Source code for statsmodels.tsa.statespace.sarimax

"""
SARIMAX Model

"""
from __future__ import division, absolute_import, print_function
from statsmodels.compat.python import long

from warnings import warn

import numpy as np
import pandas as pd
from .kalman_filter import KalmanFilter, FilterResults
from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper
from .tools import (
companion_matrix, diff, is_invertible, constrain_stationary_univariate,
unconstrain_stationary_univariate, solve_discrete_lyapunov
)
from statsmodels.tools.tools import Bunch
from statsmodels.tools.data import _is_using_pandas
from statsmodels.tsa.tsatools import lagmat
from statsmodels.tools.sm_exceptions import ValueWarning
import statsmodels.base.wrapper as wrap

[docs]class SARIMAX(MLEModel):
r"""
Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors
model

Parameters
----------
endog : array_like
The observed time-series process :math:y
exog : array_like, optional
Array of exogenous regressors, shaped nobs x k.
order : iterable or iterable of iterables, optional
The (p,d,q) order of the model for the number of AR parameters,
differences, and MA parameters. d must be an integer
indicating the integration order of the process, while
p and q may either be an integers indicating the AR and MA
orders (so that all lags up to those orders are included) or else
iterables giving specific AR and / or MA lags to include. Default is
an AR(1) model: (1,0,0).
seasonal_order : iterable, optional
The (P,D,Q,s) order of the seasonal component of the model for the
AR parameters, differences, MA parameters, and periodicity.
d must be an integer indicating the integration order of the process,
while p and q may either be an integers indicating the AR and MA
orders (so that all lags up to those orders are included) or else
iterables giving specific AR and / or MA lags to include. s is an
integer giving the periodicity (number of periods in season), often it
is 4 for quarterly data or 12 for monthly data. Default is no seasonal
effect.
trend : str{'n','c','t','ct'} or iterable, optional
Parameter controlling the deterministic trend polynomial :math:A(t).
Can be specified as a string where 'c' indicates a constant (i.e. a
degree zero component of the trend polynomial), 't' indicates a
linear trend with time, and 'ct' is both. Can also be specified as an
iterable defining the polynomial as in numpy.poly1d, where
[1,1,0,1] would denote :math:a + bt + ct^3. Default is to not
include a trend component.
measurement_error : boolean, optional
Whether or not to assume the endogenous observations endog were
measured with error. Default is False.
time_varying_regression : boolean, optional
Used when an explanatory variables, exog, are provided provided
to select whether or not coefficients on the exogenous regressors are
allowed to vary over time. Default is False.
mle_regression : boolean, optional
Whether or not to use estimate the regression coefficients for the
exogenous variables as part of maximum likelihood estimation or through
the Kalman filter (i.e. recursive least squares). If
time_varying_regression is True, this must be set to False. Default
is True.
simple_differencing : boolean, optional
Whether or not to use partially conditional maximum likelihood
estimation. If True, differencing is performed prior to estimation,
which discards the first :math:s D + d initial rows but reuslts in a
smaller state-space formulation. If False, the full SARIMAX model is
put in state-space form so that all datapoints can be used in
estimation. Default is False.
enforce_stationarity : boolean, optional
Whether or not to transform the AR parameters to enforce stationarity
in the autoregressive component of the model. Default is True.
enforce_invertibility : boolean, optional
Whether or not to transform the MA parameters to enforce invertibility
in the moving average component of the model. Default is True.
hamilton_representation : boolean, optional
Whether or not to use the Hamilton representation of an ARMA process
(if True) or the Harvey representation (if False). Default is False.
**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
----------
measurement_error : boolean
Whether or not to assume the endogenous
observations endog were measured with error.
state_error : boolean
Whether or not the transition equation has an error component.
mle_regression : boolean
Whether or not the regression coefficients for
the exogenous variables were estimated via maximum
likelihood estimation.
state_regression : boolean
Whether or not the regression coefficients for
the exogenous variables are included as elements
of the state space and estimated via the Kalman
filter.
time_varying_regression : boolean
Whether or not coefficients on the exogenous
regressors are allowed to vary over time.
simple_differencing : boolean
Whether or not to use partially conditional maximum likelihood
estimation.
enforce_stationarity : boolean
Whether or not to transform the AR parameters
to enforce stationarity in the autoregressive
component of the model.
enforce_invertibility : boolean
Whether or not to transform the MA parameters
to enforce invertibility in the moving average
component of the model.
hamilton_representation : boolean
Whether or not to use the Hamilton representation of an ARMA process.
trend : str{'n','c','t','ct'} or iterable
Parameter controlling the deterministic
trend polynomial :math:A(t). See the class
polynomial_ar : array
Array containing autoregressive lag polynomial
coefficients, ordered from lowest degree to highest.
Initialized with ones, unless a coefficient is
constrained to be zero (in which case it is zero).
polynomial_ma : array
Array containing moving average lag polynomial
coefficients, ordered from lowest degree to highest.
Initialized with ones, unless a coefficient is
constrained to be zero (in which case it is zero).
polynomial_seasonal_ar : array
Array containing seasonal moving average lag
polynomial coefficients, ordered from lowest degree
to highest. Initialized with ones, unless a
coefficient is constrained to be zero (in which
case it is zero).
polynomial_seasonal_ma : array
Array containing seasonal moving average lag
polynomial coefficients, ordered from lowest degree
to highest. Initialized with ones, unless a
coefficient is constrained to be zero (in which
case it is zero).
polynomial_trend : array
Array containing trend polynomial coefficients,
ordered from lowest degree to highest. Initialized
with ones, unless a coefficient is constrained to be
zero (in which case it is zero).
k_ar : int
Highest autoregressive order in the model, zero-indexed.
k_ar_params : int
Number of autoregressive parameters to be estimated.
k_diff : int
Order of intergration.
k_ma : int
Highest moving average order in the model, zero-indexed.
k_ma_params : int
Number of moving average parameters to be estimated.
seasonal_periods : int
Number of periods in a season.
k_seasonal_ar : int
Highest seasonal autoregressive order in the model, zero-indexed.
k_seasonal_ar_params : int
Number of seasonal autoregressive parameters to be estimated.
k_seasonal_diff : int
Order of seasonal intergration.
k_seasonal_ma : int
Highest seasonal moving average order in the model, zero-indexed.
k_seasonal_ma_params : int
Number of seasonal moving average parameters to be estimated.
k_trend : int
Order of the trend polynomial plus one (i.e. the constant polynomial
would have k_trend=1).
k_exog : int
Number of exogenous regressors.

