# -*- coding: utf-8 -*-
"""
State Space Model
Author: Chad Fulton
License: Simplified-BSD
"""
from statsmodels.compat.pandas import is_int_index
import contextlib
import warnings
import datetime as dt
from types import SimpleNamespace
import numpy as np
import pandas as pd
from scipy.stats import norm
from statsmodels.tools.tools import pinv_extended, Bunch
from statsmodels.tools.sm_exceptions import PrecisionWarning, ValueWarning
from statsmodels.tools.numdiff import (_get_epsilon, approx_hess_cs,
approx_fprime_cs, approx_fprime)
from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.eval_measures import aic, aicc, bic, hqic
import statsmodels.base.wrapper as wrap
import statsmodels.tsa.base.prediction as pred
from statsmodels.base.data import PandasData
import statsmodels.tsa.base.tsa_model as tsbase
from .news import NewsResults
from .simulation_smoother import SimulationSmoother
from .kalman_smoother import SmootherResults
from .kalman_filter import INVERT_UNIVARIATE, SOLVE_LU, MEMORY_CONSERVE
from .initialization import Initialization
from .tools import prepare_exog, concat, _safe_cond
def _handle_args(names, defaults, *args, **kwargs):
output_args = []
# We need to handle positional arguments in two ways, in case this was
# called by a Scipy optimization routine
if len(args) > 0:
# the fit() method will pass a dictionary
if isinstance(args[0], dict):
flags = args[0]
# otherwise, a user may have just used positional arguments...
else:
flags = dict(zip(names, args))
for i in range(len(names)):
output_args.append(flags.get(names[i], defaults[i]))
for name, value in flags.items():
if name in kwargs:
raise TypeError("loglike() got multiple values for keyword"
" argument '%s'" % name)
else:
for i in range(len(names)):
output_args.append(kwargs.pop(names[i], defaults[i]))
return tuple(output_args) + (kwargs,)
def _check_index(desired_index, dta, title='data'):
given_index = None
if isinstance(dta, (pd.Series, pd.DataFrame)):
given_index = dta.index
if given_index is not None and not desired_index.equals(given_index):
desired_freq = getattr(desired_index, 'freq', None)
given_freq = getattr(given_index, 'freq', None)
if ((desired_freq is not None or given_freq is not None) and
desired_freq != given_freq):
raise ValueError('Given %s does not have an index'
' that extends the index of the'
' model. Expected index frequency is'
' "%s", but got "%s".'
% (title, desired_freq, given_freq))
else:
raise ValueError('Given %s does not have an index'
' that extends the index of the'
' model.' % title)
[docs]class MLEModel(tsbase.TimeSeriesModel):
r"""
State space model for maximum likelihood estimation
Parameters
----------
endog : array_like
The observed time-series process :math:`y`
k_states : int
The dimension of the unobserved state process.
exog : array_like, optional
Array of exogenous regressors, shaped nobs x k. Default is no
exogenous regressors.
dates : array_like of datetime, optional
An array-like object of datetime objects. If a Pandas object is given
for endog, it is assumed to have a DateIndex.
freq : str, optional
The frequency of the time-series. A Pandas offset or 'B', 'D', 'W',
'M', 'A', or 'Q'. This is optional if dates are given.
**kwargs
Keyword arguments may be used to provide default values for state space
matrices or for Kalman filtering options. See `Representation`, and
`KalmanFilter` for more details.
Attributes
----------
ssm : statsmodels.tsa.statespace.kalman_filter.KalmanFilter
Underlying state space representation.
See Also
--------
statsmodels.tsa.statespace.mlemodel.MLEResults
statsmodels.tsa.statespace.kalman_filter.KalmanFilter
statsmodels.tsa.statespace.representation.Representation
Notes
-----
This class wraps the state space model with Kalman filtering to add in
functionality for maximum likelihood estimation. In particular, it adds
the concept of updating the state space representation based on a defined
set of parameters, through the `update` method or `updater` attribute (see
below for more details on which to use when), and it adds a `fit` method
which uses a numerical optimizer to select the parameters that maximize
the likelihood of the model.
The `start_params` `update` method must be overridden in the
child class (and the `transform` and `untransform` methods, if needed).
"""
def __init__(self, endog, k_states, exog=None, dates=None, freq=None,
**kwargs):
# Initialize the model base
super(MLEModel, self).__init__(endog=endog, exog=exog,
dates=dates, freq=freq,
missing='none')
# Store kwargs to recreate model
self._init_kwargs = kwargs
# Prepared the endog array: C-ordered, shape=(nobs x k_endog)
self.endog, self.exog = self.prepare_data()
# Dimensions
self.nobs = self.endog.shape[0]
self.k_states = k_states
# Initialize the state-space representation
self.initialize_statespace(**kwargs)
# Setup holder for fixed parameters
self._has_fixed_params = False
self._fixed_params = None
self._params_index = None
self._fixed_params_index = None
self._free_params_index = None
[docs] def prepare_data(self):
"""
Prepare data for use in the state space representation
"""
endog = np.array(self.data.orig_endog, order='C')
exog = self.data.orig_exog
if exog is not None:
exog = np.array(exog)
# Base class may allow 1-dim data, whereas we need 2-dim
if endog.ndim == 1:
endog.shape = (endog.shape[0], 1) # this will be C-contiguous
return endog, exog
[docs] def initialize_statespace(self, **kwargs):
"""
Initialize the state space representation
Parameters
----------
**kwargs
Additional keyword arguments to pass to the state space class
constructor.
"""
# (Now self.endog is C-ordered and in long format (nobs x k_endog). To
# get F-ordered and in wide format just need to transpose)
endog = self.endog.T
# Instantiate the state space object
self.ssm = SimulationSmoother(endog.shape[0], self.k_states,
nobs=endog.shape[1], **kwargs)
# Bind the data to the model
self.ssm.bind(endog)
# Other dimensions, now that `ssm` is available
self.k_endog = self.ssm.k_endog
def _get_index_with_final_state(self):
# The index we inherit from `TimeSeriesModel` will only cover the
# data sample itself, but we will also need an index value for the
# final state which is the next time step to the last datapoint.
# This method figures out an appropriate value for the three types of
# supported indexes: date-based, Int64Index, or RangeIndex
if self._index_dates:
if isinstance(self._index, pd.DatetimeIndex):
index = pd.date_range(
start=self._index[0], periods=len(self._index) + 1,
freq=self._index.freq)
elif isinstance(self._index, pd.PeriodIndex):
index = pd.period_range(
start=self._index[0], periods=len(self._index) + 1,
freq=self._index.freq)
else:
raise NotImplementedError
elif isinstance(self._index, pd.RangeIndex):
# COMPAT: pd.RangeIndex does not have start, stop, step prior to
# pandas 0.25
try:
start = self._index.start
stop = self._index.stop
step = self._index.step
except AttributeError:
start = self._index._start
stop = self._index._stop
step = self._index._step
index = pd.RangeIndex(start, stop + step, step)
elif is_int_index(self._index):
# The only valid Int64Index is a full, incrementing index, so this
# is general
value = self._index[-1] + 1
index = pd.Index(self._index.tolist() + [value])
else:
raise NotImplementedError
return index
def __setitem__(self, key, value):
return self.ssm.__setitem__(key, value)
def __getitem__(self, key):
return self.ssm.__getitem__(key)
def _get_init_kwds(self):
# Get keywords based on model attributes
kwds = super(MLEModel, self)._get_init_kwds()
for key, value in kwds.items():
if value is None and hasattr(self.ssm, key):
kwds[key] = getattr(self.ssm, key)
return kwds
[docs] def clone(self, endog, exog=None, **kwargs):
"""
Clone state space model with new data and optionally new specification
Parameters
----------
endog : array_like
The observed time-series process :math:`y`
k_states : int
The dimension of the unobserved state process.
exog : array_like, optional
Array of exogenous regressors, shaped nobs x k. Default is no
exogenous regressors.
kwargs
Keyword arguments to pass to the new model class to change the
model specification.
Returns
-------
model : MLEModel subclass
Notes
-----
This method must be implemented
"""
raise NotImplementedError('This method is not implemented in the base'
' class and must be set up by each specific'
' model.')
def _clone_from_init_kwds(self, endog, **kwargs):
# Cannot make this the default, because there is extra work required
# for subclasses to make _get_init_kwds useful.
use_kwargs = self._get_init_kwds()
use_kwargs.update(kwargs)
# Check for `exog`
if getattr(self, 'k_exog', 0) > 0 and kwargs.get('exog', None) is None:
raise ValueError('Cloning a model with an exogenous component'
' requires specifying a new exogenous array using'
' the `exog` argument.')
mod = self.__class__(endog, **use_kwargs)
return mod
[docs] def set_filter_method(self, filter_method=None, **kwargs):
"""
Set the filtering method
The filtering method controls aspects of which Kalman filtering
approach will be used.
Parameters
----------
filter_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
-----
This method is rarely used. See the corresponding function in the
`KalmanFilter` class for details.
"""
self.ssm.set_filter_method(filter_method, **kwargs)
[docs] def set_inversion_method(self, inversion_method=None, **kwargs):
"""
Set the inversion method
The Kalman filter may contain one matrix inversion: that of the
forecast error covariance matrix. The inversion method controls how and
if that inverse is performed.
Parameters
----------
inversion_method : int, optional
Bitmask value to set the inversion method to. See notes for
details.
**kwargs
Keyword arguments may be used to influence the inversion method by
setting individual boolean flags. See notes for details.
Notes
-----
This method is rarely used. See the corresponding function in the
`KalmanFilter` class for details.
"""
self.ssm.set_inversion_method(inversion_method, **kwargs)
[docs] def set_stability_method(self, stability_method=None, **kwargs):
"""
Set the numerical stability method
The Kalman filter is a recursive algorithm that may in some cases
suffer issues with numerical stability. The stability method controls
what, if any, measures are taken to promote stability.
Parameters
----------
stability_method : int, optional
Bitmask value to set the stability method to. See notes for
details.
**kwargs
Keyword arguments may be used to influence the stability method by
setting individual boolean flags. See notes for details.
Notes
-----
This method is rarely used. See the corresponding function in the
`KalmanFilter` class for details.
"""
self.ssm.set_stability_method(stability_method, **kwargs)
[docs] def set_conserve_memory(self, conserve_memory=None, **kwargs):
"""
Set the memory conservation method
By default, the Kalman filter computes a number of intermediate
matrices at each iteration. The memory conservation options control
which of those matrices are stored.
Parameters
----------
conserve_memory : int, optional
Bitmask value to set the memory conservation method to. See notes
for details.
**kwargs
Keyword arguments may be used to influence the memory conservation
method by setting individual boolean flags.
Notes
-----
This method is rarely used. See the corresponding function in the
`KalmanFilter` class for details.
"""
self.ssm.set_conserve_memory(conserve_memory, **kwargs)
[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 : 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.
Notes
-----
This method is rarely used. See the corresponding function in the
`KalmanSmoother` class for details.
"""
self.ssm.set_smoother_output(smoother_output, **kwargs)
[docs] def initialize_known(self, initial_state, initial_state_cov):
"""Initialize known"""
self.ssm.initialize_known(initial_state, initial_state_cov)
[docs] def initialize_approximate_diffuse(self, variance=None):
"""Initialize approximate diffuse"""
self.ssm.initialize_approximate_diffuse(variance)
[docs] def initialize_stationary(self):
"""Initialize stationary"""
self.ssm.initialize_stationary()
@property
def initialization(self):
return self.ssm.initialization
@initialization.setter
def initialization(self, value):
self.ssm.initialization = value
@property
def initial_variance(self):
return self.ssm.initial_variance
@initial_variance.setter
def initial_variance(self, value):
self.ssm.initial_variance = value
@property
def loglikelihood_burn(self):
return self.ssm.loglikelihood_burn
@loglikelihood_burn.setter
def loglikelihood_burn(self, value):
self.ssm.loglikelihood_burn = value
@property
def tolerance(self):
return self.ssm.tolerance
@tolerance.setter
def tolerance(self, value):
self.ssm.tolerance = value
def _validate_can_fix_params(self, param_names):
for param_name in param_names:
if param_name not in self.param_names:
raise ValueError('Invalid parameter name passed: "%s".'
% param_name)
[docs] @contextlib.contextmanager
def fix_params(self, params):
"""
Fix parameters to specific values (context manager)
Parameters
----------
params : dict
Dictionary describing the fixed parameter values, of the form
`param_name: fixed_value`. See the `param_names` property for valid
parameter names.
Examples
--------
>>> mod = sm.tsa.SARIMAX(endog, order=(1, 0, 1))
>>> with mod.fix_params({'ar.L1': 0.5}):
res = mod.fit()
"""
k_params = len(self.param_names)
# Initialization (this is done here rather than in the constructor
# because param_names may not be available at that point)
if self._fixed_params is None:
self._fixed_params = {}
self._params_index = dict(
zip(self.param_names, np.arange(k_params)))
# Cache the current fixed parameters
cache_fixed_params = self._fixed_params.copy()
cache_has_fixed_params = self._has_fixed_params
cache_fixed_params_index = self._fixed_params_index
cache_free_params_index = self._free_params_index
# Validate parameter names and values
all_fixed_param_names = (
set(params.keys()) | set(self._fixed_params.keys())
)
self._validate_can_fix_params(all_fixed_param_names)
# Set the new fixed parameters, keeping the order as given by
# param_names
self._fixed_params.update(params)
self._fixed_params = dict([
(name, self._fixed_params[name]) for name in self.param_names
if name in self._fixed_params])
# Update associated values
self._has_fixed_params = True
self._fixed_params_index = [self._params_index[key]
for key in self._fixed_params.keys()]
self._free_params_index = list(
set(np.arange(k_params)).difference(self._fixed_params_index))
try:
yield
finally:
# Reset the fixed parameters
self._has_fixed_params = cache_has_fixed_params
self._fixed_params = cache_fixed_params
self._fixed_params_index = cache_fixed_params_index
self._free_params_index = cache_free_params_index
[docs] def fit(self, start_params=None, transformed=True, includes_fixed=False,
cov_type=None, cov_kwds=None, method='lbfgs', maxiter=50,
full_output=1, disp=5, callback=None, return_params=False,
optim_score=None, optim_complex_step=None, optim_hessian=None,
flags=None, low_memory=False, **kwargs):
"""
Fits the model by maximum likelihood via Kalman filter.
Parameters
----------
start_params : array_like, optional
Initial guess of the solution for the loglikelihood maximization.
If None, the default is given by Model.start_params.
transformed : bool, optional
Whether or not `start_params` is already transformed. Default is
True.
includes_fixed : bool, optional
If parameters were previously fixed with the `fix_params` method,
this argument describes whether or not `start_params` also includes
the fixed parameters, in addition to the free parameters. Default
is False.
cov_type : str, optional
The `cov_type` keyword governs the method for calculating the
covariance matrix of parameter estimates. Can be one of:
- 'opg' for the outer product of gradient estimator
- 'oim' for the observed information matrix estimator, calculated
using the method of Harvey (1989)
- 'approx' for the observed information matrix estimator,
calculated using a numerical approximation of the Hessian matrix.
- 'robust' for an approximate (quasi-maximum likelihood) covariance
matrix that may be valid even in the presence of some
misspecifications. Intermediate calculations use the 'oim'
method.
- 'robust_approx' is the same as 'robust' except that the
intermediate calculations use the 'approx' method.
- 'none' for no covariance matrix calculation.
Default is 'opg' unless memory conservation is used to avoid
computing the loglikelihood values for each observation, in which
case the default is 'approx'.
cov_kwds : dict or None, optional
A dictionary of arguments affecting covariance matrix computation.
**opg, oim, approx, robust, robust_approx**
- 'approx_complex_step' : bool, optional - If True, numerical
approximations are computed using complex-step methods. If False,
numerical approximations are computed using finite difference
methods. Default is True.
- 'approx_centered' : bool, optional - If True, numerical
approximations computed using finite difference methods use a
centered approximation. Default is False.
method : str, optional
The `method` determines which solver from `scipy.optimize`
is used, and it can be chosen from among the following strings:
- 'newton' for Newton-Raphson
- 'nm' for Nelder-Mead
- 'bfgs' for Broyden-Fletcher-Goldfarb-Shanno (BFGS)
- 'lbfgs' for limited-memory BFGS with optional box constraints
- 'powell' for modified Powell's method
- 'cg' for conjugate gradient
- 'ncg' for Newton-conjugate gradient
- 'basinhopping' for global basin-hopping solver
The explicit arguments in `fit` are passed to the solver,
with the exception of the basin-hopping solver. Each
solver has several optional arguments that are not the same across
solvers. See the notes section below (or scipy.optimize) for the
available arguments and for the list of explicit arguments that the
basin-hopping solver supports.
maxiter : int, optional
The maximum number of iterations to perform.
full_output : bool, optional
Set to True to have all available output in the Results object's
mle_retvals attribute. The output is dependent on the solver.
See LikelihoodModelResults notes section for more information.
disp : bool, optional
Set to True to print convergence messages.
callback : callable callback(xk), optional
Called after each iteration, as callback(xk), where xk is the
current parameter vector.
return_params : bool, optional
Whether or not to return only the array of maximizing parameters.
