time-varying models,
this can also be done, but updated state space matrices for the
out-of-sample time points must be provided via the `extend_kwargs`
argument.
See [1]_, Chapter 4.7, for all details about how these autocovariances
are computed.
The `t` and `start`/`end` parameters compute and return only the
requested autocovariances. As a result, using these parameters is
recommended to reduce the computational burden, particularly if the
number of observations and/or the dimension of the state vector is
large.
References
----------
.. [1] Durbin, James, and Siem Jan Koopman. 2012.
Time Series Analysis by State Space Methods: Second Edition.
Oxford University Press.
"""
# We can cache the results for time-invariant models
cache_key = None
if extend_kwargs is None or len(extend_kwargs) == 0:
cache_key = (lag, t, start, end)
# Short-circuit for a cache-hit
if (cache_key is not None and
cache_key in self.__smoothed_state_autocovariance):
return self.__smoothed_state_autocovariance[cache_key]
# Switch to only positive values for `lag`
forward_autocovariances = False
if lag < 0:
lag = -lag
forward_autocovariances = True
# Handle `t`
if t is not None and (start is not None or end is not None):
raise ValueError('Cannot specify both `t` and `start` or `end`.')
if t is not None:
start = t
end = t + 1
# Defaults
if start is None:
start = 0
if end is None:
if forward_autocovariances and lag > 1 and extend_kwargs is None:
end = self.nobs - lag + 1
else:
end = self.nobs
if extend_kwargs is None:
extend_kwargs = {}
# Sanity checks
if start < 0 or end < 0:
raise ValueError('Negative `t`, `start`, or `end` is not allowed.')
if end < start:
raise ValueError('`end` must be after `start`')
if lag == 0 and self.smoothed_state_cov is None:
raise RuntimeError('Cannot return smoothed state covariances'
' if those values have not been computed by'
' Kalman smoothing.')
# We already have in-sample (+1 out-of-sample) smoothed covariances
if lag == 0 and end <= self.nobs + 1:
acov = self.smoothed_state_cov
if end == self.nobs + 1:
acov = np.concatenate(
(acov[..., start:], self.predicted_state_cov[..., -1:]),
axis=2).T
else:
acov = acov.T[start:end]
# In-sample, we can compute up to Cov(T, T+1) or Cov(T+1, T) and down
# to Cov(1, 2) or Cov(2, 1). So:
# - For lag=1 we set Cov(1, 0) = np.nan and then can compute up to T-1
# in-sample values Cov(2, 1), ..., Cov(T, T-1) and the first
# out-of-sample value Cov(T+1, T)
elif (lag == 1 and self.smoothed_state_autocov is not None and
not forward_autocovariances and end <= self.nobs + 1):
# nans = np.zeros((self.k_states, self.k_states, lag)) * np.nan
# acov = np.concatenate((nans, self.smoothed_state_autocov),
# axis=2).transpose(2, 0, 1)[start:end]
if start == 0:
nans = np.zeros((self.k_states, self.k_states, lag)) * np.nan
acov = np.concatenate(
(nans, self.smoothed_state_autocov[..., :end - 1]),
axis=2)
else:
acov = self.smoothed_state_autocov[..., start - 1:end - 1]
acov = acov.transpose(2, 0, 1)
# - For lag=-1 we can compute T in-sample values, Cov(1, 2), ...,
# Cov(T, T+1) but we cannot compute the first out-of-sample value
# Cov(T+1, T+2).
elif (lag == 1 and self.smoothed_state_autocov is not None and
forward_autocovariances and end < self.nobs + 1):
acov = self.smoothed_state_autocov.T[start:end]
# Otherwise, we need to compute additional values at the end of the
# sample
else:
if forward_autocovariances:
# Cov(t, t + lag), t = start, ..., end
acov = self._smoothed_state_autocovariance(
lag, start, end, extend_kwargs=extend_kwargs)
else:
# Cov(t, t + lag)' = Cov(t + lag, t),
# with t = start - lag, ..., end - lag
out = self._smoothed_state_autocovariance(
lag, start - lag, end - lag, extend_kwargs=extend_kwargs)
acov = out.transpose(0, 2, 1)
# Squeeze the last axis or else reshape to have the same axis
# definitions as e.g. smoothed_state_cov
if t is not None:
acov = acov[0]
else:
acov = acov.transpose(1, 2, 0)
# Fill in the cache, if applicable
if cache_key is not None:
self.__smoothed_state_autocovariance[cache_key] = acov
return acov
def news(self, previous, t=None, start=None, end=None,
revised=None, design=None):
r"""
Compute the news and impacts associated with a data release
Parameters
----------
previous : SmootherResults
Prior results object relative to which to compute the news. This
results object must have identical state space representation for
the prior sample period so that the only difference is that this
results object has updates to the observed data.
t : int, optional
A specific period for which to compute the news. Cannot be used in
combination with `start` or `end`.
start : int, optional
The start of the interval (inclusive) of news to compute. Cannot be
used in combination with the `t` argument. Default is the last
period of the sample (`nobs - 1`).
end : int, optional
The end of the interval (exclusive) of news to compute. Note that
since it is an exclusive endpoint, the returned news do not include
the value at this index. Cannot be used in combination with the `t`
argument.
design : array, optional
Design matrix for the period `t` in time-varying models. If this
model has a time-varying design matrix, and the argument `t` is out
of this model's sample, then a new design matrix for period `t`
must be provided. Unused otherwise.