Notes
-----
The SARIMA model is specified :math:(p, d, q) \times (P, D, Q)_s.

.. math::

\phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D y_t = A(t) +
\theta_q (L) \tilde \theta_Q (L^s) \zeta_t

In terms of a univariate structural model, this can be represented as

.. math::

y_t & = u_t + \eta_t \\
\phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) +
\theta_q (L) \tilde \theta_Q (L^s) \zeta_t

where :math:\eta_t is only applicable in the case of measurement error
(although it is also used in the case of a pure regression model, i.e. if
p=q=0).

In terms of this model, regression with SARIMA errors can be represented
easily as

.. math::

y_t & = \beta_t x_t + u_t \\
\phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) +
\theta_q (L) \tilde \theta_Q (L^s) \zeta_t

this model is the one used when exogenous regressors are provided.

Note that the reduced form lag polynomials will be written as:

.. math::

\Phi (L) \equiv \phi_p (L) \tilde \phi_P (L^s) \\
\Theta (L) \equiv \theta_q (L) \tilde \theta_Q (L^s)

If mle_regression is True, regression coefficients are treated as
additional parameters to be estimated via maximum likelihood. Otherwise
they are included as part of the state with a diffuse initialization.
In this case, however, with approximate diffuse initialization, results
can be sensitive to the initial variance.

This class allows two different underlying representations of ARMA models
as state space models: that of Hamilton and that of Harvey. Both are
equivalent in the sense that they are analytical representations of the
ARMA model, but the state vectors of each have different meanings. For
this reason, maximum likelihood does not result in identical parameter
estimates and even the same set of parameters will result in different
loglikelihoods.

The Harvey representation is convenient because it allows integrating
differencing into the state vector to allow using all observations for
estimation.

In this implementation of differenced models, the Hamilton representation
is not able to accomodate differencing in the state vector, so
simple_differencing (which performs differencing prior to estimation so
that the first d + sD observations are lost) must be used.

Many other packages use the Hamilton representation, so that tests against
Stata and R require using it along with simple differencing (as Stata
does).

Detailed information about state space models can be found in [1]_. Some
specific references are:

- Chapter 3.4 describes ARMA and ARIMA models in state space form (using
the Harvey representation), and gives references for basic seasonal
models and models with a multiplicative form (for example the airline
model). It also shows a state space model for a full ARIMA process (this
is what is done here if simple_differencing=False).
- Chapter 3.6 describes estimating regression effects via the Kalman filter
(this is performed if mle_regression is False), regression with
time-varying coefficients, and regression with ARMA errors (recall from
above that if regression effects are present, the model estimated by this
class is regression with SARIMA errors).
- Chapter 8.4 describes the application of an ARMA model to an example
dataset. A replication of this section is available in an example
IPython notebook in the documentation.

References
----------
.. [1] Durbin, James, and Siem Jan Koopman. 2012.
Time Series Analysis by State Space Methods: Second Edition.
Oxford University Press.
"""

def __init__(self, endog, exog=None, order=(1, 0, 0),
seasonal_order=(0, 0, 0, 0), trend=None,
measurement_error=False, time_varying_regression=False,
mle_regression=True, simple_differencing=False,
enforce_stationarity=True, enforce_invertibility=True,
hamilton_representation=False, **kwargs):

# Model parameters
self.seasonal_periods = seasonal_order[3]
self.measurement_error = measurement_error
self.time_varying_regression = time_varying_regression
self.mle_regression = mle_regression
self.simple_differencing = simple_differencing
self.enforce_stationarity = enforce_stationarity
self.enforce_invertibility = enforce_invertibility
self.hamilton_representation = hamilton_representation

# Save given orders
self.order = order
self.seasonal_order = seasonal_order

# Enforce non-MLE coefficients if time varying coefficients is
# specified
if self.time_varying_regression and self.mle_regression:
raise ValueError('Models with time-varying regression coefficients'
' must integrate the coefficients as part of the'
' state vector, so that mle_regression must'
' be set to False.')

# Lag polynomials
# Assume that they are given from lowest degree to highest, that all
# degrees except for the constant are included, and that they are
# boolean vectors (0 for not included, 1 for included).
if isinstance(order[0], (int, long)):
self.polynomial_ar = np.r_[1., np.ones(order[0])]
else:
self.polynomial_ar = np.r_[1., order[0]]
if isinstance(order[2], (int, long)):
self.polynomial_ma = np.r_[1., np.ones(order[2])]
else:
self.polynomial_ma = np.r_[1., order[2]]
# Assume that they are given from lowest degree to highest, that the
# degrees correspond to (1*s, 2*s, ..., P*s), and that they are
# boolean vectors (0 for not included, 1 for included).
if isinstance(seasonal_order[0], (int, long)):
self.polynomial_seasonal_ar = np.r_[
1.,  # constant
([0] * (self.seasonal_periods - 1) + [1]) * seasonal_order[0]
]
else:
self.polynomial_seasonal_ar = np.r_[
1., [0] * self.seasonal_periods * len(seasonal_order[0])
]
for i in range(len(seasonal_order[0])):
self.polynomial_seasonal_ar[(i + 1) * self.seasonal_periods] = (
seasonal_order[0][i]
)
if isinstance(seasonal_order[2], (int, long)):
self.polynomial_seasonal_ma = np.r_[
1.,  # constant
([0] * (self.seasonal_periods - 1) + [1]) * seasonal_order[2]
]
else:
self.polynomial_seasonal_ma = np.r_[
1., [0] * self.seasonal_periods * len(seasonal_order[2])
]
for i in range(len(seasonal_order[2])):
self.polynomial_seasonal_ma[(i + 1) * self.seasonal_periods] = (
seasonal_order[2][i]
)

# Deterministic trend polynomial
self.trend = trend
if trend is None or trend == 'n':
self.polynomial_trend = np.ones((0))
elif trend == 'c':
self.polynomial_trend = np.r_[1]
elif trend == 't':
self.polynomial_trend = np.r_[0, 1]
elif trend == 'ct':
self.polynomial_trend = np.r_[1, 1]
else:
self.polynomial_trend = (np.array(trend) > 0).astype(int)