Default is False.
optim_score : {'harvey', 'approx'} or None, optional
The method by which the score vector is calculated. 'harvey' uses
the method from Harvey (1989), 'approx' uses either finite
difference or complex step differentiation depending upon the
value of `optim_complex_step`, and None uses the built-in gradient
approximation of the optimizer. Default is None. This keyword is
only relevant if the optimization method uses the score.
optim_complex_step : bool, optional
Whether or not to use complex step differentiation when
approximating the score; if False, finite difference approximation
is used. Default is True. This keyword is only relevant if
`optim_score` is set to 'harvey' or 'approx'.
optim_hessian : {'opg','oim','approx'}, optional
The method by which the Hessian is numerically approximated. 'opg'
uses outer product of gradients, 'oim' uses the information
matrix formula from Harvey (1989), and 'approx' uses numerical
approximation. This keyword is only relevant if the
optimization method uses the Hessian matrix.
low_memory : bool, optional
If set to True, techniques are applied to substantially reduce
memory usage. If used, some features of the results object will
not be available (including smoothed results and in-sample
prediction), although out-of-sample forecasting is possible.
Default is False.
**kwargs
Additional keyword arguments to pass to the optimizer.
Returns
-------
results
Results object holding results from fitting a state space model.
See Also
--------
statsmodels.base.model.LikelihoodModel.fit
statsmodels.tsa.statespace.mlemodel.MLEResults
statsmodels.tsa.statespace.structural.UnobservedComponentsResults
"""
if start_params is None:
start_params = self.start_params
transformed = True
includes_fixed = True
# Update the score method
if optim_score is None and method == 'lbfgs':
kwargs.setdefault('approx_grad', True)
kwargs.setdefault('epsilon', 1e-5)
elif optim_score is None:
optim_score = 'approx'
# Check for complex step differentiation
if optim_complex_step is None:
optim_complex_step = not self.ssm._complex_endog
elif optim_complex_step and self.ssm._complex_endog:
raise ValueError('Cannot use complex step derivatives when data'
' or parameters are complex.')
# Standardize starting parameters
start_params = self.handle_params(start_params, transformed=True,
includes_fixed=includes_fixed)
# Unconstrain the starting parameters
if transformed:
start_params = self.untransform_params(start_params)
# Remove any fixed parameters
if self._has_fixed_params:
start_params = start_params[self._free_params_index]
# If all parameters are fixed, we are done
if self._has_fixed_params and len(start_params) == 0:
mlefit = Bunch(params=[], mle_retvals=None,
mle_settings=None)
else:
# Remove disallowed kwargs
disallow = (
"concentrate_scale",
"enforce_stationarity",
"enforce_invertibility"
)
kwargs = {k: v for k, v in kwargs.items() if k not in disallow}
# Maximum likelihood estimation
if flags is None:
flags = {}
flags.update({
'transformed': False,
'includes_fixed': False,
'score_method': optim_score,
'approx_complex_step': optim_complex_step
})
if optim_hessian is not None:
flags['hessian_method'] = optim_hessian
fargs = (flags,)
mlefit = super(MLEModel, self).fit(start_params, method=method,
fargs=fargs,
maxiter=maxiter,
full_output=full_output,
disp=disp, callback=callback,
skip_hessian=True, **kwargs)
# Just return the fitted parameters if requested
if return_params:
return self.handle_params(mlefit.params, transformed=False,
includes_fixed=False)
# Otherwise construct the results class if desired
else:
# Handle memory conservation option
if low_memory:
conserve_memory = self.ssm.conserve_memory
self.ssm.set_conserve_memory(MEMORY_CONSERVE)
# Perform filtering / smoothing
if (self.ssm.memory_no_predicted or self.ssm.memory_no_gain
or self.ssm.memory_no_smoothing):
func = self.filter
else:
func = self.smooth
res = func(mlefit.params, transformed=False, includes_fixed=False,
cov_type=cov_type, cov_kwds=cov_kwds)
res.mlefit = mlefit
res.mle_retvals = mlefit.mle_retvals
res.mle_settings = mlefit.mle_settings
# Reset memory conservation
if low_memory:
self.ssm.set_conserve_memory(conserve_memory)
return res
[docs] def fit_constrained(self, constraints, start_params=None, **fit_kwds):
"""
Fit the model with some parameters subject to equality constraints.
Parameters
----------
constraints : dict
Dictionary of constraints, of the form `param_name: fixed_value`.
See the `param_names` property for valid parameter names.
start_params : array_like, optional
Initial guess of the solution for the loglikelihood maximization.
If None, the default is given by Model.start_params.
**fit_kwds : keyword arguments
fit_kwds are used in the optimization of the remaining parameters.
Returns
-------
results : Results instance
Examples
--------
>>> mod = sm.tsa.SARIMAX(endog, order=(1, 0, 1))
>>> res = mod.fit_constrained({'ar.L1': 0.5})
"""
with self.fix_params(constraints):
res = self.fit(start_params, **fit_kwds)
return res
@property
def _res_classes(self):
return {'fit': (MLEResults, MLEResultsWrapper)}
def _wrap_results(self, params, result, return_raw, cov_type=None,
cov_kwds=None, results_class=None, wrapper_class=None):
if not return_raw:
# Wrap in a results object
result_kwargs = {}
if cov_type is not None:
result_kwargs['cov_type'] = cov_type
if cov_kwds is not None:
result_kwargs['cov_kwds'] = cov_kwds
if results_class is None:
results_class = self._res_classes['fit'][0]
if wrapper_class is None:
wrapper_class = self._res_classes['fit'][1]
res = results_class(self, params, result, **result_kwargs)
result = wrapper_class(res)
return result
[docs] def filter(self, params, transformed=True, includes_fixed=False,
complex_step=False, cov_type=None, cov_kwds=None,
return_ssm=False, results_class=None,
results_wrapper_class=None, low_memory=False, **kwargs):
"""
Kalman filtering
Parameters
----------
params : array_like
Array of parameters at which to evaluate the loglikelihood
function.
transformed : bool, optional
Whether or not `params` is already transformed. Default is True.
return_ssm : bool,optional
Whether or not to return only the state space output or a full
results object. Default is to return a full results object.
cov_type : str, optional
See `MLEResults.fit` for a description of covariance matrix types
for results object.
cov_kwds : dict or None, optional
See `MLEResults.get_robustcov_results` for a description required
keywords for alternative covariance estimators
low_memory : bool, optional
If set to True, techniques are applied to substantially reduce
memory usage. If used, some features of the results object will
not be available (including in-sample prediction), although
out-of-sample forecasting is possible. Default is False.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
"""
params = self.handle_params(params, transformed=transformed,
includes_fixed=includes_fixed)
self.update(params, transformed=True, includes_fixed=True,
complex_step=complex_step)
# Save the parameter names
self.data.param_names = self.param_names
if complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
# Handle memory conservation
if low_memory:
kwargs['conserve_memory'] = MEMORY_CONSERVE
# Get the state space output
result = self.ssm.filter(complex_step=complex_step, **kwargs)
# Wrap in a results object
return self._wrap_results(params, result, return_ssm, cov_type,
cov_kwds, results_class,
results_wrapper_class)
[docs] def smooth(self, params, transformed=True, includes_fixed=False,
complex_step=False, cov_type=None, cov_kwds=None,
return_ssm=False, results_class=None,
results_wrapper_class=None, **kwargs):
"""
Kalman smoothing
Parameters
----------
params : array_like
Array of parameters at which to evaluate the loglikelihood
function.
transformed : bool, optional
Whether or not `params` is already transformed. Default is True.
return_ssm : bool,optional
Whether or not to return only the state space output or a full
results object. Default is to return a full results object.
cov_type : str, optional
See `MLEResults.fit` for a description of covariance matrix types
for results object.
cov_kwds : dict or None, optional
See `MLEResults.get_robustcov_results` for a description required
keywords for alternative covariance estimators
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
"""
params = self.handle_params(params, transformed=transformed,
includes_fixed=includes_fixed)
self.update(params, transformed=True, includes_fixed=True,
complex_step=complex_step)
# Save the parameter names
self.data.param_names = self.param_names
if complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
# Get the state space output
result = self.ssm.smooth(complex_step=complex_step, **kwargs)
# Wrap in a results object
return self._wrap_results(params, result, return_ssm, cov_type,
cov_kwds, results_class,
results_wrapper_class)
_loglike_param_names = ['transformed', 'includes_fixed', 'complex_step']
_loglike_param_defaults = [True, False, False]
[docs] def loglike(self, params, *args, **kwargs):
"""
Loglikelihood evaluation
Parameters
----------
params : array_like
Array of parameters at which to evaluate the loglikelihood
function.
transformed : bool, optional
Whether or not `params` is already transformed. Default is True.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
See Also
--------
update : modifies the internal state of the state space model to
reflect new params
Notes
-----
[1]_ recommend maximizing the average likelihood to avoid scale issues;
this is done automatically by the base Model fit method.
References
----------
.. [1] Koopman, Siem Jan, Neil Shephard, and Jurgen A. Doornik. 1999.
Statistical Algorithms for Models in State Space Using SsfPack 2.2.
Econometrics Journal 2 (1): 107-60. doi:10.1111/1368-423X.00023.
"""
transformed, includes_fixed, complex_step, kwargs = _handle_args(
MLEModel._loglike_param_names, MLEModel._loglike_param_defaults,
*args, **kwargs)
params = self.handle_params(params, transformed=transformed,
includes_fixed=includes_fixed)
self.update(params, transformed=True, includes_fixed=True,
complex_step=complex_step)
if complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
loglike = self.ssm.loglike(complex_step=complex_step, **kwargs)
# Koopman, Shephard, and Doornik recommend maximizing the average
# likelihood to avoid scale issues, but the averaging is done
# automatically in the base model `fit` method
return loglike
[docs] def loglikeobs(self, params, transformed=True, includes_fixed=False,
complex_step=False, **kwargs):
"""
Loglikelihood evaluation
Parameters
----------
params : array_like
Array of parameters at which to evaluate the loglikelihood
function.
transformed : bool, optional
Whether or not `params` is already transformed. Default is True.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
See Also
--------
update : modifies the internal state of the Model to reflect new params
Notes
-----
[1]_ recommend maximizing the average likelihood to avoid scale issues;
this is done automatically by the base Model fit method.
References
----------
.. [1] Koopman, Siem Jan, Neil Shephard, and Jurgen A. Doornik. 1999.
Statistical Algorithms for Models in State Space Using SsfPack 2.2.
Econometrics Journal 2 (1): 107-60. doi:10.1111/1368-423X.00023.
"""
params = self.handle_params(params, transformed=transformed,
includes_fixed=includes_fixed)
# If we're using complex-step differentiation, then we cannot use
# Cholesky factorization
if complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
self.update(params, transformed=True, includes_fixed=True,
complex_step=complex_step)
return self.ssm.loglikeobs(complex_step=complex_step, **kwargs)
[docs] def simulation_smoother(self, simulation_output=None, **kwargs):
r"""
Retrieve a simulation smoother for the state space model.
Parameters
----------
simulation_output : int, optional
Determines which simulation smoother output is calculated.
Default is all (including state and disturbances).
**kwargs
Additional keyword arguments, used to set the simulation output.
See `set_simulation_output` for more details.
Returns
-------
SimulationSmoothResults
"""
return self.ssm.simulation_smoother(
simulation_output=simulation_output, **kwargs)
def _forecasts_error_partial_derivatives(self, params, transformed=True,
includes_fixed=False,
approx_complex_step=None,
approx_centered=False,
res=None, **kwargs):
params = np.array(params, ndmin=1)
# We cannot use complex-step differentiation with non-transformed
# parameters
if approx_complex_step is None:
approx_complex_step = transformed
if not transformed and approx_complex_step:
raise ValueError("Cannot use complex-step approximations to"
" calculate the observed_information_matrix"
" with untransformed parameters.")
# If we're using complex-step differentiation, then we cannot use
# Cholesky factorization
if approx_complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
# Get values at the params themselves
if res is None:
self.update(params, transformed=transformed,
includes_fixed=includes_fixed,
complex_step=approx_complex_step)
res = self.ssm.filter(complex_step=approx_complex_step, **kwargs)
# Setup
n = len(params)
# Compute partial derivatives w.r.t. forecast error and forecast
# error covariance
partials_forecasts_error = (
np.zeros((self.k_endog, self.nobs, n))
)
partials_forecasts_error_cov = (
np.zeros((self.k_endog, self.k_endog, self.nobs, n))
)
if approx_complex_step:
epsilon = _get_epsilon(params, 2, None, n)
increments = np.identity(n) * 1j * epsilon
for i, ih in enumerate(increments):
self.update(params + ih, transformed=transformed,
includes_fixed=includes_fixed,
complex_step=True)
_res = self.ssm.filter(complex_step=True, **kwargs)
partials_forecasts_error[:, :, i] = (
_res.forecasts_error.imag / epsilon[i]
)
partials_forecasts_error_cov[:, :, :, i] = (
_res.forecasts_error_cov.imag / epsilon[i]
)
elif not approx_centered:
epsilon = _get_epsilon(params, 2, None, n)
ei = np.zeros((n,), float)
for i in range(n):
ei[i] = epsilon[i]
self.update(params + ei, transformed=transformed,
includes_fixed=includes_fixed, complex_step=False)
_res = self.ssm.filter(complex_step=False, **kwargs)
partials_forecasts_error[:, :, i] = (
_res.forecasts_error - res.forecasts_error) / epsilon[i]
partials_forecasts_error_cov[:, :, :, i] = (
_res.forecasts_error_cov -
res.forecasts_error_cov) / epsilon[i]
ei[i] = 0.0
else:
epsilon = _get_epsilon(params, 3, None, n) / 2.
ei = np.zeros((n,), float)
for i in range(n):
ei[i] = epsilon[i]
self.update(params + ei, transformed=transformed,
includes_fixed=includes_fixed, complex_step=False)
_res1 = self.ssm.filter(complex_step=False, **kwargs)
self.update(params - ei, transformed=transformed,
includes_fixed=includes_fixed, complex_step=False)
_res2 = self.ssm.filter(complex_step=False, **kwargs)
partials_forecasts_error[:, :, i] = (
(_res1.forecasts_error - _res2.forecasts_error) /
(2 * epsilon[i]))
partials_forecasts_error_cov[:, :, :, i] = (
(_res1.forecasts_error_cov - _res2.forecasts_error_cov) /
(2 * epsilon[i]))
ei[i] = 0.0
return partials_forecasts_error, partials_forecasts_error_cov
def _score_complex_step(self, params, **kwargs):
# the default epsilon can be too small
# inversion_method = INVERT_UNIVARIATE | SOLVE_LU
epsilon = _get_epsilon(params, 2., None, len(params))
kwargs['transformed'] = True
kwargs['complex_step'] = True
return approx_fprime_cs(params, self.loglike, epsilon=epsilon,
kwargs=kwargs)
def _score_finite_difference(self, params, approx_centered=False,
**kwargs):
kwargs['transformed'] = True
return approx_fprime(params, self.loglike, kwargs=kwargs,
centered=approx_centered)
def _score_harvey(self, params, approx_complex_step=True, **kwargs):
score_obs = self._score_obs_harvey(
params, approx_complex_step=approx_complex_step, **kwargs)
return np.sum(score_obs, axis=0)
def _score_obs_harvey(self, params, approx_complex_step=True,
approx_centered=False, includes_fixed=False,
**kwargs):
"""
Score
Parameters
----------
params : array_like, optional
Array of parameters at which to evaluate the loglikelihood
function.
**kwargs
Additional keyword arguments to pass to the Kalman filter. See
`KalmanFilter.filter` for more details.
Notes
-----
This method is from Harvey (1989), section 3.4.5
References
----------
Harvey, Andrew C. 1990.
Forecasting, Structural Time Series Models and the Kalman Filter.
Cambridge University Press.
"""
params = np.array(params, ndmin=1)
n = len(params)
# Get values at the params themselves
self.update(params, transformed=True, includes_fixed=includes_fixed,
complex_step=approx_complex_step)
if approx_complex_step:
kwargs['inversion_method'] = INVERT_UNIVARIATE | SOLVE_LU
if 'transformed' in kwargs:
del kwargs['transformed']
res = self.ssm.filter(complex_step=approx_complex_step, **kwargs)
# Get forecasts error partials
partials_forecasts_error, partials_forecasts_error_cov = (
self._forecasts_error_partial_derivatives(
params, transformed=True, includes_fixed=includes_fixed,
approx_complex_step=approx_complex_step,
approx_centered=approx_centered, res=res, **kwargs))
# Compute partial derivatives w.r.t. likelihood function
partials = np.zeros((self.nobs, n))
k_endog = self.k_endog
for t in range(self.nobs):
inv_forecasts_error_cov = np.linalg.inv(
res.forecasts_error_cov[:, :, t])
for i in range(n):
partials[t, i] += np.trace(np.dot(
np.dot(inv_forecasts_error_cov,
partials_forecasts_error_cov[:, :, t, i]),
(np.eye(k_endog) -
np.dot(inv_forecasts_error_cov,
np.outer(res.forecasts_error[:, t],
res.forecasts_error[:, t])))))
# 2 * dv / di * F^{-1} v_t
# where x = F^{-1} v_t or F x = v
partials[t, i] += 2 * np.dot(
partials_forecasts_error[:, t, i],
np.dot(inv_forecasts_error_cov, res.forecasts_error[:, t]))
return -partials / 2.
_score_param_names = ['transformed', 'includes_fixed', 'score_method',
'approx_complex_step', 'approx_centered']
_score_param_defaults = [True, False, 'approx', None, False]
[docs] def score(self, params, *args, **kwargs):
"""
Compute the score function at params.
Parameters
----------
params : array_like
Array of parameters at which to evaluate the score.
*args
Additional positional arguments to the `loglike` method.
**kwargs
Additional keyword arguments to the `loglike` method.
Returns
-------
score : ndarray
Score, evaluated at `params`.
Notes
-----
This is a numerical approximation, calculated using first-order complex
step differentiation on the `loglike` method.
Both args and kwargs are necessary because the optimizer from
`fit` must call this function and only supports passing arguments via
args (for example `scipy.optimize.fmin_l_bfgs`).