Returns
-------
news_results : SimpleNamespace
News and impacts associated with a data release. Includes the
following attributes:
- `update_impacts`: update to forecasts of impacted variables from
the news. It is equivalent to E[y^i | post] - E[y^i | revision],
where y^i are the variables of interest. In [1]_, this is
described as "revision" in equation (17).
- `revision_impacts`: update to forecasts of variables impacted
variables from data revisions. It is
E[y^i | revision] - E[y^i | previous], and does not have a
specific notation in [1]_, since there for simplicity they assume
that there are no revisions.
- `news`: the unexpected component of the updated data. Denoted
I = y^u - E[y^u | previous], where y^u are the data points that
were newly incorporated in a data release (but not including
revisions to data points that already existed in the previous
release). In [1]_, this is described as "news" in equation (17).
- `gain`: the gain matrix associated with the "Kalman-like" update
from the news, E[y I'] E[I I']^{-1}. In [1]_, this can be found
in the equation For E[y_{k,t_k} \mid I_{v+1}] in the middle of
page 17.
- `update_forecasts`: forecasts of the updated periods used to
construct the news, E[y^u | previous].
- `update_realized`: realizations of the updated periods used to
construct the news, y^u.
- `prev_impacted_forecasts`: previous forecast of the periods of
interest, E[y^i | previous].
- `post_impacted_forecasts`: forecast of the periods of interest
after taking into account both revisions and updates,
E[y^i | post].
- `revision_results`: results object that updates the `previous`
results to take into account data revisions.
- `revisions_ix`: list of `(t, i)` positions of revisions in endog
- `updates_ix`: list of `(t, i)` positions of updates to endog
Notes
-----
This method computes the effect of new data (e.g. from a new data
release) on smoothed forecasts produced by a state space model, as
described in [1]_.
References
----------
.. [1] Bańbura, Marta and Modugno, Michele. 2010.
"Maximum likelihood estimation of factor models on data sets
with arbitrary pattern of missing data."
No 1189, Working Paper Series, European Central Bank.
https://EconPapers.repec.org/RePEc:ecb:ecbwps:20101189.
.. [2] Bańbura, Marta, and Michele Modugno.
"Maximum likelihood estimation of factor models on datasets with
arbitrary pattern of missing data."
Journal of Applied Econometrics 29, no. 1 (2014): 133-160.
"""
# Handle `t`
if t is not None and (start is not None or end is not None):
raise ValueError('Cannot specify both `t` and `start` or `end`.')
if t is not None:
start = t
end = t + 1
# Defaults
if start is None:
start = self.nobs - 1
if end is None:
end = self.nobs
# Sanity checks
if start < 0 or end < 0:
raise ValueError('Negative `t`, `start`, or `end` is not allowed.')
if end <= start:
raise ValueError('`end` must be after `start`')
if self.smoothed_state_cov is None:
raise ValueError('Cannot compute news without having applied the'
' Kalman smoother first.')
error_ss = ('This results object has %s and so it does not appear to'
' by an extension of `previous`. Can only compute the'
' news by comparing this results set to previous results'
' objects.')
if self.nobs < previous.nobs:
raise ValueError(error_ss % 'fewer observations than'
' `previous`')
if not (self.k_endog == previous.k_endog and
self.k_states == previous.k_states and
self.k_posdef == previous.k_posdef):
raise ValueError(error_ss % 'different state space dimensions than'
' `previous`')
for key in self.model.shapes.keys():
if key == 'obs':
continue
tv = getattr(self, key).shape[-1] > 1
tv_prev = getattr(previous, key).shape[-1] > 1
if tv and not tv_prev:
raise ValueError(error_ss % f'time-varying {key} while'
' `previous` does not')
if not tv and tv_prev:
raise ValueError(error_ss % f'time-invariant {key} while'
' `previous` does not')
# We cannot forecast out-of-sample periods in a time-varying models
if end > self.nobs and not self.model.time_invariant:
raise RuntimeError('Cannot compute the impacts of news on periods'
' outside of the sample in time-varying'
' models.')