# Model orders
# Note: k_ar, k_ma, k_seasonal_ar, k_seasonal_ma do not include the
# constant term, so they may be zero.
# Note: for a typical ARMA(p,q) model, p = k_ar_params = k_ar - 1 and
# q = k_ma_params = k_ma - 1, although this may not be true for models
# with arbitrary log polynomials.
self.k_ar = int(self.polynomial_ar.shape[0] - 1)
self.k_ar_params = int(np.sum(self.polynomial_ar) - 1)
self.k_diff = int(order[1])
self.k_ma = int(self.polynomial_ma.shape[0] - 1)
self.k_ma_params = int(np.sum(self.polynomial_ma) - 1)

self.k_seasonal_ar = int(self.polynomial_seasonal_ar.shape[0] - 1)
self.k_seasonal_ar_params = (
int(np.sum(self.polynomial_seasonal_ar) - 1)
)
self.k_seasonal_diff = int(seasonal_order[1])
self.k_seasonal_ma = int(self.polynomial_seasonal_ma.shape[0] - 1)
self.k_seasonal_ma_params = (
int(np.sum(self.polynomial_seasonal_ma) - 1)
)

# Make internal copies of the differencing orders because if we use
# simple differencing, then we will need to internally use zeros after
# the simple differencing has been performed
self._k_diff = self.k_diff
self._k_seasonal_diff = self.k_seasonal_diff

# We can only use the Hamilton representation if differencing is not
# performed as a part of the state space
if (self.hamilton_representation and not (self.simple_differencing or
self._k_diff == self._k_seasonal_diff == 0)):
raise ValueError('The Hamilton representation is only available'
' for models in which there is no differencing'
' integrated into the state vector. Set'
' simple_differencing to True or set'
' hamilton_representation to False')

# Note: k_trend is not the degree of the trend polynomial, because e.g.
# k_trend = 1 corresponds to the degree zero polynomial (with only a
# constant term).
self.k_trend = int(np.sum(self.polynomial_trend))

# Model order
# (this is used internally in a number of locations)
self._k_order = max(self.k_ar + self.k_seasonal_ar,
self.k_ma + self.k_seasonal_ma + 1)
if self._k_order == 1 and self.k_ar + self.k_seasonal_ar == 0:
self._k_order = 0

# Exogenous data
self.k_exog = 0
if exog is not None:
exog_is_using_pandas = _is_using_pandas(exog, None)
if not exog_is_using_pandas:
exog = np.asarray(exog)

# Make sure we have 2-dimensional array
if exog.ndim < 2:
if not exog_is_using_pandas:
exog = np.atleast_2d(exog).T
else:
exog = pd.DataFrame(exog)

self.k_exog = exog.shape[1]
# Redefine mle_regression to be true only if it was previously set to
# true and there are exogenous regressors
self.mle_regression = (
self.mle_regression and exog is not None and self.k_exog > 0
)
# State regression is regression with coefficients estiamted within
# the state vector
self.state_regression = (
not self.mle_regression and exog is not None and self.k_exog > 0
)
# If all we have is a regression (so k_ar = k_ma = 0), then put the
# error term as measurement error
if self.state_regression and self._k_order == 0:
self.measurement_error = True

# Number of states
k_states = self._k_order
if not self.simple_differencing:
k_states += self.seasonal_periods * self._k_seasonal_diff + self._k_diff
if self.state_regression:
k_states += self.k_exog

# Number of diffuse states
k_diffuse_states = k_states
if self.enforce_stationarity:
k_diffuse_states -= self._k_order

# Number of positive definite elements of the state covariance matrix
k_posdef = int(self._k_order > 0)
# Only have an error component to the states if k_posdef > 0
self.state_error = k_posdef > 0
if self.state_regression and self.time_varying_regression:
k_posdef += self.k_exog

# Diffuse initialization can be more sensistive to the variance value
# in the case of state regression, so set a higher than usual default
# variance
if self.state_regression:
kwargs.setdefault('initial_variance', 1e10)

# Number of parameters
self.k_params = (
self.k_ar_params + self.k_ma_params +
self.k_seasonal_ar_params + self.k_seasonal_ar_params +
self.k_trend +
self.measurement_error + 1
)
if self.mle_regression:
self.k_params += self.k_exog

# We need to have an array or pandas at this point
self.orig_endog = endog
self.orig_exog = exog
if not _is_using_pandas(endog, None):
endog = np.asanyarray(endog)

# Update the differencing dimensions if simple differencing is applied
self.orig_k_diff = self._k_diff
self.orig_k_seasonal_diff = self._k_seasonal_diff
if (self.simple_differencing and
(self._k_diff > 0 or self._k_seasonal_diff > 0)):
self._k_diff = 0
self._k_seasonal_diff = 0

# Internally used in several locations
self._k_states_diff = (
self._k_diff + self.seasonal_periods * self._k_seasonal_diff
)

# Set some model variables now so they will be available for the
# initialize() method, below
self.nobs = len(endog)
self.k_states = k_states
self.k_posdef = k_posdef

# By default, do not calculate likelihood while it is controlled by
# diffuse initial conditions.
kwargs.setdefault('loglikelihood_burn', k_diffuse_states)

# Initialize the statespace
super(SARIMAX, self).__init__(
endog, exog=exog, k_states=k_states, k_posdef=k_posdef, **kwargs
)

# Set as time-varying model if we have time-trend or exog
if self.k_exog > 0 or len(self.polynomial_trend) > 1:
self.ssm._time_invariant = False

# Handle kwargs specified initialization
if self.ssm.initialization is not None:
self._manual_initialization = True

# Initialize the fixed components of the statespace model
self.ssm.design = self.initial_design
self.ssm.state_intercept = self.initial_state_intercept
self.ssm.transition = self.initial_transition
self.ssm.selection = self.initial_selection

# If we are estimating a simple ARMA model, then we can use a faster
# initialization method (unless initialization was already specified).
if k_diffuse_states == 0 and not self._manual_initialization:
self.initialize_stationary()

# update _init_keys attached by super
self._init_keys += ['order',  'seasonal_order', 'trend',
'measurement_error', 'time_varying_regression',
'mle_regression', 'simple_differencing',
'enforce_stationarity', 'enforce_invertibility',
'hamilton_representation'] + list(kwargs.keys())
# TODO: I think the kwargs or not attached, need to recover from ???