"""
(transformed, includes_fixed, method, approx_complex_step,
approx_centered, kwargs) = (
_handle_args(MLEModel._score_param_names,
MLEModel._score_param_defaults, *args, **kwargs))
# For fit() calls, the method is called 'score_method' (to distinguish
# it from the method used for fit) but generally in kwargs the method
# will just be called 'method'
if 'method' in kwargs:
method = kwargs.pop('method')
if approx_complex_step is None:
approx_complex_step = not self.ssm._complex_endog
if approx_complex_step and self.ssm._complex_endog:
raise ValueError('Cannot use complex step derivatives when data'
' or parameters are complex.')
out = self.handle_params(
params, transformed=transformed, includes_fixed=includes_fixed,
return_jacobian=not transformed)
if transformed:
params = out
else:
params, transform_score = out
if method == 'harvey':
kwargs['includes_fixed'] = True
score = self._score_harvey(
params, approx_complex_step=approx_complex_step, **kwargs)
elif method == 'approx' and approx_complex_step:
kwargs['includes_fixed'] = True
score = self._score_complex_step(params, **kwargs)
elif method == 'approx':
kwargs['includes_fixed'] = True
score = self._score_finite_difference(
params, approx_centered=approx_centered, **kwargs)
else:
raise NotImplementedError('Invalid score method.')
if not transformed:
score = np.dot(transform_score, score)
if self._has_fixed_params and not includes_fixed:
score = score[self._free_params_index]
return score
[docs] def score_obs(self, params, method='approx', transformed=True,
includes_fixed=False, approx_complex_step=None,
approx_centered=False, **kwargs):
"""
Compute the score per observation, evaluated at params
Parameters
----------
params : array_like
Array of parameters at which to evaluate the score.
**kwargs
Additional arguments to the `loglike` method.
Returns
-------
score : ndarray
Score per observation, evaluated at `params`.
Notes
-----
This is a numerical approximation, calculated using first-order complex
step differentiation on the `loglikeobs` method.
"""
if not transformed and approx_complex_step:
raise ValueError("Cannot use complex-step approximations to"
" calculate the score at each observation"
" with untransformed parameters.")
if approx_complex_step is None:
approx_complex_step = not self.ssm._complex_endog
if approx_complex_step and self.ssm._complex_endog:
raise ValueError('Cannot use complex step derivatives when data'
' or parameters are complex.')
params = self.handle_params(params, transformed=True,
includes_fixed=includes_fixed)
kwargs['transformed'] = transformed
kwargs['includes_fixed'] = True
if method == 'harvey':
score = self._score_obs_harvey(
params, approx_complex_step=approx_complex_step, **kwargs)
elif method == 'approx' and approx_complex_step:
# the default epsilon can be too small
epsilon = _get_epsilon(params, 2., None, len(params))
kwargs['complex_step'] = True
score = approx_fprime_cs(params, self.loglikeobs, epsilon=epsilon,
kwargs=kwargs)
elif method == 'approx':
score = approx_fprime(params, self.loglikeobs, kwargs=kwargs,
centered=approx_centered)
else:
raise NotImplementedError('Invalid scoreobs method.')
return score
_hessian_param_names = ['transformed', 'hessian_method',
'approx_complex_step', 'approx_centered']
_hessian_param_defaults = [True, 'approx', None, False]
[docs] def hessian(self, params, *args, **kwargs):
r"""
Hessian matrix of the likelihood function, evaluated at the given
parameters
Parameters
----------
params : array_like
Array of parameters at which to evaluate the hessian.
*args
Additional positional arguments to the `loglike` method.
**kwargs
Additional keyword arguments to the `loglike` method.
Returns
-------
hessian : ndarray
Hessian matrix evaluated at `params`
Notes
-----
This is a numerical approximation.
Both args and kwargs are necessary because the optimizer from
`fit` must call this function and only supports passing arguments via
args (for example `scipy.optimize.fmin_l_bfgs`).
"""
transformed, method, approx_complex_step, approx_centered, kwargs = (
_handle_args(MLEModel._hessian_param_names,
MLEModel._hessian_param_defaults,
*args, **kwargs))
# For fit() calls, the method is called 'hessian_method' (to
# distinguish it from the method used for fit) but generally in kwargs
# the method will just be called 'method'
if 'method' in kwargs:
method = kwargs.pop('method')
if not transformed and approx_complex_step:
raise ValueError("Cannot use complex-step approximations to"
" calculate the hessian with untransformed"
" parameters.")
if approx_complex_step is None:
approx_complex_step = not self.ssm._complex_endog
if approx_complex_step and self.ssm._complex_endog:
raise ValueError('Cannot use complex step derivatives when data'
' or parameters are complex.')
if method == 'oim':
hessian = self._hessian_oim(
params, transformed=transformed,
approx_complex_step=approx_complex_step,
approx_centered=approx_centered, **kwargs)
elif method == 'opg':
hessian = self._hessian_opg(
params, transformed=transformed,
approx_complex_step=approx_complex_step,
approx_centered=approx_centered, **kwargs)
elif method == 'approx' and approx_complex_step:
hessian = self._hessian_complex_step(
params, transformed=transformed, **kwargs)
elif method == 'approx':
hessian = self._hessian_finite_difference(
params, transformed=transformed,
approx_centered=approx_centered, **kwargs)
else:
raise NotImplementedError('Invalid Hessian calculation method.')
return hessian
def _hessian_oim(self, params, **kwargs):
"""
Hessian matrix computed using the Harvey (1989) information matrix
"""
return -self.observed_information_matrix(params, **kwargs)
def _hessian_opg(self, params, **kwargs):
"""
Hessian matrix computed using the outer product of gradients
information matrix
"""
return -self.opg_information_matrix(params, **kwargs)
def _hessian_finite_difference(self, params, approx_centered=False,
**kwargs):
params = np.array(params, ndmin=1)
warnings.warn('Calculation of the Hessian using finite differences'
' is usually subject to substantial approximation'
' errors.', PrecisionWarning)
if not approx_centered:
epsilon = _get_epsilon(params, 3, None, len(params))
else:
epsilon = _get_epsilon(params, 4, None, len(params)) / 2
hessian = approx_fprime(params, self._score_finite_difference,
epsilon=epsilon, kwargs=kwargs,
centered=approx_centered)
return hessian / (self.nobs - self.ssm.loglikelihood_burn)
def _hessian_complex_step(self, params, **kwargs):
"""
Hessian matrix computed by second-order complex-step differentiation
on the `loglike` function.
"""
# the default epsilon can be too small
epsilon = _get_epsilon(params, 3., None, len(params))
kwargs['transformed'] = True
kwargs['complex_step'] = True
hessian = approx_hess_cs(
params, self.loglike, epsilon=epsilon, kwargs=kwargs)
return hessian / (self.nobs - self.ssm.loglikelihood_burn)
@property
def start_params(self):
"""
(array) Starting parameters for maximum likelihood estimation.
"""
if hasattr(self, '_start_params'):
return self._start_params
else:
raise NotImplementedError
@property
def param_names(self):
"""
(list of str) List of human readable parameter names (for parameters
actually included in the model).
"""
if hasattr(self, '_param_names'):
return self._param_names
else:
try:
names = ['param.%d' % i for i in range(len(self.start_params))]
except NotImplementedError:
names = []
return names
@property
def state_names(self):
"""
(list of str) List of human readable names for unobserved states.
"""
if hasattr(self, '_state_names'):
return self._state_names
else:
names = ['state.%d' % i for i in range(self.k_states)]
return names
[docs] def handle_params(self, params, transformed=True, includes_fixed=False,
return_jacobian=False):
"""
Ensure model parameters satisfy shape and other requirements
"""
params = np.array(params, ndmin=1)
# Never want integer dtype, so convert to floats
if np.issubdtype(params.dtype, np.integer):
params = params.astype(np.float64)
if not includes_fixed and self._has_fixed_params:
k_params = len(self.param_names)
new_params = np.zeros(k_params, dtype=params.dtype) * np.nan
new_params[self._free_params_index] = params
params = new_params
if not transformed:
# It may be the case that the transformation relies on having
# "some" (non-NaN) values for the fixed parameters, even if we will
# not actually be transforming the fixed parameters (as they will)
# be set below regardless
if not includes_fixed and self._has_fixed_params:
params[self._fixed_params_index] = (
list(self._fixed_params.values()))
if return_jacobian:
transform_score = self.transform_jacobian(params)
params = self.transform_params(params)
if not includes_fixed and self._has_fixed_params:
params[self._fixed_params_index] = (
list(self._fixed_params.values()))
return (params, transform_score) if return_jacobian else params
[docs] def update(self, params, transformed=True, includes_fixed=False,
complex_step=False):
"""
Update the parameters of the model
Parameters
----------
params : array_like
Array of new parameters.
transformed : bool, optional
Whether or not `params` is already transformed. If set to False,
`transform_params` is called. Default is True.
Returns
-------
params : array_like
Array of parameters.
Notes
-----
Since Model is a base class, this method should be overridden by
subclasses to perform actual updating steps.
"""
return self.handle_params(params=params, transformed=transformed,
includes_fixed=includes_fixed)
def _validate_out_of_sample_exog(self, exog, out_of_sample):
"""
Validate given `exog` as satisfactory for out-of-sample operations
Parameters
----------
exog : array_like or None
New observations of exogenous regressors, if applicable.
out_of_sample : int
Number of new observations required.
Returns
-------
exog : array or None
A numpy array of shape (out_of_sample, k_exog) if the model
contains an `exog` component, or None if it does not.
"""
if out_of_sample and self.k_exog > 0:
if exog is None:
raise ValueError('Out-of-sample operations in a model'
' with a regression component require'
' additional exogenous values via the'
' `exog` argument.')
exog = np.array(exog)
required_exog_shape = (out_of_sample, self.k_exog)
try:
exog = exog.reshape(required_exog_shape)
except ValueError:
raise ValueError('Provided exogenous values are not of the'
' appropriate shape. Required %s, got %s.'
% (str(required_exog_shape),
str(exog.shape)))
elif self.k_exog > 0 and exog is not None:
exog = None
warnings.warn('Exogenous array provided, but additional data'
' is not required. `exog` argument ignored.',
ValueWarning)
return exog
def _get_extension_time_varying_matrices(
self, params, exog, out_of_sample, extend_kwargs=None,
transformed=True, includes_fixed=False, **kwargs):
"""
Get updated time-varying state space system matrices
Parameters
----------
params : array_like
Array of parameters used to construct the time-varying system
matrices.
exog : array_like or None
New observations of exogenous regressors, if applicable.
out_of_sample : int
Number of new observations required.
extend_kwargs : dict, optional
Dictionary of keyword arguments to pass to the state space model
constructor. For example, for an SARIMAX state space model, this
could be used to pass the `concentrate_scale=True` keyword
argument. Any arguments that are not explicitly set in this
dictionary will be copied from the current model instance.
transformed : bool, optional
Whether or not `start_params` is already transformed. Default is
True.
includes_fixed : bool, optional
If parameters were previously fixed with the `fix_params` method,
this argument describes whether or not `start_params` also includes
the fixed parameters, in addition to the free parameters. Default
is False.
"""
# Get the appropriate exog for the extended sample
exog = self._validate_out_of_sample_exog(exog, out_of_sample)
# Create extended model
if extend_kwargs is None:
extend_kwargs = {}
# Handle trend offset for extended model
if getattr(self, 'k_trend', 0) > 0 and hasattr(self, 'trend_offset'):
extend_kwargs.setdefault(
'trend_offset', self.trend_offset + self.nobs)
mod_extend = self.clone(
endog=np.zeros((out_of_sample, self.k_endog)), exog=exog,
**extend_kwargs)
mod_extend.update(params, transformed=transformed,
includes_fixed=includes_fixed)
# Retrieve the extensions to the time-varying system matrices and
# put them in kwargs
for name in self.ssm.shapes.keys():
if name == 'obs' or name in kwargs:
continue
original = getattr(self.ssm, name)
extended = getattr(mod_extend.ssm, name)
so = original.shape[-1]
se = extended.shape[-1]
if ((so > 1 or se > 1) or (
so == 1 and self.nobs == 1 and
np.any(original[..., 0] != extended[..., 0]))):
kwargs[name] = extended[..., -out_of_sample:]
return kwargs
[docs] def simulate(self, params, nsimulations, measurement_shocks=None,
state_shocks=None, initial_state=None, anchor=None,
repetitions=None, exog=None, extend_model=None,
extend_kwargs=None, transformed=True, includes_fixed=False,
**kwargs):
r"""
Simulate a new time series following the state space model
Parameters
----------
params : array_like
Array of parameters to use in constructing the state space
representation to use when simulating.
nsimulations : int
The number of observations to simulate. If the model is
time-invariant this can be any number. If the model is
time-varying, then this number must be less than or equal to the
number of observations.
measurement_shocks : array_like, optional
If specified, these are the shocks to the measurement equation,
:math:`\varepsilon_t`. If unspecified, these are automatically
generated using a pseudo-random number generator. If specified,
must be shaped `nsimulations` x `k_endog`, where `k_endog` is the
same as in the state space model.
state_shocks : array_like, optional
If specified, these are the shocks to the state equation,
:math:`\eta_t`. If unspecified, these are automatically
generated using a pseudo-random number generator. If specified,
must be shaped `nsimulations` x `k_posdef` where `k_posdef` is the
same as in the state space model.
initial_state : array_like, optional
If specified, this is the initial state vector to use in
simulation, which should be shaped (`k_states` x 1), where
`k_states` is the same as in the state space model. If unspecified,
but the model has been initialized, then that initialization is
used. This must be specified if `anchor` is anything other than
"start" or 0 (or else you can use the `simulate` method on a
results object rather than on the model object).
anchor : int, str, or datetime, optional
First period for simulation. The simulation will be conditional on
all existing datapoints prior to the `anchor`. Type depends on the
index of the given `endog` in the model. Two special cases are the
strings 'start' and 'end'. `start` refers to beginning the
simulation at the first period of the sample, and `end` refers to
beginning the simulation at the first period after the sample.
Integer values can run from 0 to `nobs`, or can be negative to
apply negative indexing. Finally, if a date/time index was provided
to the model, then this argument can be a date string to parse or a
datetime type. Default is 'start'.
repetitions : int, optional
Number of simulated paths to generate. Default is 1 simulated path.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
transformed : bool, optional
Whether or not `params` is already transformed. Default is
True.
includes_fixed : bool, optional
If parameters were previously fixed with the `fix_params` method,
this argument describes whether or not `params` also includes
the fixed parameters, in addition to the free parameters. Default
is False.
Returns
-------
simulated_obs : ndarray
An array of simulated observations. If `repetitions=None`, then it
will be shaped (nsimulations x k_endog) or (nsimulations,) if
`k_endog=1`. Otherwise it will be shaped
(nsimulations x k_endog x repetitions). If the model was given
Pandas input then the output will be a Pandas object. If
`k_endog > 1` and `repetitions` is not None, then the output will
be a Pandas DataFrame that has a MultiIndex for the columns, with
the first level containing the names of the `endog` variables and
the second level containing the repetition number.
See Also
--------
impulse_responses
Impulse response functions
"""
# Make sure the model class has the current parameters
self.update(params, transformed=transformed,
includes_fixed=includes_fixed)
# Get the starting location
if anchor is None or anchor == 'start':
iloc = 0
elif anchor == 'end':
iloc = self.nobs
else:
iloc, _, _ = self._get_index_loc(anchor)
if isinstance(iloc, slice):
iloc = iloc.start
if iloc < 0:
iloc = self.nobs + iloc
if iloc > self.nobs:
raise ValueError('Cannot anchor simulation outside of the sample.')
if iloc > 0 and initial_state is None:
raise ValueError('If `anchor` is after the start of the sample,'
' must provide a value for `initial_state`.')
# Get updated time-varying system matrices in **kwargs, if necessary
out_of_sample = max(iloc + nsimulations - self.nobs, 0)
if extend_model is None:
extend_model = self.exog is not None or not self.ssm.time_invariant
if out_of_sample and extend_model:
kwargs = self._get_extension_time_varying_matrices(
params, exog, out_of_sample, extend_kwargs,
transformed=transformed, includes_fixed=includes_fixed,
**kwargs)
# Standardize the dimensions of the initial state
if initial_state is not None:
initial_state = np.array(initial_state)
if initial_state.ndim < 2:
initial_state = np.atleast_2d(initial_state).T
# Construct a model that represents the simulation period
end = min(self.nobs, iloc + nsimulations)
nextend = iloc + nsimulations - end
sim_model = self.ssm.extend(np.empty((nextend, self.k_endog)),
start=iloc, end=end, **kwargs)
# Simulate the data
_repetitions = 1 if repetitions is None else repetitions
sim = np.zeros((nsimulations, self.k_endog, _repetitions))
for i in range(_repetitions):
initial_state_variates = None
if initial_state is not None:
if initial_state.shape[1] == 1:
initial_state_variates = initial_state[:, 0]
else:
initial_state_variates = initial_state[:, i]
# TODO: allow specifying measurement / state shocks for each
# repetition?
out, _ = sim_model.simulate(
nsimulations, measurement_shocks, state_shocks,
initial_state_variates)
sim[:, :, i] = out
# Wrap data / squeeze where appropriate
use_pandas = isinstance(self.data, PandasData)
index = None
if use_pandas:
_, _, _, index = self._get_prediction_index(
iloc, iloc + nsimulations - 1)
# If `repetitions` isn't set, we squeeze the last dimension(s)
if repetitions is None:
if self.k_endog == 1:
sim = sim[:, 0, 0]
if use_pandas:
sim = pd.Series(sim, index=index, name=self.endog_names)
else:
sim = sim[:, :, 0]
if use_pandas:
sim = pd.DataFrame(sim, index=index,
columns=self.endog_names)
elif use_pandas:
shape = sim.shape
endog_names = self.endog_names
if not isinstance(endog_names, list):
endog_names = [endog_names]
columns = pd.MultiIndex.from_product([endog_names,
np.arange(shape[2])])
sim = pd.DataFrame(sim.reshape(shape[0], shape[1] * shape[2]),
index=index, columns=columns)
return sim
[docs] def impulse_responses(self, params, steps=1, impulse=0,
orthogonalized=False, cumulative=False, anchor=None,
exog=None, extend_model=None, extend_kwargs=None,
transformed=True, includes_fixed=False, **kwargs):
"""
Impulse response function
Parameters
----------
params : array_like
Array of model parameters.
steps : int, optional
The number of steps for which impulse responses are calculated.