# For time-varying case, figure out extension kwargs
extend_kwargs = {}
for key in self.model.shapes.keys():
if key == 'obs':
continue
mat = getattr(self, key)
prev_mat = getattr(previous, key)
if mat.shape[-1] > prev_mat.shape[-1]:
extend_kwargs[key] = mat[..., prev_mat.shape[-1]:]
# Figure out which indices have changed
revisions_ix, updates_ix = previous.model.diff_endog(self.endog.T)
# Compute prev / post impact forecasts
prev_impacted_forecasts = (
previous.smoothed_forecasts[..., start:end])
if end > previous.nobs:
predict_start = max(start, previous.nobs)
p = previous.predict(
start=predict_start, end=end, **extend_kwargs)
prev_impacted_forecasts = np.concatenate(
(prev_impacted_forecasts, p.forecasts), axis=1)
post_impacted_forecasts = (
self.smoothed_forecasts[..., start:end])
if end > self.nobs:
predict_start = max(start, self.nobs)
p = self.predict(start=predict_start, end=end, **extend_kwargs)
post_impacted_forecasts = np.concatenate(
(post_impacted_forecasts, p.forecasts), axis=1)
# If we have revisions to previous data, then we need to construct a
# new results set that only includes those revisions
if len(revisions_ix) > 0 and revised is None:
# Copy time-varying matrices (required by clone)
clone_kwargs = {}
for key in self.model.shapes.keys():
if key == 'obs':
continue
prev_mat = getattr(previous, key)
if prev_mat.shape[-1] > 1:
clone_kwargs[key] = prev_mat
rev_endog = self.endog.T[:previous.nobs].copy()
rev_endog[previous.missing.astype(bool).T] = np.nan
rev_mod = previous.model.clone(rev_endog, **clone_kwargs)
# TODO: performance: can get a performance improvement for large
# models with `update_filter=False, update_smoother=False`,
# but then will need to manually populate the fields that we
# need for what we do below (i.e. we call
# `smoothed_forecasts` and `predict`)
# TODO: performance: we don't need to smooth back through the
# entire sample for what we're doing, since we only need
# smoothed_forecasts for the impact period
revised = rev_mod.smooth()
# Compute impacts from the revisions, if any
if len(revisions_ix) > 0:
# Compute the effect of revisions on forecasts of the impacted
# variables
revised_impact_forecasts = (
revised.smoothed_forecasts[..., start:end])
if end > revised.nobs:
predict_start = max(start, revised.nobs)
p = revised.predict(
start=predict_start, end=end, **extend_kwargs)
revised_impact_forecasts = np.concatenate(
(revised_impact_forecasts, p.forecasts), axis=1)
revision_impacts = (revised_impact_forecasts -
prev_impacted_forecasts).T
if t is not None:
revision_impacts = revision_impacts[0]
else:
revised = previous
revision_impacts = None
# Now handle updates
if len(updates_ix) > 0:
# Figure out which time points we need forecast errors for
update_t, update_k = zip(*updates_ix)
update_start_t = np.min(update_t)
update_end_t = np.max(update_t)
update_end_insample_t = np.minimum(revised.nobs - 1, update_end_t)
update_nforecast = update_end_t - update_end_insample_t
# For the in-sample periods, get out the smoothed forecasts for
# the updated variables for each relevant time period
i1 = update_start_t
i2 = update_end_insample_t + 1
if i2 > i1:
forecasts_insample = revised.smoothed_forecasts[:, i1:i2]
else:
forecasts_insample = np.zeros((self.k_endog, 0))
# For the out-of-sample periods, predicted and smoothed forecasts
# for the updated variables are the same
if update_nforecast > 0:
forecast_start = previous.nobs
forecast_end = forecast_start + update_nforecast
p = revised.predict(
start=forecast_start, end=forecast_end, **extend_kwargs)
forecasts_oos = p.forecasts
else:
forecasts_oos = np.zeros((self.k_endog, 0))
forecasts = np.c_[forecasts_insample, forecasts_oos].T
realized = self.endog.T[update_start_t:update_end_t + 1]
forecasts_error = realized - forecasts
# Now subset forecast errors to only the (time, endog) elements
# that are updates
ix_t = update_t - update_start_t
update_realized = realized[ix_t, update_k]
update_forecasts = forecasts[ix_t, update_k]
update_forecasts_error = forecasts_error[ix_t, update_k]
# Get the gains associated with each of the periods
if self.design.shape[2] == 1:
design = self.design[..., 0][None, ...]
elif end <= self.nobs:
design = self.design[..., start:end].transpose(2, 0, 1)
else:
if design is None:
raise ValueError('Model has time-varying design matrix, so'
' an updated time-varying matrix for'
' period `t` is required.')
elif design.ndim == 2:
design = design[None, ...]