def _get_init_kwds(self):
kwds = super(SARIMAX, 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 prepare_data(self):
endog, exog = super(SARIMAX, self).prepare_data()

# Perform simple differencing if requested
if (self.simple_differencing and
(self.orig_k_diff > 0 or self.orig_k_seasonal_diff > 0)):
# Save the original length
orig_length = endog.shape[0]
# Perform simple differencing
endog = diff(endog.copy(), self.orig_k_diff,
self.orig_k_seasonal_diff, self.seasonal_periods)
if exog is not None:
exog = diff(exog.copy(), self.orig_k_diff,
self.orig_k_seasonal_diff, self.seasonal_periods)

# Reset the ModelData datasets
self.data.endog, self.data.exog = (
self.data._convert_endog_exog(endog, exog))

# Reset dates, if provided
if self.data.dates is not None:
new_length = self.data.endog.shape[0]
self.data.dates = self.data.dates[orig_length - new_length:]

# Reset the nobs
self.nobs = endog.shape[0]

# Cache the arrays for calculating the intercept from the trend
# components
time_trend = np.arange(1, self.nobs + 1)
self._trend_data = np.zeros((self.nobs, self.k_trend))
i = 0
for k in self.polynomial_trend.nonzero()[0]:
if k == 0:
self._trend_data[:, i] = np.ones(self.nobs,)
else:
self._trend_data[:, i] = time_trend**k
i += 1

return endog, exog

[docs]    def initialize(self):
"""
Initialize the SARIMAX model.

Notes
-----
These initialization steps must occur following the parent class
__init__ function calls.
"""
super(SARIMAX, self).initialize()

# Internal flag for whether the default mixed approximate diffuse /
# stationary initialization has been overridden with a user-supplied
# initialization
self._manual_initialization = False

# Cache the indexes of included polynomial orders (for update below)
# (but we do not want the index of the constant term, so exclude the
# first index)
self._polynomial_ar_idx = np.nonzero(self.polynomial_ar)[0][1:]
self._polynomial_ma_idx = np.nonzero(self.polynomial_ma)[0][1:]
self._polynomial_seasonal_ar_idx = np.nonzero(
self.polynomial_seasonal_ar
)[0][1:]
self._polynomial_seasonal_ma_idx = np.nonzero(
self.polynomial_seasonal_ma
)[0][1:]

# Save the indices corresponding to the reduced form lag polynomial
# parameters in the transition and selection matrices so that they
# don't have to be recalculated for each update()
start_row = self._k_states_diff
end_row = start_row + self.k_ar + self.k_seasonal_ar
col = self._k_states_diff
if not self.hamilton_representation:
self.transition_ar_params_idx = (
np.s_['transition', start_row:end_row, col]
)
else:
self.transition_ar_params_idx = (
np.s_['transition', col, start_row:end_row]
)

start_row += 1
end_row = start_row + self.k_ma + self.k_seasonal_ma
col = 0
if not self.hamilton_representation:
self.selection_ma_params_idx = (
np.s_['selection', start_row:end_row, col]
)
else:
self.design_ma_params_idx = (
np.s_['design', col, start_row:end_row]
)

# Cache indices for exog variances in the state covariance matrix
if self.state_regression and self.time_varying_regression:
idx = np.diag_indices(self.k_posdef)
self._exog_variance_idx = ('state_cov', idx[0][-self.k_exog:],
idx[1][-self.k_exog:])

[docs]    def initialize_known(self, initial_state, initial_state_cov):
self._manual_initialization = True
self.ssm.initialize_known(initial_state, initial_state_cov)
initialize_known.__doc__ = KalmanFilter.initialize_known.__doc__

[docs]    def initialize_approximate_diffuse(self, variance=None):
self._manual_initialization = True
self.ssm.initialize_approximate_diffuse(variance)
initialize_approximate_diffuse.__doc__ = (
KalmanFilter.initialize_approximate_diffuse.__doc__
)

[docs]    def initialize_stationary(self):
self._manual_initialization = True
self.ssm.initialize_stationary()
initialize_stationary.__doc__ = (
KalmanFilter.initialize_stationary.__doc__
)

[docs]    def initialize_state(self, variance=None, complex_step=False):
"""
Initialize state and state covariance arrays in preparation for the
Kalman filter.

Parameters
----------
variance : float, optional
The variance for approximating diffuse initial conditions. Default
can be found in the Representation class documentation.

Notes
-----
Initializes the ARMA component of the state space to the typical
stationary values and the other components as approximate diffuse.

Can be overridden be calling one of the other initialization methods
before fitting the model.
"""
# Check if a manual initialization has already been specified
if self._manual_initialization:
return

# If we're not enforcing stationarity, then we can't initialize a
# stationary component
if not self.enforce_stationarity:
self.initialize_approximate_diffuse(variance)
return

# Otherwise, create the initial state and state covariance matrix
# as from a combination of diffuse and stationary components

# Create initialized non-stationary components
if variance is None:
variance = self.ssm.initial_variance

dtype = self.ssm.transition.dtype
initial_state = np.zeros(self.k_states, dtype=dtype)
initial_state_cov = np.eye(self.k_states, dtype=dtype) * variance

# Get the offsets (from the bottom or bottom right of the vector /
# matrix) for the stationary component.
if self.state_regression:
start = -(self.k_exog + self._k_order)
end = -self.k_exog if self.k_exog > 0 else None
else:
start = -self._k_order
end = None

# Add in the initialized stationary components
if self._k_order > 0:
selection_stationary = self.ssm.selection[start:end, :, 0]
selected_state_cov_stationary = np.dot(
np.dot(selection_stationary, self.ssm.state_cov[:, :, 0]),
selection_stationary.T
)
initial_state_cov_stationary = solve_discrete_lyapunov(
self.ssm.transition[start:end, start:end, 0],
selected_state_cov_stationary,
complex_step=complex_step
)

initial_state_cov[start:end, start:end] = (
initial_state_cov_stationary
)

self.ssm.initialize_known(initial_state, initial_state_cov)

@property
def initial_design(self):
"""Initial design matrix"""
# Basic design matrix
design = np.r_[
[1] * self._k_diff,
([0] * (self.seasonal_periods - 1) + [1]) * self._k_seasonal_diff,
[1] * self.state_error, [0] * (self._k_order - 1)
]