Default is 1. Note that for time-invariant models, the initial
impulse is not counted as a step, so if `steps=1`, the output will
have 2 entries.
impulse : int, str or array_like
If an integer, the state innovation to pulse; must be between 0
and `k_posdef-1`. If a str, it indicates which column of df
the unit (1) impulse is given.
Alternatively, a custom impulse vector may be provided; must be
shaped `k_posdef x 1`.
orthogonalized : bool, optional
Whether or not to perform impulse using orthogonalized innovations.
Note that this will also affect custum `impulse` vectors. Default
is False.
cumulative : bool, optional
Whether or not to return cumulative impulse responses. Default is
False.
anchor : int, str, or datetime, optional
Time point within the sample for the state innovation impulse. Type
depends on the index of the given `endog` in the model. Two special
cases are the strings 'start' and 'end', which refer to setting the
impulse at the first and last points of the sample, respectively.
Integer values can run from 0 to `nobs - 1`, or can be negative to
apply negative indexing. Finally, if a date/time index was provided
to the model, then this argument can be a date string to parse or a
datetime type. Default is 'start'.
exog : array_like, optional
New observations of exogenous regressors for our-of-sample periods,
if applicable.
transformed : bool, optional
Whether or not `params` is already transformed. Default is
True.
includes_fixed : bool, optional
If parameters were previously fixed with the `fix_params` method,
this argument describes whether or not `params` also includes
the fixed parameters, in addition to the free parameters. Default
is False.
**kwargs
If the model has time-varying design or transition matrices and the
combination of `anchor` and `steps` implies creating impulse
responses for the out-of-sample period, then these matrices must
have updated values provided for the out-of-sample steps. For
example, if `design` is a time-varying component, `nobs` is 10,
`anchor=1`, and `steps` is 15, a (`k_endog` x `k_states` x 7)
matrix must be provided with the new design matrix values.
Returns
-------
impulse_responses : ndarray
Responses for each endogenous variable due to the impulse
given by the `impulse` argument. For a time-invariant model, the
impulse responses are given for `steps + 1` elements (this gives
the "initial impulse" followed by `steps` responses for the
important cases of VAR and SARIMAX models), while for time-varying
models the impulse responses are only given for `steps` elements
(to avoid having to unexpectedly provide updated time-varying
matrices).
See Also
--------
simulate
Simulate a time series according to the given state space model,
optionally with specified series for the innovations.
Notes
-----
Intercepts in the measurement and state equation are ignored when
calculating impulse responses.
TODO: add an option to allow changing the ordering for the
orthogonalized option. Will require permuting matrices when
constructing the extended model.
"""
# Make sure the model class has the current parameters
self.update(params, transformed=transformed,
includes_fixed=includes_fixed)
# For time-invariant models, add an additional `step`. This is the
# default for time-invariant models based on the expected behavior for
# ARIMA and VAR models: we want to record the initial impulse and also
# `steps` values of the responses afterwards.
# Note: we don't modify `steps` itself, because
# `KalmanFilter.impulse_responses` also adds an additional step in this
# case (this is so that there isn't different behavior when calling
# this method versus that method). We just need to also keep track of
# this here because we need to generate the correct extended model.
additional_steps = 0
if (self.ssm._design.shape[2] == 1 and
self.ssm._transition.shape[2] == 1 and
self.ssm._selection.shape[2] == 1):
additional_steps = 1
# Get the starting location
if anchor is None or anchor == 'start':
iloc = 0
elif anchor == 'end':
iloc = self.nobs - 1
else:
iloc, _, _ = self._get_index_loc(anchor)
if isinstance(iloc, slice):
iloc = iloc.start
if iloc < 0:
iloc = self.nobs + iloc
if iloc >= self.nobs:
raise ValueError('Cannot anchor impulse responses outside of the'
' sample.')
time_invariant = (
self.ssm._design.shape[2] == self.ssm._obs_cov.shape[2] ==
self.ssm._transition.shape[2] == self.ssm._selection.shape[2] ==
self.ssm._state_cov.shape[2] == 1)
# Get updated time-varying system matrices in **kwargs, if necessary
# (Note: KalmanFilter adds 1 to steps to account for the first impulse)
out_of_sample = max(
iloc + (steps + additional_steps + 1) - self.nobs, 0)
if extend_model is None:
extend_model = self.exog is not None and not time_invariant
if out_of_sample and extend_model:
kwargs = self._get_extension_time_varying_matrices(
params, exog, out_of_sample, extend_kwargs,
transformed=transformed, includes_fixed=includes_fixed,
**kwargs)
# Special handling for matrix terms that are time-varying but
# irrelevant for impulse response functions. Must be set since
# ssm.extend() requires that we pass new matrices for these, but they
# are ignored for IRF purposes.
end = min(self.nobs, iloc + steps + additional_steps)
nextend = iloc + (steps + additional_steps + 1) - end
if ('obs_intercept' not in kwargs and
self.ssm._obs_intercept.shape[1] > 1):
kwargs['obs_intercept'] = np.zeros((self.k_endog, nextend))
if ('state_intercept' not in kwargs and
self.ssm._state_intercept.shape[1] > 1):
kwargs['state_intercept'] = np.zeros((self.k_states, nextend))
if 'obs_cov' not in kwargs and self.ssm._obs_cov.shape[2] > 1:
kwargs['obs_cov'] = np.zeros((self.k_endog, self.k_endog, nextend))
# Special handling for matrix terms that are time-varying but
# only the value at the anchor matters for IRF purposes.
if 'state_cov' not in kwargs and self.ssm._state_cov.shape[2] > 1:
tmp = np.zeros((self.ssm.k_posdef, self.ssm.k_posdef, nextend))
tmp[:] = self['state_cov', :, :, iloc:iloc + 1]
kwargs['state_cov'] = tmp
if 'selection' not in kwargs and self.ssm._selection.shape[2] > 1:
tmp = np.zeros((self.k_states, self.ssm.k_posdef, nextend))
tmp[:] = self['selection', :, :, iloc:iloc + 1]
kwargs['selection'] = tmp
# Construct a model that represents the simulation period
sim_model = self.ssm.extend(np.empty((nextend, self.k_endog)),
start=iloc, end=end, **kwargs)
# Compute the impulse responses
# Convert endog name to index
use_pandas = isinstance(self.data, PandasData)
if type(impulse) == str:
if not use_pandas:
raise ValueError('Endog must be pd.DataFrame.')
impulse = self.endog_names.index(impulse)
irfs = sim_model.impulse_responses(
steps, impulse, orthogonalized, cumulative)
# IRF is (nobs x k_endog); do not want to squeeze in case of steps = 1
if irfs.shape[1] == 1:
irfs = irfs[:, 0]
# Wrap data / squeeze where appropriate
if use_pandas:
if self.k_endog == 1:
irfs = pd.Series(irfs, name=self.endog_names)
else:
irfs = pd.DataFrame(irfs, columns=self.endog_names)
return irfs
[docs]class MLEResults(tsbase.TimeSeriesModelResults):
r"""
Class to hold results from fitting a state space model.
Parameters
----------
model : MLEModel instance
The fitted model instance
params : ndarray
Fitted parameters
filter_results : KalmanFilter instance
The underlying state space model and Kalman filter output
Attributes
----------
model : Model instance
A reference to the model that was fit.
filter_results : KalmanFilter instance
The underlying state space model and Kalman filter output
nobs : float
The number of observations used to fit the model.
params : ndarray
The parameters of the model.
scale : float
This is currently set to 1.0 unless the model uses concentrated
filtering.
See Also
--------
MLEModel
statsmodels.tsa.statespace.kalman_filter.FilterResults
statsmodels.tsa.statespace.representation.FrozenRepresentation
"""
def __init__(self, model, params, results, cov_type=None, cov_kwds=None,
**kwargs):
self.data = model.data
scale = results.scale
tsbase.TimeSeriesModelResults.__init__(self, model, params,
normalized_cov_params=None,
scale=scale)
# Save the fixed parameters
self._has_fixed_params = self.model._has_fixed_params
self._fixed_params_index = self.model._fixed_params_index
self._free_params_index = self.model._free_params_index
# TODO: seems like maybe self.fixed_params should be the dictionary
# itself, not just the keys?
if self._has_fixed_params:
self._fixed_params = self.model._fixed_params.copy()
self.fixed_params = list(self._fixed_params.keys())
else:
self._fixed_params = None
self.fixed_params = []
self.param_names = [
'%s (fixed)' % name if name in self.fixed_params else name
for name in (self.data.param_names or [])]
# Save the state space representation output
self.filter_results = results
if isinstance(results, SmootherResults):
self.smoother_results = results
else:
self.smoother_results = None
# Dimensions
self.nobs = self.filter_results.nobs
self.nobs_diffuse = self.filter_results.nobs_diffuse
if self.nobs_diffuse > 0 and self.loglikelihood_burn > 0:
warnings.warn('Care should be used when applying a loglikelihood'
' burn to a model with exact diffuse initialization.'
' Some results objects, e.g. degrees of freedom,'
' expect only one of the two to be set.')
# This only excludes explicitly burned (usually approximate diffuse)
# periods but does not exclude exact diffuse periods. This is
# because the loglikelihood remains valid for the initial periods in
# the exact diffuse case (see DK, 2012, section 7.2) and so also do
# e.g. information criteria (see DK, 2012, section 7.4) and the score
# vector (see DK, 2012, section 7.3.3, equation 7.15).
# However, other objects should be excluded in the diffuse periods
# (e.g. the diffuse forecast errors, so in some cases a different
# nobs_effective will have to be computed and used)
self.nobs_effective = self.nobs - self.loglikelihood_burn
P = self.filter_results.initial_diffuse_state_cov
self.k_diffuse_states = 0 if P is None else np.sum(np.diagonal(P) == 1)
# Degrees of freedom (see DK 2012, section 7.4)
k_free_params = self.params.size - len(self.fixed_params)
self.df_model = (k_free_params + self.k_diffuse_states
+ self.filter_results.filter_concentrated)
self.df_resid = self.nobs_effective - self.df_model
# Setup covariance matrix notes dictionary
if not hasattr(self, 'cov_kwds'):
self.cov_kwds = {}
if cov_type is None:
cov_type = 'approx' if results.memory_no_likelihood else 'opg'
self.cov_type = cov_type
# Setup the cache
self._cache = {}
# Handle covariance matrix calculation
if cov_kwds is None:
cov_kwds = {}
self._cov_approx_complex_step = (
cov_kwds.pop('approx_complex_step', True))
self._cov_approx_centered = cov_kwds.pop('approx_centered', False)
try:
self._rank = None
self._get_robustcov_results(cov_type=cov_type, use_self=True,
**cov_kwds)
except np.linalg.LinAlgError:
self._rank = 0
k_params = len(self.params)
self.cov_params_default = np.zeros((k_params, k_params)) * np.nan
self.cov_kwds['cov_type'] = (
'Covariance matrix could not be calculated: singular.'
' information matrix.')
self.model.update(self.params, transformed=True, includes_fixed=True)
# References of filter and smoother output
extra_arrays = [
'filtered_state', 'filtered_state_cov', 'predicted_state',
'predicted_state_cov', 'forecasts', 'forecasts_error',
'forecasts_error_cov', 'standardized_forecasts_error',
'forecasts_error_diffuse_cov', 'predicted_diffuse_state_cov',
'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']
for name in extra_arrays:
setattr(self, name, getattr(self.filter_results, name, None))
# Remove too-short results when memory conservation was used
if self.filter_results.memory_no_forecast_mean:
self.forecasts = None
self.forecasts_error = None
if self.filter_results.memory_no_forecast_cov:
self.forecasts_error_cov = None
if self.filter_results.memory_no_predicted_mean:
self.predicted_state = None
if self.filter_results.memory_no_predicted_cov:
self.predicted_state_cov = None
if self.filter_results.memory_no_filtered_mean:
self.filtered_state = None
if self.filter_results.memory_no_filtered_cov:
self.filtered_state_cov = None
if self.filter_results.memory_no_gain:
pass
if self.filter_results.memory_no_smoothing:
pass
if self.filter_results.memory_no_std_forecast:
self.standardized_forecasts_error = None
# Save more convenient access to states
# (will create a private attribute _states here and provide actual
# access via a getter, so that we can e.g. issue a warning in the case
# that a useless Pandas index was given in the model specification)
self._states = SimpleNamespace()
use_pandas = isinstance(self.data, PandasData)
index = self.model._index
columns = self.model.state_names
# Predicted states
# Note: a complication here is that we also include the initial values
# here, so that we need an extended index in the Pandas case
if (self.predicted_state is None or
self.filter_results.memory_no_predicted_mean):
self._states.predicted = None
elif use_pandas:
extended_index = self.model._get_index_with_final_state()
self._states.predicted = pd.DataFrame(
self.predicted_state.T, index=extended_index, columns=columns)
else:
self._states.predicted = self.predicted_state.T
if (self.predicted_state_cov is None or
self.filter_results.memory_no_predicted_cov):
self._states.predicted_cov = None
elif use_pandas:
extended_index = self.model._get_index_with_final_state()
tmp = np.transpose(self.predicted_state_cov, (2, 0, 1))
self._states.predicted_cov = pd.DataFrame(
np.reshape(tmp, (tmp.shape[0] * tmp.shape[1], tmp.shape[2])),
index=pd.MultiIndex.from_product(
[extended_index, columns]).swaplevel(),
columns=columns)
else:
self._states.predicted_cov = np.transpose(
self.predicted_state_cov, (2, 0, 1))
# Filtered states
if (self.filtered_state is None or
self.filter_results.memory_no_filtered_mean):
self._states.filtered = None
elif use_pandas:
self._states.filtered = pd.DataFrame(
self.filtered_state.T, index=index, columns=columns)
else:
self._states.filtered = self.filtered_state.T
if (self.filtered_state_cov is None or
self.filter_results.memory_no_filtered_cov):
self._states.filtered_cov = None
elif use_pandas:
tmp = np.transpose(self.filtered_state_cov, (2, 0, 1))
self._states.filtered_cov = pd.DataFrame(
np.reshape(tmp, (tmp.shape[0] * tmp.shape[1], tmp.shape[2])),
index=pd.MultiIndex.from_product([index, columns]).swaplevel(),
columns=columns)
else:
self._states.filtered_cov = np.transpose(
self.filtered_state_cov, (2, 0, 1))
# Smoothed states
if self.smoothed_state is None:
self._states.smoothed = None
elif use_pandas:
self._states.smoothed = pd.DataFrame(
self.smoothed_state.T, index=index, columns=columns)
else:
self._states.smoothed = self.smoothed_state.T
if self.smoothed_state_cov is None:
self._states.smoothed_cov = None
elif use_pandas:
tmp = np.transpose(self.smoothed_state_cov, (2, 0, 1))
self._states.smoothed_cov = pd.DataFrame(
np.reshape(tmp, (tmp.shape[0] * tmp.shape[1], tmp.shape[2])),
index=pd.MultiIndex.from_product([index, columns]).swaplevel(),
columns=columns)
else:
self._states.smoothed_cov = np.transpose(
self.smoothed_state_cov, (2, 0, 1))
# Handle removing data
self._data_attr_model = getattr(self, '_data_attr_model', [])
self._data_attr_model.extend(['ssm'])
self._data_attr.extend(extra_arrays)
self._data_attr.extend(['filter_results', 'smoother_results'])
def _get_robustcov_results(self, cov_type='opg', **kwargs):
"""
Create new results instance with specified covariance estimator as
default
Note: creating new results instance currently not supported.
Parameters
----------
cov_type : str
the type of covariance matrix estimator to use. See Notes below
kwargs : depends on cov_type
Required or optional arguments for covariance calculation.
See Notes below.
Returns
-------
results : results instance
This method creates a new results instance with the requested
covariance as the default covariance of the parameters.
Inferential statistics like p-values and hypothesis tests will be
based on this covariance matrix.
Notes
-----
The following covariance types and required or optional arguments are
currently available:
- 'opg' for the outer product of gradient estimator
- 'oim' for the observed information matrix estimator, calculated
using the method of Harvey (1989)
- 'approx' for the observed information matrix estimator,
calculated using a numerical approximation of the Hessian matrix.
Uses complex step approximation by default, or uses finite
differences if `approx_complex_step=False` in the `cov_kwds`
dictionary.
- 'robust' for an approximate (quasi-maximum likelihood) covariance
matrix that may be valid even in the presence of some
misspecifications. Intermediate calculations use the 'oim'
method.
- 'robust_approx' is the same as 'robust' except that the
intermediate calculations use the 'approx' method.
- 'none' for no covariance matrix calculation.