else:
design = design.transpose(2, 0, 1)
state_gain = revised.smoothed_state_gain(
updates_ix, start=start, end=end, extend_kwargs=extend_kwargs)
obs_gain = design @ state_gain
# Get the news
update_impacts = obs_gain @ update_forecasts_error
# Squeeze if `t` argument used
if t is not None:
obs_gain = obs_gain[0]
update_impacts = update_impacts[0]
else:
update_impacts = None
update_forecasts = None
update_realized = None
update_forecasts_error = None
obs_gain = None
# Results
out = SimpleNamespace(
# update to forecast of impacted variables from news
# = E[y^i | post] - E[y^i | revision] = weight @ news
update_impacts=update_impacts,
# update to forecast of variables of interest from revisions
# = E[y^i | revision] - E[y^i | previous]
revision_impacts=revision_impacts,
# news = A = y^u - E[y^u | previous]
news=update_forecasts_error,
# gain matrix = E[y A'] E[A A']^{-1}
gain=obs_gain,
# forecasts of the updated periods used to construct the news
# = E[y^u | previous]
update_forecasts=update_forecasts,
# realizations of the updated periods used to construct the news
# = y^u
update_realized=update_realized,
# previous forecast of the periods of interest, E[y^i | previous]
prev_impacted_forecasts=prev_impacted_forecasts,
# post. forecast of the periods of interest, E[y^i | post]
post_impacted_forecasts=post_impacted_forecasts,
# results object associated with the revision
revision_results=None,
# list of (x, y) positions of revisions to endog
revisions_ix=revisions_ix,
# list of (x, y) positions of updates to endog
updates_ix=updates_ix)
if len(revisions_ix) > 0:
out.revision_results = revised
return out
def smoothed_state_gain(self, updates_ix, t=None, start=None,
end=None, extend_kwargs=None):
r"""
Cov(\tilde \alpha_{t}, I) Var(I, I)^{-1}
where I is a vector of forecast errors associated with
`update_indices`.
Parameters
----------
updates_ix : list
List of indices `(t, i)`, where `t` denotes a zero-indexed time
location and `i` denotes a zero-indexed endog variable.
"""
# Handle `t`
if t is not None and (start is not None or end is not None):
raise ValueError('Cannot specify both `t` and `start` or `end`.')
if t is not None:
start = t
end = t + 1
# Defaults
if start is None:
start = self.nobs - 1
if end is None:
end = self.nobs
if extend_kwargs is None:
extend_kwargs = {}
# Sanity checks
if start < 0 or end < 0:
raise ValueError('Negative `t`, `start`, or `end` is not allowed.')
if end <= start:
raise ValueError('`end` must be after `start`')
# Dimensions
n_periods = end - start
n_updates = len(updates_ix)
# Helper to get possibly matrix that is possibly time-varying
def get_mat(which, t):
mat = getattr(self, which)
if mat.shape[-1] > 1:
if t < self.nobs:
out = mat[..., t]
else:
if (which not in extend_kwargs or
extend_kwargs[which].shape[-1] <= t - self.nobs):
raise ValueError(f'Model has time-varying {which}'
' matrix, so an updated time-varying'
' matrix for the extension period is'
' required.')
out = extend_kwargs[which][..., t - self.nobs]
else:
out = mat[..., 0]
return out
# Helper to get Cov(\tilde \alpha_{t}, I)
def get_cov_state_revision(t):
tmp1 = np.zeros((self.k_states, n_updates))
for i in range(n_updates):
t_i, k_i = updates_ix[i]
acov = self.smoothed_state_autocovariance(
lag=t - t_i, t=t, extend_kwargs=extend_kwargs)
Z_i = get_mat('design', t_i)
tmp1[:, i:i + 1] = acov @ Z_i[k_i:k_i + 1].T
return tmp1
# Compute Cov(\tilde \alpha_{t}, I)
tmp1 = np.zeros((n_periods, self.k_states, n_updates))
for s in range(start, end):
tmp1[s - start] = get_cov_state_revision(s)
# Compute Var(I)
tmp2 = np.zeros((n_updates, n_updates))
for i in range(n_updates):
t_i, k_i = updates_ix[i]
for j in range(i + 1):
t_j, k_j = updates_ix[j]
Z_i = get_mat('design', t_i)
Z_j = get_mat('design', t_j)
acov = self.smoothed_state_autocovariance(
lag=t_i - t_j, t=t_i, extend_kwargs=extend_kwargs)
tmp2[i, j] = tmp2[j, i] = (
Z_i[k_i:k_i + 1] @ acov @ Z_j[k_j:k_j + 1].T)
if t_i == t_j:
H = get_mat('obs_cov', t_i)
if i == j:
tmp2[i, j] += H[k_i, k_j]
else:
tmp2[i, j] += H[k_i, k_j]
tmp2[j, i] += H[k_i, k_j]
# Gain
gain = tmp1 @ np.linalg.inv(tmp2)
if t is not None:
gain = gain[0]
return gain
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]