# If we have exogenous regressors included as part of the state vector
# then the exogenous data is incorporated as a time-varying component
# of the design matrix
if self.state_regression:
if self._k_order > 0:
design = np.c_[
np.reshape(
np.repeat(design, self.nobs),
(design.shape[0], self.nobs)
).T,
self.exog
].T[None, :, :]
else:
design = self.exog.T[None, :, :]
return design

@property
def initial_state_intercept(self):
"""Initial state intercept vector"""
# TODO make this self.k_trend > 1 and adjust the update to take
# into account that if the trend is a constant, it is not time-varying
if self.k_trend > 0:
state_intercept = np.zeros((self.k_states, self.nobs))
else:
state_intercept = np.zeros((self.k_states,))
return state_intercept

@property
def initial_transition(self):
"""Initial transition matrix"""
transition = np.zeros((self.k_states, self.k_states))

# Exogenous regressors component
if self.state_regression:
start = -self.k_exog
# T_\beta
transition[start:, start:] = np.eye(self.k_exog)

# Autoregressive component
start = -(self.k_exog + self._k_order)
end = -self.k_exog if self.k_exog > 0 else None
else:
# Autoregressive component
start = -self._k_order
end = None

# T_c
transition[start:end, start:end] = companion_matrix(self._k_order)
if self.hamilton_representation:
transition[start:end, start:end] = np.transpose(
companion_matrix(self._k_order)
)

# Seasonal differencing component
# T^*
if self._k_seasonal_diff > 0:
seasonal_companion = companion_matrix(self.seasonal_periods).T
seasonal_companion[0, -1] = 1
for d in range(self._k_seasonal_diff):
start = self._k_diff + d * self.seasonal_periods
end = self._k_diff + (d + 1) * self.seasonal_periods

# T_c^*
transition[start:end, start:end] = seasonal_companion

# i
for i in range(d + 1, self._k_seasonal_diff):
transition[start, end + self.seasonal_periods - 1] = 1

# \iota
transition[start, self._k_states_diff] = 1

# Differencing component
if self._k_diff > 0:
idx = np.triu_indices(self._k_diff)
# T^**
transition[idx] = 1
# [0 1]
if self.seasonal_periods > 0:
start = self._k_diff
end = self._k_states_diff
transition[:self._k_diff, start:end] = (
([0] * (self.seasonal_periods - 1) + [1]) * self._k_seasonal_diff
)
# [1 0]
column = self._k_states_diff
transition[:self._k_diff, column] = 1

return transition

@property
def initial_selection(self):
"""Initial selection matrix"""
if not (self.state_regression and self.time_varying_regression):
if self.k_posdef > 0:
selection = np.r_[
[0] * (self._k_states_diff),
[1] * (self._k_order > 0), [0] * (self._k_order - 1),
[0] * ((1 - self.mle_regression) * self.k_exog)
][:, None]
else:
selection = np.zeros((self.k_states, 0))
else:
selection = np.zeros((self.k_states, self.k_posdef))
# Typical state variance
if self._k_order > 0:
selection[0, 0] = 1
# Time-varying regression coefficient variances
for i in range(self.k_exog, 0, -1):
selection[-i, -i] = 1
return selection

[docs]    def filter(self, params, **kwargs):
kwargs.setdefault('results_class', SARIMAXResults)
kwargs.setdefault('results_wrapper_class', SARIMAXResultsWrapper)
return super(SARIMAX, self).filter(params, **kwargs)

[docs]    def smooth(self, params, **kwargs):
kwargs.setdefault('results_class', SARIMAXResults)
kwargs.setdefault('results_wrapper_class', SARIMAXResultsWrapper)
return super(SARIMAX, self).smooth(params, **kwargs)

@staticmethod
def _conditional_sum_squares(endog, k_ar, polynomial_ar, k_ma,
polynomial_ma, k_trend=0, trend_data=None):
k = 2 * k_ma
r = max(k + k_ma, k_ar)

k_params_ar = 0 if k_ar == 0 else len(polynomial_ar.nonzero()[0]) - 1
k_params_ma = 0 if k_ma == 0 else len(polynomial_ma.nonzero()[0]) - 1

residuals = None
if k_ar + k_ma + k_trend > 0:
# If we have MA terms, get residuals from an AR(k) model to use
# as data for conditional sum of squares estimates of the MA
# parameters
if k_ma > 0:
Y = endog[k:]
X = lagmat(endog, k, trim='both')
params_ar = np.linalg.pinv(X).dot(Y)
residuals = Y - np.dot(X, params_ar)

# Run an ARMA(p,q) model using the just computed residuals as data
Y = endog[r:]

X = np.empty((Y.shape[0], 0))
if k_trend > 0:
if trend_data is None:
raise ValueError('Trend data must be provided if'
' k_trend > 0.')
X = np.c_[X, trend_data[:(-r if r > 0 else None), :]]
if k_ar > 0:
cols = polynomial_ar.nonzero()[0][1:] - 1
X = np.c_[X, lagmat(endog, k_ar)[r:, cols]]
if k_ma > 0:
cols = polynomial_ma.nonzero()[0][1:] - 1
X = np.c_[X, lagmat(residuals, k_ma)[r-k:, cols]]

# Get the array of [ar_params, ma_params]
params = np.linalg.pinv(X).dot(Y)
residuals = Y - np.dot(X, params)

# Default output
params_trend = []
params_ar = []
params_ma = []
params_variance = []

# Get the params
offset = 0
if k_trend > 0:
params_trend = params[offset:k_trend + offset]
offset += k_trend
if k_ar > 0:
params_ar = params[offset:k_params_ar + offset]
offset += k_params_ar
if k_ma > 0:
params_ma = params[offset:k_params_ma + offset]
offset += k_params_ma
if residuals is not None:
params_variance = (residuals[k_params_ma:]**2).mean()

return (params_trend, params_ar, params_ma,
params_variance)

@property
def start_params(self):
"""
Starting parameters for maximum likelihood estimation
"""