"""
from statsmodels.base.covtype import descriptions
use_self = kwargs.pop('use_self', False)
if use_self:
res = self
else:
raise NotImplementedError
res = self.__class__(
self.model, self.params,
normalized_cov_params=self.normalized_cov_params,
scale=self.scale)
# Set the new covariance type
res.cov_type = cov_type
res.cov_kwds = {}
# Calculate the new covariance matrix
approx_complex_step = self._cov_approx_complex_step
if approx_complex_step:
approx_type_str = 'complex-step'
elif self._cov_approx_centered:
approx_type_str = 'centered finite differences'
else:
approx_type_str = 'finite differences'
k_params = len(self.params)
if k_params == 0:
res.cov_params_default = np.zeros((0, 0))
res._rank = 0
res.cov_kwds['description'] = 'No parameters estimated.'
elif cov_type == 'custom':
res.cov_type = kwargs['custom_cov_type']
res.cov_params_default = kwargs['custom_cov_params']
res.cov_kwds['description'] = kwargs['custom_description']
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
else:
mask = np.s_[...]
res._rank = np.linalg.matrix_rank(res.cov_params_default[mask])
elif cov_type == 'none':
res.cov_params_default = np.zeros((k_params, k_params)) * np.nan
res._rank = np.nan
res.cov_kwds['description'] = descriptions['none']
elif self.cov_type == 'approx':
res.cov_params_default = res.cov_params_approx
res.cov_kwds['description'] = descriptions['approx'].format(
approx_type=approx_type_str)
elif self.cov_type == 'oim':
res.cov_params_default = res.cov_params_oim
res.cov_kwds['description'] = descriptions['OIM'].format(
approx_type=approx_type_str)
elif self.cov_type == 'opg':
res.cov_params_default = res.cov_params_opg
res.cov_kwds['description'] = descriptions['OPG'].format(
approx_type=approx_type_str)
elif self.cov_type == 'robust' or self.cov_type == 'robust_oim':
res.cov_params_default = res.cov_params_robust_oim
res.cov_kwds['description'] = descriptions['robust-OIM'].format(
approx_type=approx_type_str)
elif self.cov_type == 'robust_approx':
res.cov_params_default = res.cov_params_robust_approx
res.cov_kwds['description'] = descriptions['robust-approx'].format(
approx_type=approx_type_str)
else:
raise NotImplementedError('Invalid covariance matrix type.')
return res
@cache_readonly
def aic(self):
"""
(float) Akaike Information Criterion
"""
return aic(self.llf, self.nobs_effective, self.df_model)
@cache_readonly
def aicc(self):
"""
(float) Akaike Information Criterion with small sample correction
"""
return aicc(self.llf, self.nobs_effective, self.df_model)
@cache_readonly
def bic(self):
"""
(float) Bayes Information Criterion
"""
return bic(self.llf, self.nobs_effective, self.df_model)
def _cov_params_approx(self, approx_complex_step=True,
approx_centered=False):
evaluated_hessian = self.nobs_effective * self.model.hessian(
params=self.params, transformed=True, includes_fixed=True,
method='approx', approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
# TODO: Case with "not approx_complex_step" is not hit in
# tests as of 2017-05-19
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
(tmp, singular_values) = pinv_extended(evaluated_hessian[mask])
neg_cov = np.zeros_like(evaluated_hessian) * np.nan
neg_cov[mask] = tmp
else:
(neg_cov, singular_values) = pinv_extended(evaluated_hessian)
self.model.update(self.params, transformed=True, includes_fixed=True)
if self._rank is None:
self._rank = np.linalg.matrix_rank(np.diag(singular_values))
return -neg_cov
@cache_readonly
def cov_params_approx(self):
"""
(array) The variance / covariance matrix. Computed using the numerical
Hessian approximated by complex step or finite differences methods.
"""
return self._cov_params_approx(self._cov_approx_complex_step,
self._cov_approx_centered)
def _cov_params_oim(self, approx_complex_step=True, approx_centered=False):
evaluated_hessian = self.nobs_effective * self.model.hessian(
self.params, hessian_method='oim', transformed=True,
includes_fixed=True, approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
(tmp, singular_values) = pinv_extended(evaluated_hessian[mask])
neg_cov = np.zeros_like(evaluated_hessian) * np.nan
neg_cov[mask] = tmp
else:
(neg_cov, singular_values) = pinv_extended(evaluated_hessian)
self.model.update(self.params, transformed=True, includes_fixed=True)
if self._rank is None:
self._rank = np.linalg.matrix_rank(np.diag(singular_values))
return -neg_cov
@cache_readonly
def cov_params_oim(self):
"""
(array) The variance / covariance matrix. Computed using the method
from Harvey (1989).
"""
return self._cov_params_oim(self._cov_approx_complex_step,
self._cov_approx_centered)
def _cov_params_opg(self, approx_complex_step=True, approx_centered=False):
evaluated_hessian = self.nobs_effective * self.model._hessian_opg(
self.params, transformed=True, includes_fixed=True,
approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
no_free_params = (self._free_params_index is not None and
len(self._free_params_index) == 0)
if no_free_params:
neg_cov = np.zeros_like(evaluated_hessian) * np.nan
singular_values = np.empty(0)
elif len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
(tmp, singular_values) = pinv_extended(evaluated_hessian[mask])
neg_cov = np.zeros_like(evaluated_hessian) * np.nan
neg_cov[mask] = tmp
else:
(neg_cov, singular_values) = pinv_extended(evaluated_hessian)
self.model.update(self.params, transformed=True, includes_fixed=True)
if self._rank is None:
if no_free_params:
self._rank = 0
else:
self._rank = np.linalg.matrix_rank(np.diag(singular_values))
return -neg_cov
@cache_readonly
def cov_params_opg(self):
"""
(array) The variance / covariance matrix. Computed using the outer
product of gradients method.
"""
return self._cov_params_opg(self._cov_approx_complex_step,
self._cov_approx_centered)
@cache_readonly
def cov_params_robust(self):
"""
(array) The QMLE variance / covariance matrix. Alias for
`cov_params_robust_oim`
"""
return self.cov_params_robust_oim
def _cov_params_robust_oim(self, approx_complex_step=True,
approx_centered=False):
cov_opg = self._cov_params_opg(approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
evaluated_hessian = self.nobs_effective * self.model.hessian(
self.params, hessian_method='oim', transformed=True,
includes_fixed=True, approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
cov_params = np.zeros_like(evaluated_hessian) * np.nan
cov_opg = cov_opg[mask]
evaluated_hessian = evaluated_hessian[mask]
tmp, singular_values = pinv_extended(
np.dot(np.dot(evaluated_hessian, cov_opg), evaluated_hessian))
cov_params[mask] = tmp
else:
(cov_params, singular_values) = pinv_extended(
np.dot(np.dot(evaluated_hessian, cov_opg), evaluated_hessian))
self.model.update(self.params, transformed=True, includes_fixed=True)
if self._rank is None:
self._rank = np.linalg.matrix_rank(np.diag(singular_values))
return cov_params
@cache_readonly
def cov_params_robust_oim(self):
"""
(array) The QMLE variance / covariance matrix. Computed using the
method from Harvey (1989) as the evaluated hessian.
"""
return self._cov_params_robust_oim(self._cov_approx_complex_step,
self._cov_approx_centered)
def _cov_params_robust_approx(self, approx_complex_step=True,
approx_centered=False):
cov_opg = self._cov_params_opg(approx_complex_step=approx_complex_step,
approx_centered=approx_centered)
evaluated_hessian = self.nobs_effective * self.model.hessian(
self.params, transformed=True, includes_fixed=True,
method='approx', approx_complex_step=approx_complex_step)
# TODO: Case with "not approx_complex_step" is not
# hit in tests as of 2017-05-19
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
cov_params = np.zeros_like(evaluated_hessian) * np.nan
cov_opg = cov_opg[mask]
evaluated_hessian = evaluated_hessian[mask]
tmp, singular_values = pinv_extended(
np.dot(np.dot(evaluated_hessian, cov_opg), evaluated_hessian))
cov_params[mask] = tmp
else:
(cov_params, singular_values) = pinv_extended(
np.dot(np.dot(evaluated_hessian, cov_opg), evaluated_hessian))
self.model.update(self.params, transformed=True, includes_fixed=True)
if self._rank is None:
self._rank = np.linalg.matrix_rank(np.diag(singular_values))
return cov_params
@cache_readonly
def cov_params_robust_approx(self):
"""
(array) The QMLE variance / covariance matrix. Computed using the
numerical Hessian as the evaluated hessian.
"""
return self._cov_params_robust_approx(self._cov_approx_complex_step,
self._cov_approx_centered)
[docs] def info_criteria(self, criteria, method='standard'):
r"""
Information criteria
Parameters
----------
criteria : {'aic', 'bic', 'hqic'}
The information criteria to compute.
method : {'standard', 'lutkepohl'}
The method for information criteria computation. Default is
'standard' method; 'lutkepohl' computes the information criteria
as in Lütkepohl (2007). See Notes for formulas.
Notes
-----
The `'standard'` formulas are:
.. math::
AIC & = -2 \log L(Y_n | \hat \psi) + 2 k \\
BIC & = -2 \log L(Y_n | \hat \psi) + k \log n \\
HQIC & = -2 \log L(Y_n | \hat \psi) + 2 k \log \log n \\
where :math:`\hat \psi` are the maximum likelihood estimates of the
parameters, :math:`n` is the number of observations, and `k` is the
number of estimated parameters.
Note that the `'standard'` formulas are returned from the `aic`, `bic`,
and `hqic` results attributes.
The `'lutkepohl'` formulas are (Lütkepohl, 2010):
.. math::
AIC_L & = \log | Q | + \frac{2 k}{n} \\
BIC_L & = \log | Q | + \frac{k \log n}{n} \\
HQIC_L & = \log | Q | + \frac{2 k \log \log n}{n} \\
where :math:`Q` is the state covariance matrix. Note that the Lütkepohl
definitions do not apply to all state space models, and should be used
with care outside of SARIMAX and VARMAX models.
References
----------
.. [*] Lütkepohl, Helmut. 2007. *New Introduction to Multiple Time*
*Series Analysis.* Berlin: Springer.
"""
criteria = criteria.lower()
method = method.lower()
if method == 'standard':
out = getattr(self, criteria)
elif method == 'lutkepohl':
if self.filter_results.state_cov.shape[-1] > 1:
raise ValueError('Cannot compute Lütkepohl statistics for'
' models with time-varying state covariance'
' matrix.')
cov = self.filter_results.state_cov[:, :, 0]
if criteria == 'aic':
out = np.squeeze(np.linalg.slogdet(cov)[1] +
2 * self.df_model / self.nobs_effective)
elif criteria == 'bic':
out = np.squeeze(np.linalg.slogdet(cov)[1] +
self.df_model * np.log(self.nobs_effective) /
self.nobs_effective)
elif criteria == 'hqic':
out = np.squeeze(np.linalg.slogdet(cov)[1] +
2 * self.df_model *
np.log(np.log(self.nobs_effective)) /
self.nobs_effective)
else:
raise ValueError('Invalid information criteria')
else:
raise ValueError('Invalid information criteria computation method')
return out
@cache_readonly
def fittedvalues(self):
"""
(array) The predicted values of the model. An (nobs x k_endog) array.
"""
# This is a (k_endog x nobs array; do not want to squeeze in case of
# the corner case where nobs = 1 (mostly a concern in the predict or
# forecast functions, but here also to maintain consistency)
fittedvalues = self.forecasts
if fittedvalues is None:
pass
elif fittedvalues.shape[0] == 1:
fittedvalues = fittedvalues[0, :]
else:
fittedvalues = fittedvalues.T
return fittedvalues
@cache_readonly
def hqic(self):
"""
(float) Hannan-Quinn Information Criterion
"""
# return (-2 * self.llf +
# 2 * np.log(np.log(self.nobs_effective)) * self.df_model)
return hqic(self.llf, self.nobs_effective, self.df_model)
@cache_readonly
def llf_obs(self):
"""
(float) The value of the log-likelihood function evaluated at `params`.
"""
return self.filter_results.llf_obs
@cache_readonly
def llf(self):
"""
(float) The value of the log-likelihood function evaluated at `params`.
"""
return self.filter_results.llf
@cache_readonly
def loglikelihood_burn(self):
"""
(float) The number of observations during which the likelihood is not
evaluated.
"""
return self.filter_results.loglikelihood_burn
@cache_readonly
def mae(self):
"""
(float) Mean absolute error
"""
return np.mean(np.abs(self.resid))
@cache_readonly
def mse(self):
"""
(float) Mean squared error
"""
return self.sse / self.nobs
@cache_readonly
def pvalues(self):
"""
(array) The p-values associated with the z-statistics of the
coefficients. Note that the coefficients are assumed to have a Normal
distribution.
"""
pvalues = np.zeros_like(self.zvalues) * np.nan
mask = np.ones_like(pvalues, dtype=bool)
mask[self._free_params_index] = True
mask &= ~np.isnan(self.zvalues)
pvalues[mask] = norm.sf(np.abs(self.zvalues[mask])) * 2
return pvalues
@cache_readonly
def resid(self):
"""
(array) The model residuals. An (nobs x k_endog) array.
"""
# This is a (k_endog x nobs array; do not want to squeeze in case of
# the corner case where nobs = 1 (mostly a concern in the predict or
# forecast functions, but here also to maintain consistency)
resid = self.forecasts_error
if resid is None:
pass
elif resid.shape[0] == 1:
resid = resid[0, :]
else:
resid = resid.T
return resid
@property
def states(self):
if self.model._index_generated and not self.model._index_none:
warnings.warn('No supported index is available. The `states`'
' DataFrame uses a generated integer index',
ValueWarning)
return self._states
@cache_readonly
def sse(self):
"""
(float) Sum of squared errors
"""
return np.sum(self.resid**2)
@cache_readonly
def zvalues(self):
"""
(array) The z-statistics for the coefficients.
"""
return self.params / self.bse
[docs] def test_normality(self, method):
"""
Test for normality of standardized residuals.
Null hypothesis is normality.
Parameters
----------
method : {'jarquebera', None}
The statistical test for normality. Must be 'jarquebera' for
Jarque-Bera normality test. If None, an attempt is made to select
an appropriate test.
See Also
--------
statsmodels.stats.stattools.jarque_bera
The Jarque-Bera test of normality.
Notes
-----
Let `d` = max(loglikelihood_burn, nobs_diffuse); this test is
calculated ignoring the first `d` residuals.
In the case of missing data, the maintained hypothesis is that the
data are missing completely at random. This test is then run on the
standardized residuals excluding those corresponding to missing
observations.
"""
if method is None:
method = 'jarquebera'
if self.standardized_forecasts_error is None:
raise ValueError('Cannot compute test statistic when standardized'
' forecast errors have not been computed.')
if method == 'jarquebera':
from statsmodels.stats.stattools import jarque_bera
d = np.maximum(self.loglikelihood_burn, self.nobs_diffuse)
output = []
for i in range(self.model.k_endog):
resid = self.filter_results.standardized_forecasts_error[i, d:]
mask = ~np.isnan(resid)
output.append(jarque_bera(resid[mask]))
else:
raise NotImplementedError('Invalid normality test method.')
return np.array(output)
[docs] def test_heteroskedasticity(self, method, alternative='two-sided',
use_f=True):
r"""
Test for heteroskedasticity of standardized residuals
Tests whether the sum-of-squares in the first third of the sample is
significantly different than the sum-of-squares in the last third
of the sample. Analogous to a Goldfeld-Quandt test. The null hypothesis
is of no heteroskedasticity.
Parameters
----------
method : {'breakvar', None}
The statistical test for heteroskedasticity. Must be 'breakvar'
for test of a break in the variance. If None, an attempt is
made to select an appropriate test.
alternative : str, 'increasing', 'decreasing' or 'two-sided'
This specifies the alternative for the p-value calculation. Default
is two-sided.
use_f : bool, optional
Whether or not to compare against the asymptotic distribution
(chi-squared) or the approximate small-sample distribution (F).
Default is True (i.e. default is to compare against an F
distribution).
Returns
-------
output : ndarray
An array with `(test_statistic, pvalue)` for each endogenous
variable. The array is then sized `(k_endog, 2)`. If the method is
called as `het = res.test_heteroskedasticity()`, then `het[0]` is
an array of size 2 corresponding to the first endogenous variable,
where `het[0][0]` is the test statistic, and `het[0][1]` is the
p-value.
See Also
--------
statsmodels.tsa.stattools.breakvar_heteroskedasticity_test
Notes
-----
The null hypothesis is of no heteroskedasticity.
For :math:`h = [T/3]`, the test statistic is:
.. math::
H(h) = \sum_{t=T-h+1}^T \tilde v_t^2
\Bigg / \sum_{t=d+1}^{d+1+h} \tilde v_t^2
where :math:`d` = max(loglikelihood_burn, nobs_diffuse)` (usually
corresponding to diffuse initialization under either the approximate
or exact approach).
This statistic can be tested against an :math:`F(h,h)` distribution.
Alternatively, :math:`h H(h)` is asymptotically distributed according
to :math:`\chi_h^2`; this second test can be applied by passing
`use_f=True` as an argument.
See section 5.4 of [1]_ for the above formula and discussion, as well
as additional details.
TODO
- Allow specification of :math:`h`
References
----------
.. [1] Harvey, Andrew C. 1990. *Forecasting, Structural Time Series*
*Models and the Kalman Filter.* Cambridge University Press.