# Perform differencing if necessary (i.e. if simple differencing is
# false so that the state-space model will use the entire dataset)
trend_data = self._trend_data
if not self.simple_differencing and (
self._k_diff > 0 or self._k_seasonal_diff > 0):
endog = diff(self.endog, self._k_diff,
self._k_seasonal_diff, self.seasonal_periods)
if self.exog is not None:
exog = diff(self.exog, self._k_diff,
self._k_seasonal_diff, self.seasonal_periods)
else:
exog = None
trend_data = trend_data[:endog.shape[0], :]
else:
endog = self.endog.copy()
exog = self.exog.copy() if self.exog is not None else None
endog = endog.squeeze()

# Although the Kalman filter can deal with missing values in endog,
# conditional sum of squares cannot
if np.any(np.isnan(endog)):
if exog is not None:
if trend_data is not None:

# Regression effects via OLS
params_exog = []
if self.k_exog > 0:
params_exog = np.linalg.pinv(exog).dot(endog)
endog = endog - np.dot(exog, params_exog)
if self.state_regression:
params_exog = []

# Non-seasonal ARMA component and trend
(params_trend, params_ar, params_ma,
params_variance) = self._conditional_sum_squares(
endog, self.k_ar, self.polynomial_ar, self.k_ma,
self.polynomial_ma, self.k_trend, trend_data
)

# If we have estimated non-stationary start parameters but enforce
# stationarity is on, raise an error
invalid_ar = (
self.k_ar > 0 and
self.enforce_stationarity and
not is_invertible(np.r_[1, -params_ar])
)
if invalid_ar:
raise ValueError('Non-stationary starting autoregressive'
' parameters found with enforce_stationarity'
' set to True.')

# If we have estimated non-invertible start parameters but enforce
# invertibility is on, raise an error
invalid_ma = (
self.k_ma > 0 and
self.enforce_invertibility and
not is_invertible(np.r_[1, params_ma])
)
if invalid_ma:
raise ValueError('non-invertible starting MA parameters found'
' with enforce_invertibility set to True.')

# Seasonal Parameters
_, params_seasonal_ar, params_seasonal_ma, params_seasonal_variance = (
self._conditional_sum_squares(
endog, self.k_seasonal_ar, self.polynomial_seasonal_ar,
self.k_seasonal_ma, self.polynomial_seasonal_ma
)
)

# If we have estimated non-stationary start parameters but enforce
# stationarity is on, raise an error
invalid_seasonal_ar = (
self.k_seasonal_ar > 0 and
self.enforce_stationarity and
not is_invertible(np.r_[1, -params_seasonal_ar])
)
if invalid_seasonal_ar:
raise ValueError('Non-stationary starting autoregressive'
' parameters found with enforce_stationarity'
' set to True.')

# If we have estimated non-invertible start parameters but enforce
# invertibility is on, raise an error
invalid_seasonal_ma = (
self.k_seasonal_ma > 0 and
self.enforce_invertibility and
not is_invertible(np.r_[1, params_seasonal_ma])
)
if invalid_seasonal_ma:
raise ValueError('non-invertible starting seasonal moving average'
' parameters found with enforce_invertibility'
' set to True.')

# Variances
params_exog_variance = []
if self.state_regression and self.time_varying_regression:
# TODO how to set the initial variance parameters?
params_exog_variance = [1] * self.k_exog
if self.state_error and params_variance == []:
if not params_seasonal_variance == []:
params_variance = params_seasonal_variance
elif self.k_exog > 0:
params_variance = np.dot(endog, endog)
else:
params_variance = 1
params_measurement_variance = 1 if self.measurement_error else []

# Combine all parameters
return np.r_[
params_trend,
params_exog,
params_ar,
params_ma,
params_seasonal_ar,
params_seasonal_ma,
params_exog_variance,
params_measurement_variance,
params_variance
]

@property
def endog_names(self, latex=False):
"""Names of endogenous variables"""
diff = ''
if self.k_diff > 0:
if self.k_diff == 1:
diff = '\Delta' if latex else 'D'
else:
diff = ('\Delta^%d' if latex else 'D%d') % self.k_diff

seasonal_diff = ''
if self.k_seasonal_diff > 0:
if self.k_seasonal_diff == 1:
seasonal_diff = (('\Delta_%d' if latex else 'DS%d') %
(self.seasonal_periods))
else:
seasonal_diff = (('\Delta_%d^%d' if latex else 'D%dS%d') %
(self.k_seasonal_diff, self.seasonal_periods))
endog_diff = self.simple_differencing
if endog_diff and self.k_diff > 0 and self.k_seasonal_diff > 0:
return (('%s%s %s' if latex else '%s.%s.%s') %
(diff, seasonal_diff, self.data.ynames))
elif endog_diff and self.k_diff > 0:
return (('%s %s' if latex else '%s.%s') %
(diff, self.data.ynames))
elif endog_diff and self.k_seasonal_diff > 0:
return (('%s %s' if latex else '%s.%s') %
(seasonal_diff, self.data.ynames))
else:
return self.data.ynames

params_complete = [
'trend', 'exog', 'ar', 'ma', 'seasonal_ar', 'seasonal_ma',
'exog_variance', 'measurement_variance', 'variance'
]

@property
def param_terms(self):
"""
List of parameters actually included in the model, in sorted order.

TODO Make this an OrderedDict with slice or indices as the values.
"""
model_orders = self.model_orders
# Get basic list from model orders
params = [
order for order in self.params_complete
if model_orders[order] > 0
]
# k_exog may be positive without associated parameters if it is in the
# state vector
if 'exog' in params and not self.mle_regression:
params.remove('exog')

return params

@property
def param_names(self):
"""
List of human readable parameter names (for parameters actually
included in the model).
"""
params_sort_order = self.param_terms
model_names = self.model_names
return [
name for param in params_sort_order for name in model_names[param]
]

@property
def model_orders(self):
"""
The orders of each of the polynomials in the model.
"""
return {
'trend': self.k_trend,
'exog': self.k_exog,
'ar': self.k_ar,
'ma': self.k_ma,
'seasonal_ar': self.k_seasonal_ar,
'seasonal_ma': self.k_seasonal_ma,
'reduced_ar': self.k_ar + self.k_seasonal_ar,
'reduced_ma': self.k_ma + self.k_seasonal_ma,
'exog_variance': self.k_exog if (
self.state_regression and self.time_varying_regression) else 0,
'measurement_variance': int(self.measurement_error),
'variance': int(self.state_error),
}