"""
if method is None:
method = 'breakvar'
if self.standardized_forecasts_error is None:
raise ValueError('Cannot compute test statistic when standardized'
' forecast errors have not been computed.')
if method == 'breakvar':
from statsmodels.tsa.stattools import (
breakvar_heteroskedasticity_test
)
# Store some values
resid = self.filter_results.standardized_forecasts_error
d = np.maximum(self.loglikelihood_burn, self.nobs_diffuse)
# This differs from self.nobs_effective because here we want to
# exclude exact diffuse periods, whereas self.nobs_effective only
# excludes explicitly burned (usually approximate diffuse) periods.
nobs_effective = self.nobs - d
h = int(np.round(nobs_effective / 3))
test_statistics = []
p_values = []
for i in range(self.model.k_endog):
test_statistic, p_value = breakvar_heteroskedasticity_test(
resid[i, d:],
subset_length=h,
alternative=alternative,
use_f=use_f
)
test_statistics.append(test_statistic)
p_values.append(p_value)
output = np.c_[test_statistics, p_values]
else:
raise NotImplementedError('Invalid heteroskedasticity test'
' method.')
return output
[docs] def test_serial_correlation(self, method, df_adjust=False, lags=None):
"""
Ljung-Box test for no serial correlation of standardized residuals
Null hypothesis is no serial correlation.
Parameters
----------
method : {'ljungbox','boxpierece', None}
The statistical test for serial correlation. If None, an attempt is
made to select an appropriate test.
lags : None, int or array_like
If lags is an integer then this is taken to be the largest lag
that is included, the test result is reported for all smaller lag
length.
If lags is a list or array, then all lags are included up to the
largest lag in the list, however only the tests for the lags in the
list are reported.
If lags is None, then the default maxlag is 12*(nobs/100)^{1/4}.
After 0.12 the default maxlag will change to min(10, nobs // 5) for
non-seasonal models and min(2*m, nobs // 5) for seasonal time
series where m is the seasonal period.
df_adjust : bool, optional
If True, the degrees of freedom consumed by the model is subtracted
from the degrees-of-freedom used in the test so that the adjusted
dof for the statistics are lags - model_df. In an ARMA model, this
value is usually p+q where p is the AR order and q is the MA order.
When using df_adjust, it is not possible to use tests based on
fewer than model_df lags.
Returns
-------
output : ndarray
An array with `(test_statistic, pvalue)` for each endogenous
variable and each lag. The array is then sized
`(k_endog, 2, lags)`. If the method is called as
`ljungbox = res.test_serial_correlation()`, then `ljungbox[i]`
holds the results of the Ljung-Box test (as would be returned by
`statsmodels.stats.diagnostic.acorr_ljungbox`) for the `i` th
endogenous variable.
See Also
--------
statsmodels.stats.diagnostic.acorr_ljungbox
Ljung-Box test for serial correlation.
Notes
-----
Let `d` = max(loglikelihood_burn, nobs_diffuse); this test is
calculated ignoring the first `d` residuals.
Output is nan for any endogenous variable which has missing values.
"""
if method is None:
method = 'ljungbox'
if self.standardized_forecasts_error is None:
raise ValueError('Cannot compute test statistic when standardized'
' forecast errors have not been computed.')
if method == 'ljungbox' or method == 'boxpierce':
from statsmodels.stats.diagnostic import acorr_ljungbox
d = np.maximum(self.loglikelihood_burn, self.nobs_diffuse)
# This differs from self.nobs_effective because here we want to
# exclude exact diffuse periods, whereas self.nobs_effective only
# excludes explicitly burned (usually approximate diffuse) periods.
nobs_effective = self.nobs - d
output = []
# Default lags for acorr_ljungbox is 40, but may not always have
# that many observations
if lags is None:
seasonal_periods = getattr(self.model, "seasonal_periods", 0)
if seasonal_periods:
lags = min(2 * seasonal_periods, nobs_effective // 5)
else:
lags = min(10, nobs_effective // 5)
model_df = 0
if df_adjust:
model_df = max(0, self.df_model - self.k_diffuse_states - 1)
for i in range(self.model.k_endog):
results = acorr_ljungbox(
self.filter_results.standardized_forecasts_error[i][d:],
lags=lags, boxpierce=(method == 'boxpierce'),
model_df=model_df, return_df=False)
if method == 'ljungbox':
output.append(results[0:2])
else:
output.append(results[2:])
output = np.c_[output]
else:
raise NotImplementedError('Invalid serial correlation test'
' method.')
return output
[docs] def get_prediction(self, start=None, end=None, dynamic=False,
index=None, exog=None, extend_model=None,
extend_kwargs=None, **kwargs):
"""
In-sample prediction and out-of-sample forecasting
Parameters
----------
start : int, str, or datetime, optional
Zero-indexed observation number at which to start forecasting,
i.e., the first forecast is start. Can also be a date string to
parse or a datetime type. Default is the the zeroth observation.
end : int, str, or datetime, optional
Zero-indexed observation number at which to end forecasting, i.e.,
the last forecast is end. Can also be a date string to
parse or a datetime type. However, if the dates index does not
have a fixed frequency, end must be an integer index if you
want out of sample prediction. Default is the last observation in
the sample.
dynamic : bool, int, str, or datetime, optional
Integer offset relative to `start` at which to begin dynamic
prediction. Can also be an absolute date string to parse or a
datetime type (these are not interpreted as offsets).
Prior to this observation, true endogenous values will be used for
prediction; starting with this observation and continuing through
the end of prediction, forecasted endogenous values will be used
instead.
**kwargs
Additional arguments may required for forecasting beyond the end
of the sample. See `FilterResults.predict` for more details.
Returns
-------
predictions : PredictionResults
PredictionResults instance containing in-sample predictions /
out-of-sample forecasts and results including confidence intervals.
See Also
--------
forecast
Out-of-sample forecasts.
predict
In-sample predictions and out-of-sample forecasts.
get_forecast
Out-of-sample forecasts and results including confidence intervals.
"""
if start is None:
start = 0
# Handle start, end, dynamic
start, end, out_of_sample, prediction_index = (
self.model._get_prediction_index(start, end, index))
# Handle `dynamic`
if isinstance(dynamic, (str, dt.datetime, pd.Timestamp)):
dynamic, _, _ = self.model._get_index_loc(dynamic)
# Convert to offset relative to start
dynamic = dynamic - start
# If we have out-of-sample forecasting and `exog` or in general any
# kind of time-varying state space model, then we need to create an
# extended model to get updated state space system matrices
if extend_model is None:
extend_model = (self.model.exog is not None or
not self.filter_results.time_invariant)
if out_of_sample and extend_model:
kwargs = self.model._get_extension_time_varying_matrices(
self.params, exog, out_of_sample, extend_kwargs,
transformed=True, includes_fixed=True, **kwargs)
# Make sure the model class has the current parameters
self.model.update(self.params, transformed=True, includes_fixed=True)
# Perform the prediction
# This is a (k_endog x npredictions) array; do not want to squeeze in
# case of npredictions = 1
prediction_results = self.filter_results.predict(
start, end + out_of_sample + 1, dynamic, **kwargs)
# Return a new mlemodel.PredictionResults object
return PredictionResultsWrapper(PredictionResults(
self, prediction_results, row_labels=prediction_index))
[docs] def get_forecast(self, steps=1, **kwargs):
"""
Out-of-sample forecasts and prediction intervals
Parameters
----------
steps : int, str, or datetime, optional
If an integer, the number of steps to forecast from the end of the
sample. Can also be a date string to parse or a datetime type.
However, if the dates index does not have a fixed frequency, steps
must be an integer. Default
**kwargs
Additional arguments may required for forecasting beyond the end
of the sample. See `FilterResults.predict` for more details.
Returns
-------
forecasts : PredictionResults
PredictionResults instance containing out-of-sample forecasts and
results including confidence intervals.
See also
--------
forecast
Out-of-sample forecasts.
predict
In-sample predictions and out-of-sample forecasts.
get_prediction
In-sample predictions / out-of-sample forecasts and results
including confidence intervals.
"""
if isinstance(steps, int):
end = self.nobs + steps - 1
else:
end = steps
return self.get_prediction(start=self.nobs, end=end, **kwargs)
[docs] def predict(self, start=None, end=None, dynamic=False, **kwargs):
"""
In-sample prediction and out-of-sample forecasting
Parameters
----------
start : {int, str,datetime}, optional
Zero-indexed observation number at which to start forecasting,
i.e., the first forecast is start. Can also be a date string to
parse or a datetime type. Default is the zeroth observation.
end : {int, str,datetime}, optional
Zero-indexed observation number at which to end forecasting, i.e.,
the last forecast is end. Can also be a date string to
parse or a datetime type. However, if the dates index does not
have a fixed frequency, end must be an integer index if you
want out of sample prediction. Default is the last observation in
the sample.
dynamic : {bool, int, str,datetime}, optional
Integer offset relative to `start` at which to begin dynamic
prediction. Can also be an absolute date string to parse or a
datetime type (these are not interpreted as offsets).
Prior to this observation, true endogenous values will be used for
prediction; starting with this observation and continuing through
the end of prediction, forecasted endogenous values will be used
instead.
**kwargs
Additional arguments may be required for forecasting beyond the end
of the sample. See ``FilterResults.predict`` for more details.
Returns
-------
predictions : array_like
In-sample predictions / Out-of-sample forecasts. (Numpy array or
Pandas Series or DataFrame, depending on input and dimensions).
Dimensions are `(npredict x k_endog)`.
See Also
--------
forecast
Out-of-sample forecasts.
get_forecast
Out-of-sample forecasts and results including confidence intervals.
get_prediction
In-sample predictions / out-of-sample forecasts and results
including confidence intervals.
"""
# Perform the prediction
prediction_results = self.get_prediction(start, end, dynamic, **kwargs)
return prediction_results.predicted_mean
[docs] def forecast(self, steps=1, **kwargs):
"""
Out-of-sample forecasts
Parameters
----------
steps : int, str, or datetime, optional
If an integer, the number of steps to forecast from the end of the
sample. Can also be a date string to parse or a datetime type.
However, if the dates index does not have a fixed frequency, steps
must be an integer. Default
**kwargs
Additional arguments may required for forecasting beyond the end
of the sample. See `FilterResults.predict` for more details.
Returns
-------
forecast : array_like
Out-of-sample forecasts (Numpy array or Pandas Series or DataFrame,
depending on input and dimensions).
Dimensions are `(steps x k_endog)`.
See Also
--------
predict
In-sample predictions and out-of-sample forecasts.
get_forecast
Out-of-sample forecasts and results including confidence intervals.
get_prediction
In-sample predictions / out-of-sample forecasts and results
including confidence intervals.
"""
if isinstance(steps, int):
end = self.nobs + steps - 1
else:
end = steps
return self.predict(start=self.nobs, end=end, **kwargs)
[docs] def simulate(self, nsimulations, measurement_shocks=None,
state_shocks=None, initial_state=None, anchor=None,
repetitions=None, exog=None, extend_model=None,
extend_kwargs=None, **kwargs):
r"""
Simulate a new time series following the state space model
Parameters
----------
nsimulations : int
The number of observations to simulate. If the model is
time-invariant this can be any number. If the model is
time-varying, then this number must be less than or equal to the
number
measurement_shocks : array_like, optional
If specified, these are the shocks to the measurement equation,
:math:`\varepsilon_t`. If unspecified, these are automatically
generated using a pseudo-random number generator. If specified,
must be shaped `nsimulations` x `k_endog`, where `k_endog` is the
same as in the state space model.
state_shocks : array_like, optional
If specified, these are the shocks to the state equation,
:math:`\eta_t`. If unspecified, these are automatically
generated using a pseudo-random number generator. If specified,
must be shaped `nsimulations` x `k_posdef` where `k_posdef` is the
same as in the state space model.
initial_state : array_like, optional
If specified, this is the initial state vector to use in
simulation, which should be shaped (`k_states` x 1), where
`k_states` is the same as in the state space model. If unspecified,
but the model has been initialized, then that initialization is
used. This must be specified if `anchor` is anything other than
"start" or 0.
anchor : int, str, or datetime, optional
Starting point from which to begin the simulations; type depends on
the index of the given `endog` model. Two special cases are the
strings 'start' and 'end', which refer to starting at the beginning
and end of the sample, respectively. If a date/time index was
provided to the model, then this argument can be a date string to
parse or a datetime type. Otherwise, an integer index should be
given. Default is 'start'.
repetitions : int, optional
Number of simulated paths to generate. Default is 1 simulated path.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
Returns
-------
simulated_obs : ndarray
An array of simulated observations. If `repetitions=None`, then it
will be shaped (nsimulations x k_endog) or (nsimulations,) if
`k_endog=1`. Otherwise it will be shaped
(nsimulations x k_endog x repetitions). If the model was given
Pandas input then the output will be a Pandas object. If
`k_endog > 1` and `repetitions` is not None, then the output will
be a Pandas DataFrame that has a MultiIndex for the columns, with
the first level containing the names of the `endog` variables and
the second level containing the repetition number.
See Also
--------
impulse_responses
Impulse response functions
"""
# Get the starting location
if anchor is None or anchor == 'start':
iloc = 0
elif anchor == 'end':
iloc = self.nobs
else:
iloc, _, _ = self.model._get_index_loc(anchor)
if isinstance(iloc, slice):
iloc = iloc.start
if iloc < 0:
iloc = self.nobs + iloc
if iloc > self.nobs:
raise ValueError('Cannot anchor simulation outside of the sample.')
# Setup the initial state
if initial_state is None:
initial_state_moments = (
self.predicted_state[:, iloc],
self.predicted_state_cov[:, :, iloc])
_repetitions = 1 if repetitions is None else repetitions
initial_state = np.random.multivariate_normal(
*initial_state_moments, size=_repetitions).T
scale = self.scale if self.filter_results.filter_concentrated else None
with self.model.ssm.fixed_scale(scale):
sim = self.model.simulate(
self.params, nsimulations,
measurement_shocks=measurement_shocks,
state_shocks=state_shocks, initial_state=initial_state,
anchor=anchor, repetitions=repetitions, exog=exog,
transformed=True, includes_fixed=True,
extend_model=extend_model, extend_kwargs=extend_kwargs,
**kwargs)
return sim
[docs] def impulse_responses(self, steps=1, impulse=0, orthogonalized=False,
cumulative=False, **kwargs):
"""
Impulse response function
Parameters
----------
steps : int, optional
The number of steps for which impulse responses are calculated.
Default is 1. Note that for time-invariant models, the initial
impulse is not counted as a step, so if `steps=1`, the output will
have 2 entries.
impulse : int, str or array_like
If an integer, the state innovation to pulse; must be between 0
and `k_posdef-1`. If a str, it indicates which column of df
the unit (1) impulse is given.
Alternatively, a custom impulse vector may be provided; must be
shaped `k_posdef x 1`.
orthogonalized : bool, optional
Whether or not to perform impulse using orthogonalized innovations.
Note that this will also affect custum `impulse` vectors. Default
is False.
cumulative : bool, optional
Whether or not to return cumulative impulse responses. Default is
False.
anchor : int, str, or datetime, optional
Time point within the sample for the state innovation impulse. Type
depends on the index of the given `endog` in the model. Two special
cases are the strings 'start' and 'end', which refer to setting the
impulse at the first and last points of the sample, respectively.
Integer values can run from 0 to `nobs - 1`, or can be negative to
apply negative indexing. Finally, if a date/time index was provided
to the model, then this argument can be a date string to parse or a
datetime type. Default is 'start'.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
**kwargs
If the model has time-varying design or transition matrices and the
combination of `anchor` and `steps` implies creating impulse
responses for the out-of-sample period, then these matrices must
have updated values provided for the out-of-sample steps. For
example, if `design` is a time-varying component, `nobs` is 10,
`anchor=1`, and `steps` is 15, a (`k_endog` x `k_states` x 7)
matrix must be provided with the new design matrix values.
Returns
-------
impulse_responses : ndarray
Responses for each endogenous variable due to the impulse
given by the `impulse` argument. For a time-invariant model, the
impulse responses are given for `steps + 1` elements (this gives
the "initial impulse" followed by `steps` responses for the
important cases of VAR and SARIMAX models), while for time-varying
models the impulse responses are only given for `steps` elements
(to avoid having to unexpectedly provide updated time-varying
matrices).
See Also
--------
simulate
Simulate a time series according to the given state space model,
optionally with specified series for the innovations.
Notes
-----
Intercepts in the measurement and state equation are ignored when
calculating impulse responses.
"""
scale = self.scale if self.filter_results.filter_concentrated else None
with self.model.ssm.fixed_scale(scale):
irfs = self.model.impulse_responses(self.params, steps, impulse,
orthogonalized, cumulative,
**kwargs)
# These are wrapped automatically, so just return the array
if isinstance(irfs, (pd.Series, pd.DataFrame)):
irfs = irfs.values
return irfs
def _apply(self, mod, refit=False, fit_kwargs=None, **kwargs):
if fit_kwargs is None:
fit_kwargs = {}
if refit:
fit_kwargs.setdefault('start_params', self.params)
if self._has_fixed_params:
fit_kwargs.setdefault('includes_fixed', True)
res = mod.fit_constrained(self._fixed_params, **fit_kwargs)
else:
res = mod.fit(**fit_kwargs)
else:
if 'cov_type' in fit_kwargs:
raise ValueError('Cannot specify covariance type in'
' `fit_kwargs` unless refitting'
' parameters (not available in extend).')
if 'cov_kwds' in fit_kwargs:
raise ValueError('Cannot specify covariance keyword arguments'
' in `fit_kwargs` unless refitting'
' parameters (not available in extend).')
if self.cov_type == 'none':
fit_kwargs['cov_type'] = 'none'
else:
fit_kwargs['cov_type'] = 'custom'
fit_kwargs['cov_kwds'] = {
'custom_cov_type': self.cov_type,
'custom_cov_params': self.cov_params_default,
'custom_description': ('Parameters and standard errors'
' were estimated using a different'
' dataset and were then applied to'
' this dataset. %s'
% self.cov_kwds['description'])}
if self.smoother_results is not None:
func = mod.smooth
else:
func = mod.filter
if self._has_fixed_params:
with mod.fix_params(self._fixed_params):
fit_kwargs.setdefault('includes_fixed', True)
res = func(self.params, **fit_kwargs)
else:
res = func(self.params, **fit_kwargs)
return res
def _news_previous_results(self, previous, start, end, periods):
# Compute the news
out = self.smoother_results.news(previous.smoother_results,
start=start, end=end)
return out
def _news_updated_results(self, updated, start, end, periods):
return updated._news_previous_results(self, start, end, periods)
def _news_previous_data(self, endog, start, end, periods, exog):
previous = self.apply(endog, exog=exog, copy_initialization=True)
return self._news_previous_results(previous, start, end, periods)
def _news_updated_data(self, endog, start, end, periods, exog):
updated = self.apply(endog, exog=exog, copy_initialization=True)
return self._news_updated_results(updated, start, end, periods)
[docs] def news(self, comparison, impact_date=None, impacted_variable=None,
start=None, end=None, periods=None, exog=None,
comparison_type=None, return_raw=False, tolerance=1e-10,
**kwargs):
"""
Compute impacts from updated data (news and revisions)
Parameters
----------
comparison : array_like or MLEResults
An updated dataset with updated and/or revised data from which the
news can be computed, or an updated or previous results object
to use in computing the news.
impact_date : int, str, or datetime, optional
A single specific period of impacts from news and revisions to
compute. Can also be a date string to parse or a datetime type.
This argument cannot be used in combination with `start`, `end`, or
`periods`. Default is the first out-of-sample observation.
impacted_variable : str, list, array, or slice, optional
Observation variable label or slice of labels specifying that only
specific impacted variables should be shown in the News output. The
impacted variable(s) describe the variables that were *affected* by
the news. If you do not know the labels for the variables, check
the `endog_names` attribute of the model instance.
start : int, str, or datetime, optional
The first period of impacts from news and revisions to compute.
Can also be a date string to parse or a datetime type. Default is
the first out-of-sample observation.
end : int, str, or datetime, optional
The last period of impacts from news and revisions to compute.
Can also be a date string to parse or a datetime type. Default is
the first out-of-sample observation.
periods : int, optional
The number of periods of impacts from news and revisions to
compute.
exog : array_like, optional
Array of exogenous regressors for the out-of-sample period, if
applicable.
comparison_type : {None, 'previous', 'updated'}
This denotes whether the `comparison` argument represents a
*previous* results object or dataset or an *updated* results object
or dataset. If not specified, then an attempt is made to determine
the comparison type.
return_raw : bool, optional
Whether or not to return only the specific output or a full
results object. Default is to return a full results object.
tolerance : float, optional
The numerical threshold for determining zero impact. Default is
that any impact less than 1e-10 is assumed to be zero.
Returns
-------
NewsResults
Impacts of data revisions and news on estimates
References
----------
.. [1] 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.
.. [2] Bańbura, Marta, Domenico Giannone, and Lucrezia Reichlin.
"Nowcasting."
The Oxford Handbook of Economic Forecasting. July 8, 2011.
.. [3] Bańbura, Marta, Domenico Giannone, Michele Modugno, and Lucrezia
Reichlin.
"Now-casting and the real-time data flow."
In Handbook of economic forecasting, vol. 2, pp. 195-237.
Elsevier, 2013.
"""
# Validate input
if self.smoother_results is None:
raise ValueError('Cannot compute news without Kalman smoother'
' results.')
# If we were given data, create a new results object
comparison_dataset = not isinstance(
comparison, (MLEResults, MLEResultsWrapper))
if comparison_dataset:
# If `exog` is longer than `comparison`, then we extend it to match
nobs_endog = len(comparison)
nobs_exog = len(exog) if exog is not None else nobs_endog
if nobs_exog > nobs_endog:
_, _, _, ix = self.model._get_prediction_index(
start=0, end=nobs_exog - 1)
# TODO: check that the index of `comparison` matches the model
comparison = np.asarray(comparison)
if comparison.ndim < 2:
comparison = np.atleast_2d(comparison).T
if (comparison.ndim != 2 or
comparison.shape[1] != self.model.k_endog):
raise ValueError('Invalid shape for `comparison`. Must'
f' contain {self.model.k_endog} columns.')
extra = np.zeros((nobs_exog - nobs_endog,
self.model.k_endog)) * np.nan
comparison = pd.DataFrame(
np.concatenate([comparison, extra], axis=0), index=ix,
columns=self.model.endog_names)
# Get the results object
comparison = self.apply(comparison, exog=exog,
copy_initialization=True, **kwargs)
# Now, figure out the `updated` versus `previous` results objects
nmissing = self.filter_results.missing.sum()
nmissing_comparison = comparison.filter_results.missing.sum()
if (comparison_type == 'updated' or (comparison_type is None and (
comparison.nobs > self.nobs or
(comparison.nobs == self.nobs and
nmissing > nmissing_comparison)))):
updated = comparison
previous = self
elif (comparison_type == 'previous' or (comparison_type is None and (
comparison.nobs < self.nobs or
(comparison.nobs == self.nobs and
nmissing < nmissing_comparison)))):
updated = self
previous = comparison
else:
raise ValueError('Could not automatically determine the type'
' of comparison requested to compute the'
' News, so it must be specified as "updated"'
' or "previous", using the `comparison_type`'
' keyword argument')
# Check that the index of `updated` is a superset of the
# index of `previous`
# Note: the try/except block is for Pandas < 0.25, in which
# `PeriodIndex.difference` raises a ValueError if the argument is not
# also a `PeriodIndex`.
diff = previous.model._index.difference(updated.model._index)
if len(diff) > 0:
raise ValueError('The index associated with the updated results is'
' not a superset of the index associated with the'
' previous results, and so these datasets do not'
' appear to be related. Can only compute the'
' news by comparing this results set to previous'
' results objects.')
# Handle start, end, periods
# There doesn't seem to be any universal defaults that both (a) make
# sense for all data update combinations, and (b) work with both
# time-invariant and time-varying models. So we require that the user
# specify exactly two of start, end, periods.
if impact_date is not None:
if not (start is None and end is None and periods is None):
raise ValueError('Cannot use the `impact_date` argument in'
' combination with `start`, `end`, or'
' `periods`.')
start = impact_date
periods = 1
if start is None and end is None and periods is None:
start = previous.nobs - 1
end = previous.nobs - 1
if int(start is None) + int(end is None) + int(periods is None) != 1:
raise ValueError('Of the three parameters: start, end, and'
' periods, exactly two must be specified')
# If we have the `periods` object, we need to convert `start`/`end` to
# integers so that we can compute the other one. That's because
# _get_prediction_index doesn't support a `periods` argument
elif start is not None and periods is not None:
start, _, _, _ = self.model._get_prediction_index(start, start)
end = start + (periods - 1)
elif end is not None and periods is not None:
_, end, _, _ = self.model._get_prediction_index(end, end)
start = end - (periods - 1)
elif start is not None and end is not None:
pass
# Get the integer-based start, end and the prediction index
start, end, out_of_sample, prediction_index = (
updated.model._get_prediction_index(start, end))
end = end + out_of_sample
# News results will always use Pandas, so if the model's data was not
# from Pandas, we'll create an index, as if the model's data had been
# given a default Pandas index.
if prediction_index is None:
prediction_index = pd.RangeIndex(start=start, stop=end + 1)
# For time-varying models try to create an appended `updated` model
# with NaN values. Do not extend the model if this was already done
# above (i.e. the case that `comparison` was a new dataset), because
# in that case `exog` and `kwargs` should have
# been set with the input `comparison` dataset in mind, and so would be
# useless here. Ultimately, we've already extended `updated` as far
# as we can. So raise an exception in that case with a useful message.
# However, we still want to try to accommodate extending the model here
# if it is possible.
# Note that we do not need to extend time-invariant models, because
# `KalmanSmoother.news` can itself handle any impact dates for
# time-invariant models.
time_varying = not (previous.filter_results.time_invariant or
updated.filter_results.time_invariant)
if time_varying and end >= updated.nobs:
# If we the given `comparison` was a dataset and either `exog` or
# `kwargs` was set, then we assume that we cannot create an updated
# time-varying model (because then we can't tell if `kwargs` and
# `exog` arguments are meant to apply to the `comparison` dataset
# or to this extension)
if comparison_dataset and (exog is not None or len(kwargs) > 0):
if comparison is updated:
raise ValueError('If providing an updated dataset as the'
' `comparison` with a time-varying model,'
' then the `end` period cannot be beyond'
' the end of that updated dataset.')
else:
raise ValueError('If providing an previous dataset as the'
' `comparison` with a time-varying model,'
' then the `end` period cannot be beyond'
' the end of the (updated) results'
' object.')
# Try to extend `updated`
updated_orig = updated
# TODO: `append` should fix this k_endog=1 issue for us
# TODO: is the + 1 necessary?
if self.model.k_endog > 1:
extra = np.zeros((end - updated.nobs + 1,
self.model.k_endog)) * np.nan
else:
extra = np.zeros((end - updated.nobs + 1,)) * np.nan
updated = updated_orig.append(extra, exog=exog, **kwargs)
# Compute the news
news_results = (
updated._news_previous_results(previous, start, end + 1, periods))
if not return_raw:
news_results = NewsResults(
news_results, self, updated, previous, impacted_variable,
tolerance, row_labels=prediction_index)
return news_results
[docs] def append(self, endog, exog=None, refit=False, fit_kwargs=None,
copy_initialization=False, **kwargs):
"""
Recreate the results object with new data appended to the original data
Creates a new result object applied to a dataset that is created by
appending new data to the end of the model's original data. The new
results can then be used for analysis or forecasting.
Parameters
----------
endog : array_like
New observations from the modeled time-series process.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
refit : bool, optional
Whether to re-fit the parameters, based on the combined dataset.
Default is False (so parameters from the current results object
are used to create the new results object).
copy_initialization : bool, optional
Whether or not to copy the initialization from the current results
set to the new model. Default is False
fit_kwargs : dict, optional
Keyword arguments to pass to `fit` (if `refit=True`) or `filter` /
`smooth`.
copy_initialization : bool, optional
**kwargs
Keyword arguments may be used to modify model specification
arguments when created the new model object.
Returns
-------
results
Updated Results object, that includes results from both the
original dataset and the new dataset.
Notes
-----
The `endog` and `exog` arguments to this method must be formatted in
the same way (e.g. Pandas Series versus Numpy array) as were the
`endog` and `exog` arrays passed to the original model.
The `endog` argument to this method should consist of new observations
that occurred directly after the last element of `endog`. For any other
kind of dataset, see the `apply` method.
This method will apply filtering to all of the original data as well
as to the new data. To apply filtering only to the new data (which
can be much faster if the original dataset is large), see the `extend`
method.
See Also
--------
statsmodels.tsa.statespace.mlemodel.MLEResults.extend
statsmodels.tsa.statespace.mlemodel.MLEResults.apply
Examples
--------
>>> index = pd.period_range(start='2000', periods=2, freq='A')
>>> original_observations = pd.Series([1.2, 1.5], index=index)
>>> mod = sm.tsa.SARIMAX(original_observations)
>>> res = mod.fit()
>>> print(res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(res.fittedvalues)
2000 0.0000
2001 1.1707
Freq: A-DEC, dtype: float64
>>> print(res.forecast(1))
2002 1.4634
Freq: A-DEC, dtype: float64
>>> new_index = pd.period_range(start='2002', periods=1, freq='A')
>>> new_observations = pd.Series([0.9], index=new_index)
>>> updated_res = res.append(new_observations)
>>> print(updated_res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(updated_res.fittedvalues)
2000 0.0000
2001 1.1707
2002 1.4634
Freq: A-DEC, dtype: float64
>>> print(updated_res.forecast(1))
2003 0.878
Freq: A-DEC, dtype: float64
"""
start = self.nobs
end = self.nobs + len(endog) - 1
_, _, _, append_ix = self.model._get_prediction_index(start, end)
# Check the index of the new data
if isinstance(self.model.data, PandasData):
_check_index(append_ix, endog, '`endog`')
# Concatenate the new data to original data
new_endog = concat([self.model.data.orig_endog, endog], axis=0,
allow_mix=True)
# Handle `exog`
if exog is not None:
_, exog = prepare_exog(exog)
_check_index(append_ix, exog, '`exog`')
new_exog = concat([self.model.data.orig_exog, exog], axis=0,
allow_mix=True)
else:
new_exog = None
# Create a continuous index for the combined data
if isinstance(self.model.data, PandasData):
start = 0
end = len(new_endog) - 1
_, _, _, new_index = self.model._get_prediction_index(start, end)
# Standardize `endog` to have the right index and columns
columns = self.model.endog_names
if not isinstance(columns, list):
columns = [columns]
new_endog = pd.DataFrame(new_endog, index=new_index,
columns=columns)
# Standardize `exog` to have the right index
if new_exog is not None:
new_exog = pd.DataFrame(new_exog, index=new_index,
columns=self.model.exog_names)
if copy_initialization:
res = self.filter_results
init = Initialization(
self.model.k_states, 'known', constant=res.initial_state,
stationary_cov=res.initial_state_cov)
kwargs.setdefault('initialization', init)
mod = self.model.clone(new_endog, exog=new_exog, **kwargs)
res = self._apply(mod, refit=refit, fit_kwargs=fit_kwargs, **kwargs)
return res
[docs] def extend(self, endog, exog=None, fit_kwargs=None, **kwargs):
"""
Recreate the results object for new data that extends the original data
Creates a new result object applied to a new dataset that is assumed to
follow directly from the end of the model's original data. The new
results can then be used for analysis or forecasting.
Parameters
----------
endog : array_like
New observations from the modeled time-series process.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
fit_kwargs : dict, optional
Keyword arguments to pass to `filter` or `smooth`.
**kwargs
Keyword arguments may be used to modify model specification
arguments when created the new model object.
Returns
-------
results
Updated Results object, that includes results only for the new
dataset.
See Also
--------
statsmodels.tsa.statespace.mlemodel.MLEResults.append
statsmodels.tsa.statespace.mlemodel.MLEResults.apply
Notes
-----
The `endog` argument to this method should consist of new observations
that occurred directly after the last element of the model's original
`endog` array. For any other kind of dataset, see the `apply` method.
This method will apply filtering only to the new data provided by the
`endog` argument, which can be much faster than re-filtering the entire
dataset. However, the returned results object will only have results
for the new data. To retrieve results for both the new data and the
original data, see the `append` method.
Examples
--------
>>> index = pd.period_range(start='2000', periods=2, freq='A')
>>> original_observations = pd.Series([1.2, 1.5], index=index)
>>> mod = sm.tsa.SARIMAX(original_observations)
>>> res = mod.fit()
>>> print(res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(res.fittedvalues)
2000 0.0000
2001 1.1707
Freq: A-DEC, dtype: float64
>>> print(res.forecast(1))
2002 1.4634
Freq: A-DEC, dtype: float64
>>> new_index = pd.period_range(start='2002', periods=1, freq='A')
>>> new_observations = pd.Series([0.9], index=new_index)
>>> updated_res = res.extend(new_observations)
>>> print(updated_res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(updated_res.fittedvalues)
2002 1.4634
Freq: A-DEC, dtype: float64
>>> print(updated_res.forecast(1))
2003 0.878
Freq: A-DEC, dtype: float64
"""
start = self.nobs
end = self.nobs + len(endog) - 1
_, _, _, extend_ix = self.model._get_prediction_index(start, end)
if isinstance(self.model.data, PandasData):
_check_index(extend_ix, endog, '`endog`')
# Standardize `endog` to have the right index and columns
columns = self.model.endog_names
if not isinstance(columns, list):
columns = [columns]
endog = pd.DataFrame(endog, index=extend_ix, columns=columns)
# Extend the current fit result to additional data
mod = self.model.clone(endog, exog=exog, **kwargs)
mod.ssm.initialization = Initialization(
mod.k_states, 'known', constant=self.predicted_state[..., -1],
stationary_cov=self.predicted_state_cov[..., -1])
res = self._apply(mod, refit=False, fit_kwargs=fit_kwargs, **kwargs)
return res
[docs] def apply(self, endog, exog=None, refit=False, fit_kwargs=None,
copy_initialization=False, **kwargs):
"""
Apply the fitted parameters to new data unrelated to the original data
Creates a new result object using the current fitted parameters,
applied to a completely new dataset that is assumed to be unrelated to
the model's original data. The new results can then be used for
analysis or forecasting.
Parameters
----------
endog : array_like
New observations from the modeled time-series process.
exog : array_like, optional
New observations of exogenous regressors, if applicable.
refit : bool, optional
Whether to re-fit the parameters, using the new dataset.
Default is False (so parameters from the current results object
are used to create the new results object).
copy_initialization : bool, optional
Whether or not to copy the initialization from the current results
set to the new model. Default is False
fit_kwargs : dict, optional
Keyword arguments to pass to `fit` (if `refit=True`) or `filter` /
`smooth`.
**kwargs
Keyword arguments may be used to modify model specification
arguments when created the new model object.
Returns
-------
results
Updated Results object, that includes results only for the new
dataset.
See Also
--------
statsmodels.tsa.statespace.mlemodel.MLEResults.append
statsmodels.tsa.statespace.mlemodel.MLEResults.apply
Notes
-----
The `endog` argument to this method should consist of new observations
that are not necessarily related to the original model's `endog`
dataset. For observations that continue that original dataset by follow
directly after its last element, see the `append` and `extend` methods.