@property
def model_names(self):
"""
The plain text names of all possible model parameters.
"""
return self._get_model_names(latex=False)

@property
def model_latex_names(self):
"""
The latex names of all possible model parameters.
"""
return self._get_model_names(latex=True)

def _get_model_names(self, latex=False):
names = {
'trend': None,
'exog': None,
'ar': None,
'ma': None,
'seasonal_ar': None,
'seasonal_ma': None,
'reduced_ar': None,
'reduced_ma': None,
'exog_variance': None,
'measurement_variance': None,
'variance': None,
}

# Trend
if self.k_trend > 0:
trend_template = 't_%d' if latex else 'trend.%d'
names['trend'] = []
for i in self.polynomial_trend.nonzero()[0]:
if i == 0:
names['trend'].append('intercept')
elif i == 1:
names['trend'].append('drift')
else:
names['trend'].append(trend_template % i)

# Exogenous coefficients
if self.k_exog > 0:
names['exog'] = self.exog_names

# Autoregressive
if self.k_ar > 0:
ar_template = '$\\phi_%d$' if latex else 'ar.L%d'
names['ar'] = []
for i in self.polynomial_ar.nonzero()[0][1:]:
names['ar'].append(ar_template % i)

# Moving Average
if self.k_ma > 0:
ma_template = '$\\theta_%d$' if latex else 'ma.L%d'
names['ma'] = []
for i in self.polynomial_ma.nonzero()[0][1:]:
names['ma'].append(ma_template % i)

# Seasonal Autoregressive
if self.k_seasonal_ar > 0:
seasonal_ar_template = (
'$\\tilde \\phi_%d$' if latex else 'ar.S.L%d'
)
names['seasonal_ar'] = []
for i in self.polynomial_seasonal_ar.nonzero()[0][1:]:
names['seasonal_ar'].append(seasonal_ar_template % i)

# Seasonal Moving Average
if self.k_seasonal_ma > 0:
seasonal_ma_template = (
'$\\tilde \\theta_%d$' if latex else 'ma.S.L%d'
)
names['seasonal_ma'] = []
for i in self.polynomial_seasonal_ma.nonzero()[0][1:]:
names['seasonal_ma'].append(seasonal_ma_template % i)

# Reduced Form Autoregressive
if self.k_ar > 0 or self.k_seasonal_ar > 0:
reduced_polynomial_ar = reduced_polynomial_ar = -np.polymul(
self.polynomial_ar, self.polynomial_seasonal_ar
)
ar_template = '$\\Phi_%d$' if latex else 'ar.R.L%d'
names['reduced_ar'] = []
for i in reduced_polynomial_ar.nonzero()[0][1:]:
names['reduced_ar'].append(ar_template % i)

# Reduced Form Moving Average
if self.k_ma > 0 or self.k_seasonal_ma > 0:
reduced_polynomial_ma = np.polymul(
self.polynomial_ma, self.polynomial_seasonal_ma
)
ma_template = '$\\Theta_%d$' if latex else 'ma.R.L%d'
names['reduced_ma'] = []
for i in reduced_polynomial_ma.nonzero()[0][1:]:
names['reduced_ma'].append(ma_template % i)

# Exogenous variances
if self.state_regression and self.time_varying_regression:
exog_var_template = '$\\sigma_\\text{%s}^2$' if latex else 'var.%s'
names['exog_variance'] = [
exog_var_template % exog_name for exog_name in self.exog_names
]

# Measurement error variance
if self.measurement_error:
meas_var_tpl = (
'$\\sigma_\\eta^2$' if latex else 'var.measurement_error'
)
names['measurement_variance'] = [meas_var_tpl]

# State variance
if self.state_error:
var_tpl = '$\\sigma_\\zeta^2$' if latex else 'sigma2'
names['variance'] = [var_tpl]

return names

[docs]    def transform_params(self, unconstrained):
"""
Transform unconstrained parameters used by the optimizer to constrained
parameters used in likelihood evaluation.

Used primarily to enforce stationarity of the autoregressive lag
polynomial, invertibility of the moving average lag polynomial, and
positive variance parameters.

Parameters
----------
unconstrained : array_like
Unconstrained parameters used by the optimizer.

Returns
-------
constrained : array_like
Constrained parameters used in likelihood evaluation.

Notes
-----
If the lag polynomial has non-consecutive powers (so that the
coefficient is zero on some element of the polynomial), then the
constraint function is not onto the entire space of invertible
polynomials, although it only excludes a very small portion very close
to the invertibility boundary.
"""
unconstrained = np.array(unconstrained, ndmin=1)
constrained = np.zeros(unconstrained.shape, unconstrained.dtype)

start = end = 0

# Retain the trend parameters
if self.k_trend > 0:
end += self.k_trend
constrained[start:end] = unconstrained[start:end]
start += self.k_trend

# Retain any MLE regression coefficients
if self.mle_regression:
end += self.k_exog
constrained[start:end] = unconstrained[start:end]
start += self.k_exog

# Transform the AR parameters (phi) to be stationary
if self.k_ar_params > 0:
end += self.k_ar_params
if self.enforce_stationarity:
constrained[start:end] = (
constrain_stationary_univariate(unconstrained[start:end])
)
else:
constrained[start:end] = unconstrained[start:end]
start += self.k_ar_params

# Transform the MA parameters (theta) to be invertible
if self.k_ma_params > 0:
end += self.k_ma_params
if self.enforce_invertibility:
constrained[start:end] = (
constrain_stationary_univariate(unconstrained[start:end])
)
else:
constrained[start:end] = unconstrained[start:end]
start += self.k_ma_params

# Transform the seasonal AR parameters (\tilde phi) to be stationary
if self.k_seasonal_ar > 0:
end += self.k_seasonal_ar_params
if self.enforce_stationarity:
constrained[start:end] = (
constrain_stationary_univariate(unconstrained[start:end])
)
else:
constrained[start:end] = unconstrained[start:end]
start += self.k_seasonal_ar_params

# Transform the seasonal MA parameters (\tilde theta) to be invertible
if self.k_seasonal_ma_params > 0:
end += self.k_seasonal_ma_params
if self.enforce_invertibility:
constrained[start:end] = (
constrain_stationary_univariate(unconstrained[start:end])
)
else:
constrained[start:end] = unconstrained[start:end]
start += self.k_seasonal_ma_params

# Transform the standard deviation parameters to be positive
if self.state_regression and self.time_varying_regression:
end += self.k_exog
constrained[start:end] = unconstrained[start:end]**2
start += self.k_exog
if self.measurement_error:
constrained[start] = unconstrained[start]**2
start += 1
end += 1
if self.state_error:
constrained[start] = unconstrained[start]**2
# start += 1
# end += 1

return constrained

[docs]    def untransform_params(self, constrained):
"""
Transform constrained parameters used in likelihood evaluation
to unconstrained parameters used by the optimizer

Used primarily to reverse enforcement of stationarity of the
autoregressive lag polynomial and invertibility of the moving average
lag polynomial.