Examples
--------
>>> index = pd.period_range(start='2000', periods=2, freq='A')
>>> original_observations = pd.Series([1.2, 1.5], index=index)
>>> mod = sm.tsa.SARIMAX(original_observations)
>>> res = mod.fit()
>>> print(res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(res.fittedvalues)
2000 0.0000
2001 1.1707
Freq: A-DEC, dtype: float64
>>> print(res.forecast(1))
2002 1.4634
Freq: A-DEC, dtype: float64
>>> new_index = pd.period_range(start='1980', periods=3, freq='A')
>>> new_observations = pd.Series([1.4, 0.3, 1.2], index=new_index)
>>> new_res = res.apply(new_observations)
>>> print(new_res.params)
ar.L1 0.9756
sigma2 0.0889
dtype: float64
>>> print(new_res.fittedvalues)
1980 1.1707
1981 1.3659
1982 0.2927
Freq: A-DEC, dtype: float64
Freq: A-DEC, dtype: float64
>>> print(new_res.forecast(1))
1983 1.1707
Freq: A-DEC, dtype: float64
"""
mod = self.model.clone(endog, exog=exog, **kwargs)
if copy_initialization:
res = self.filter_results
init = Initialization(
self.model.k_states, 'known', constant=res.initial_state,
stationary_cov=res.initial_state_cov)
mod.ssm.initialization = init
res = self._apply(mod, refit=refit, fit_kwargs=fit_kwargs, **kwargs)
return res
[docs] def plot_diagnostics(self, variable=0, lags=10, fig=None, figsize=None,
truncate_endog_names=24, auto_ylims=False,
bartlett_confint=False, acf_kwargs=None):
"""
Diagnostic plots for standardized residuals of one endogenous variable
Parameters
----------
variable : int, optional
Index of the endogenous variable for which the diagnostic plots
should be created. Default is 0.
lags : int, optional
Number of lags to include in the correlogram. Default is 10.
fig : Figure, optional
If given, subplots are created in this figure instead of in a new
figure. Note that the 2x2 grid will be created in the provided
figure using `fig.add_subplot()`.
figsize : tuple, optional
If a figure is created, this argument allows specifying a size.
The tuple is (width, height).
auto_ylims : bool, optional
If True, adjusts automatically the y-axis limits to ACF values.
bartlett_confint : bool, default True
Confidence intervals for ACF values are generally placed at 2
standard errors around r_k. The formula used for standard error
depends upon the situation. If the autocorrelations are being used
to test for randomness of residuals as part of the ARIMA routine,
the standard errors are determined assuming the residuals are white
noise. The approximate formula for any lag is that standard error
of each r_k = 1/sqrt(N). See section 9.4 of [1] for more details on
the 1/sqrt(N) result. For more elementary discussion, see section
5.3.2 in [2].
For the ACF of raw data, the standard error at a lag k is
found as if the right model was an MA(k-1). This allows the
possible interpretation that if all autocorrelations past a
certain lag are within the limits, the model might be an MA of
order defined by the last significant autocorrelation. In this
case, a moving average model is assumed for the data and the
standard errors for the confidence intervals should be
generated using Bartlett's formula. For more details on
Bartlett formula result, see section 7.2 in [1].+
acf_kwargs : dict, optional
Optional dictionary of keyword arguments that are directly passed
on to the correlogram Matplotlib plot produced by plot_acf().
Returns
-------
Figure
Figure instance with diagnostic plots
See Also
--------
statsmodels.graphics.gofplots.qqplot
statsmodels.graphics.tsaplots.plot_acf
Notes
-----
Produces a 2x2 plot grid with the following plots (ordered clockwise
from top left):
1. Standardized residuals over time
2. Histogram plus estimated density of standardized residuals, along
with a Normal(0,1) density plotted for reference.
3. Normal Q-Q plot, with Normal reference line.
4. Correlogram
References
----------
[1] Brockwell and Davis, 1987. Time Series Theory and Methods
[2] Brockwell and Davis, 2010. Introduction to Time Series and
Forecasting, 2nd edition.
"""
from statsmodels.graphics.utils import _import_mpl, create_mpl_fig
_import_mpl()
fig = create_mpl_fig(fig, figsize)
# Eliminate residuals associated with burned or diffuse likelihoods
d = np.maximum(self.loglikelihood_burn, self.nobs_diffuse)
# If given a variable name, find the index
if isinstance(variable, str):
variable = self.model.endog_names.index(variable)
# Get residuals
if hasattr(self.data, 'dates') and self.data.dates is not None:
ix = self.data.dates[d:]
else:
ix = np.arange(self.nobs - d)
resid = pd.Series(
self.filter_results.standardized_forecasts_error[variable, d:],
index=ix)
if resid.shape[0] < max(d, lags):
raise ValueError(
"Length of endogenous variable must be larger the the number "
"of lags used in the model and the number of observations "
"burned in the log-likelihood calculation."
)
# Top-left: residuals vs time
ax = fig.add_subplot(221)
resid.dropna().plot(ax=ax)
ax.hlines(0, ix[0], ix[-1], alpha=0.5)
ax.set_xlim(ix[0], ix[-1])
name = self.model.endog_names[variable]
if len(name) > truncate_endog_names:
name = name[:truncate_endog_names - 3] + '...'
ax.set_title(f'Standardized residual for "{name}"')
# Top-right: histogram, Gaussian kernel density, Normal density
# Can only do histogram and Gaussian kernel density on the non-null
# elements
resid_nonmissing = resid.dropna()
ax = fig.add_subplot(222)
ax.hist(resid_nonmissing, density=True, label='Hist',
edgecolor='#FFFFFF')
from scipy.stats import gaussian_kde, norm
kde = gaussian_kde(resid_nonmissing)
xlim = (-1.96*2, 1.96*2)
x = np.linspace(xlim[0], xlim[1])
ax.plot(x, kde(x), label='KDE')
ax.plot(x, norm.pdf(x), label='N(0,1)')
ax.set_xlim(xlim)
ax.legend()
ax.set_title('Histogram plus estimated density')
# Bottom-left: QQ plot
ax = fig.add_subplot(223)
from statsmodels.graphics.gofplots import qqplot
qqplot(resid_nonmissing, line='s', ax=ax)
ax.set_title('Normal Q-Q')
# Bottom-right: Correlogram
ax = fig.add_subplot(224)
from statsmodels.graphics.tsaplots import plot_acf
if acf_kwargs is None:
acf_kwargs = {}
plot_acf(resid, ax=ax, lags=lags, auto_ylims=auto_ylims,
bartlett_confint=bartlett_confint, **acf_kwargs)
ax.set_title('Correlogram')
return fig
[docs] def summary(self, alpha=.05, start=None, title=None, model_name=None,
display_params=True, display_diagnostics=True,
truncate_endog_names=None, display_max_endog=None,
extra_top_left=None, extra_top_right=None):
"""
Summarize the Model
Parameters
----------
alpha : float, optional
Significance level for the confidence intervals. Default is 0.05.
start : int, optional
Integer of the start observation. Default is 0.
model_name : str
The name of the model used. Default is to use model class name.
Returns
-------
summary : Summary instance
This holds the summary table and text, which can be printed or
converted to various output formats.
See Also
--------
statsmodels.iolib.summary.Summary
"""
from statsmodels.iolib.summary import Summary
from statsmodels.iolib.table import SimpleTable
from statsmodels.iolib.tableformatting import fmt_params
# Model specification results
model = self.model
if title is None:
title = 'Statespace Model Results'
if start is None:
start = 0
if self.model._index_dates:
ix = self.model._index
d = ix[start]
sample = ['%02d-%02d-%02d' % (d.month, d.day, d.year)]
d = ix[-1]
sample += ['- ' + '%02d-%02d-%02d' % (d.month, d.day, d.year)]
else:
sample = [str(start), ' - ' + str(self.nobs)]
# Standardize the model name as a list of str
if model_name is None:
model_name = model.__class__.__name__
# Truncate endog names
if truncate_endog_names is None:
truncate_endog_names = False if self.model.k_endog == 1 else 24
endog_names = self.model.endog_names
if not isinstance(endog_names, list):
endog_names = [endog_names]
endog_names = [str(name) for name in endog_names]
if truncate_endog_names is not False:
n = truncate_endog_names
endog_names = [name if len(name) <= n else name[:n] + '...'
for name in endog_names]
# Shorten the endog name list if applicable
if display_max_endog is None:
display_max_endog = np.inf
yname = None
if self.model.k_endog > display_max_endog:
k = self.model.k_endog - 1
yname = '"' + endog_names[0] + f'", and {k} more'
# Create the tables
if not isinstance(model_name, list):
model_name = [model_name]
top_left = [('Dep. Variable:', None)]
top_left.append(('Model:', [model_name[0]]))
for i in range(1, len(model_name)):
top_left.append(('', ['+ ' + model_name[i]]))
top_left += [
('Date:', None),
('Time:', None),
('Sample:', [sample[0]]),
('', [sample[1]])
]
top_right = [
('No. Observations:', [self.nobs]),
('Log Likelihood', ["%#5.3f" % self.llf]),
]
if hasattr(self, 'rsquared'):
top_right.append(('R-squared:', ["%#8.3f" % self.rsquared]))
top_right += [
('AIC', ["%#5.3f" % self.aic]),
('BIC', ["%#5.3f" % self.bic]),
('HQIC', ["%#5.3f" % self.hqic])]
if (self.filter_results is not None and
self.filter_results.filter_concentrated):
top_right.append(('Scale', ["%#5.3f" % self.scale]))
if hasattr(self, 'cov_type'):
cov_type = self.cov_type
if cov_type == 'none':
cov_type = 'Not computed'
top_left.append(('Covariance Type:', [cov_type]))
if extra_top_left is not None:
top_left += extra_top_left
if extra_top_right is not None:
top_right += extra_top_right
summary = Summary()
summary.add_table_2cols(self, gleft=top_left, gright=top_right,
title=title, yname=yname)
table_ix = 1
if len(self.params) > 0 and display_params:
summary.add_table_params(self, alpha=alpha,
xname=self.param_names, use_t=False)
table_ix += 1
# Diagnostic tests results
if display_diagnostics:
try:
het = self.test_heteroskedasticity(method='breakvar')
except Exception: # FIXME: catch something specific
het = np.zeros((self.model.k_endog, 2)) * np.nan
try:
lb = self.test_serial_correlation(method='ljungbox', lags=[1])
except Exception: # FIXME: catch something specific
lb = np.zeros((self.model.k_endog, 2, 1)) * np.nan
try:
jb = self.test_normality(method='jarquebera')
except Exception: # FIXME: catch something specific
jb = np.zeros((self.model.k_endog, 4)) * np.nan
if self.model.k_endog <= display_max_endog:
format_str = lambda array: [ # noqa:E731
', '.join(['{0:.2f}'.format(i) for i in array])
]
diagn_left = [
('Ljung-Box (L1) (Q):', format_str(lb[:, 0, -1])),
('Prob(Q):', format_str(lb[:, 1, -1])),
('Heteroskedasticity (H):', format_str(het[:, 0])),
('Prob(H) (two-sided):', format_str(het[:, 1]))]
diagn_right = [('Jarque-Bera (JB):', format_str(jb[:, 0])),
('Prob(JB):', format_str(jb[:, 1])),
('Skew:', format_str(jb[:, 2])),
('Kurtosis:', format_str(jb[:, 3]))
]
summary.add_table_2cols(self, gleft=diagn_left,
gright=diagn_right, title="")
else:
columns = ['LjungBox\n(L1) (Q)', 'Prob(Q)',
'Het.(H)', 'Prob(H)',
'Jarque\nBera(JB)', 'Prob(JB)', 'Skew', 'Kurtosis']
data = pd.DataFrame(
np.c_[lb[:, :2, -1], het[:, :2], jb[:, :4]],
index=endog_names, columns=columns).applymap(
lambda num: '' if pd.isnull(num) else '%.2f' % num)
data.index.name = 'Residual of\nDep. variable'
data = data.reset_index()
params_data = data.values
params_header = data.columns.tolist()
params_stubs = None
title = 'Residual diagnostics:'
table = SimpleTable(
params_data, params_header, params_stubs,
txt_fmt=fmt_params, title=title)
summary.tables.insert(table_ix, table)
# Add warnings/notes, added to text format only
etext = []
if hasattr(self, 'cov_type') and 'description' in self.cov_kwds:
etext.append(self.cov_kwds['description'])
if self._rank < (len(self.params) - len(self.fixed_params)):
cov_params = self.cov_params()
if len(self.fixed_params) > 0:
mask = np.ix_(self._free_params_index, self._free_params_index)
cov_params = cov_params[mask]
etext.append("Covariance matrix is singular or near-singular,"
" with condition number %6.3g. Standard errors may be"
" unstable." % _safe_cond(cov_params))
if etext:
etext = ["[{0}] {1}".format(i + 1, text)
for i, text in enumerate(etext)]
etext.insert(0, "Warnings:")
summary.add_extra_txt(etext)
return summary
class MLEResultsWrapper(wrap.ResultsWrapper):
_attrs = {
'zvalues': 'columns',
'cov_params_approx': 'cov',
'cov_params_default': 'cov',
'cov_params_oim': 'cov',
'cov_params_opg': 'cov',
'cov_params_robust': 'cov',
'cov_params_robust_approx': 'cov',
'cov_params_robust_oim': 'cov',
}
_wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
_attrs)
_methods = {
'forecast': 'dates',
'impulse_responses': 'ynames'
}
_wrap_methods = wrap.union_dicts(
tsbase.TimeSeriesResultsWrapper._wrap_methods, _methods)
wrap.populate_wrapper(MLEResultsWrapper, MLEResults) # noqa:E305
class PredictionResults(pred.PredictionResults):
"""
Parameters
----------
prediction_results : kalman_filter.PredictionResults instance
Results object from prediction after fitting or filtering a state space
model.
row_labels : iterable
Row labels for the predicted data.
Attributes
----------
"""
def __init__(self, model, prediction_results, row_labels=None):
if model.model.k_endog == 1:
endog = pd.Series(prediction_results.endog[0],
name=model.model.endog_names)
else:
endog = pd.DataFrame(prediction_results.endog.T,
columns=model.model.endog_names)
self.model = Bunch(data=model.data.__class__(
endog=endog, predict_dates=row_labels))
self.prediction_results = prediction_results
# Get required values
k_endog, nobs = prediction_results.endog.shape
if not prediction_results.results.memory_no_forecast_mean:
predicted_mean = self.prediction_results.forecasts
else:
predicted_mean = np.zeros((k_endog, nobs)) * np.nan
if predicted_mean.shape[0] == 1:
predicted_mean = predicted_mean[0, :]
else:
predicted_mean = predicted_mean.transpose()
if not prediction_results.results.memory_no_forecast_cov:
var_pred_mean = self.prediction_results.forecasts_error_cov
else:
var_pred_mean = np.zeros((k_endog, k_endog, nobs)) * np.nan
if var_pred_mean.shape[0] == 1:
var_pred_mean = var_pred_mean[0, 0, :]
else:
var_pred_mean = var_pred_mean.transpose()
# Initialize
super(PredictionResults, self).__init__(predicted_mean, var_pred_mean,
dist='norm',
row_labels=row_labels)
@property
def se_mean(self):
# Replace negative values with np.nan to avoid a RuntimeWarning
var_pred_mean = self.var_pred_mean.copy()
var_pred_mean[var_pred_mean < 0] = np.nan
if var_pred_mean.ndim == 1:
se_mean = np.sqrt(var_pred_mean)
else:
se_mean = np.sqrt(var_pred_mean.T.diagonal())
return se_mean
def conf_int(self, method='endpoint', alpha=0.05, **kwds):
# TODO: this performs metadata wrapping, and that should be handled
# by attach_* methods. However, they do not currently support
# this use case.
_use_pandas = self._use_pandas
self._use_pandas = False
conf_int = super(PredictionResults, self).conf_int(alpha, **kwds)
self._use_pandas = _use_pandas
# Create a dataframe
if self._row_labels is not None:
conf_int = pd.DataFrame(conf_int, index=self.row_labels)
# Attach the endog names
ynames = self.model.data.ynames
if not type(ynames) == list:
ynames = [ynames]
names = (['lower {0}'.format(name) for name in ynames] +
['upper {0}'.format(name) for name in ynames])
conf_int.columns = names
return conf_int
def summary_frame(self, endog=0, alpha=0.05):
# TODO: finish and cleanup
# import pandas as pd
# ci_obs = self.conf_int(alpha=alpha, obs=True) # need to split
ci_mean = np.asarray(self.conf_int(alpha=alpha))
_use_pandas = self._use_pandas
self._use_pandas = False
to_include = {}
if self.predicted_mean.ndim == 1:
yname = self.model.data.ynames
to_include['mean'] = self.predicted_mean
to_include['mean_se'] = self.se_mean
k_endog = 1
else:
yname = self.model.data.ynames[endog]
to_include['mean'] = self.predicted_mean[:, endog]
to_include['mean_se'] = self.se_mean[:, endog]
k_endog = self.predicted_mean.shape[1]
self._use_pandas = _use_pandas
to_include['mean_ci_lower'] = ci_mean[:, endog]
to_include['mean_ci_upper'] = ci_mean[:, k_endog + endog]
# pandas dict does not handle 2d_array
# data = np.column_stack(list(to_include.values()))
# names = ....
res = pd.DataFrame(to_include, index=self._row_labels,
columns=list(to_include.keys()))
res.columns.name = yname
return res
class PredictionResultsWrapper(wrap.ResultsWrapper):
_attrs = {
'predicted_mean': 'dates',
'se_mean': 'dates',
't_values': 'dates',
}
_wrap_attrs = wrap.union_dicts(_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(_methods)
wrap.populate_wrapper(PredictionResultsWrapper, PredictionResults) # noqa:E305