Parameters
----------
constrained : array_like
Constrained parameters used in likelihood evaluation.

Returns
-------
constrained : array_like
Unconstrained parameters used by the optimizer.

Notes
-----
If the lag polynomial has non-consecutive powers (so that the
coefficient is zero on some element of the polynomial), then the
constraint function is not onto the entire space of invertible
polynomials, although it only excludes a very small portion very close
to the invertibility boundary.
"""
constrained = np.array(constrained, ndmin=1)
unconstrained = np.zeros(constrained.shape, constrained.dtype)

start = end = 0

# Retain the trend parameters
if self.k_trend > 0:
end += self.k_trend
unconstrained[start:end] = constrained[start:end]
start += self.k_trend

# Retain any MLE regression coefficients
if self.mle_regression:
end += self.k_exog
unconstrained[start:end] = constrained[start:end]
start += self.k_exog

# Transform the AR parameters (phi) to be stationary
if self.k_ar_params > 0:
end += self.k_ar_params
if self.enforce_stationarity:
unconstrained[start:end] = (
unconstrain_stationary_univariate(constrained[start:end])
)
else:
unconstrained[start:end] = constrained[start:end]
start += self.k_ar_params

# Transform the MA parameters (theta) to be invertible
if self.k_ma_params > 0:
end += self.k_ma_params
if self.enforce_invertibility:
unconstrained[start:end] = (
unconstrain_stationary_univariate(constrained[start:end])
)
else:
unconstrained[start:end] = constrained[start:end]
start += self.k_ma_params

# Transform the seasonal AR parameters (\tilde phi) to be stationary
if self.k_seasonal_ar > 0:
end += self.k_seasonal_ar_params
if self.enforce_stationarity:
unconstrained[start:end] = (
unconstrain_stationary_univariate(constrained[start:end])
)
else:
unconstrained[start:end] = constrained[start:end]
start += self.k_seasonal_ar_params

# Transform the seasonal MA parameters (\tilde theta) to be invertible
if self.k_seasonal_ma_params > 0:
end += self.k_seasonal_ma_params
if self.enforce_invertibility:
unconstrained[start:end] = (
unconstrain_stationary_univariate(constrained[start:end])
)
else:
unconstrained[start:end] = constrained[start:end]
start += self.k_seasonal_ma_params

# Untransform the standard deviation
if self.state_regression and self.time_varying_regression:
end += self.k_exog
unconstrained[start:end] = constrained[start:end]**0.5
start += self.k_exog
if self.measurement_error:
unconstrained[start] = constrained[start]**0.5
start += 1
end += 1
if self.state_error:
unconstrained[start] = constrained[start]**0.5
# start += 1
# end += 1

return unconstrained

[docs]    def update(self, params, transformed=True, complex_step=False):
"""
Update the parameters of the model

Updates the representation matrices to fill in the new parameter
values.

Parameters
----------
params : array_like
Array of new parameters.
transformed : boolean, 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.
"""
params = super(SARIMAX, self).update(params, transformed=transformed,
complex_step=False)

params_trend = None
params_exog = None
params_ar = None
params_ma = None
params_seasonal_ar = None
params_seasonal_ma = None
params_exog_variance = None
params_measurement_variance = None
params_variance = None

# Extract the parameters
start = end = 0
end += self.k_trend
params_trend = params[start:end]
start += self.k_trend
if self.mle_regression:
end += self.k_exog
params_exog = params[start:end]
start += self.k_exog
end += self.k_ar_params
params_ar = params[start:end]
start += self.k_ar_params
end += self.k_ma_params
params_ma = params[start:end]
start += self.k_ma_params
end += self.k_seasonal_ar_params
params_seasonal_ar = params[start:end]
start += self.k_seasonal_ar_params
end += self.k_seasonal_ma_params
params_seasonal_ma = params[start:end]
start += self.k_seasonal_ma_params
if self.state_regression and self.time_varying_regression:
end += self.k_exog
params_exog_variance = params[start:end]
start += self.k_exog
if self.measurement_error:
params_measurement_variance = params[start]
start += 1
end += 1
if self.state_error:
params_variance = params[start]
# start += 1
# end += 1

# Update lag polynomials
if self.k_ar > 0:
if self.polynomial_ar.dtype == params.dtype:
self.polynomial_ar[self._polynomial_ar_idx] = -params_ar
else:
polynomial_ar = self.polynomial_ar.real.astype(params.dtype)
polynomial_ar[self._polynomial_ar_idx] = -params_ar
self.polynomial_ar = polynomial_ar

if self.k_ma > 0:
if self.polynomial_ma.dtype == params.dtype:
self.polynomial_ma[self._polynomial_ma_idx] = params_ma
else:
polynomial_ma = self.polynomial_ma.real.astype(params.dtype)
polynomial_ma[self._polynomial_ma_idx] = params_ma
self.polynomial_ma = polynomial_ma

if self.k_seasonal_ar > 0:
idx = self._polynomial_seasonal_ar_idx
if self.polynomial_seasonal_ar.dtype == params.dtype:
self.polynomial_seasonal_ar[idx] = -params_seasonal_ar
else:
polynomial_seasonal_ar = (
self.polynomial_seasonal_ar.real.astype(params.dtype)
)
polynomial_seasonal_ar[idx] = -params_seasonal_ar
self.polynomial_seasonal_ar = polynomial_seasonal_ar

if self.k_seasonal_ma > 0:
idx = self._polynomial_seasonal_ma_idx
if self.polynomial_seasonal_ma.dtype == params.dtype:
self.polynomial_seasonal_ma[idx] = params_seasonal_ma
else:
polynomial_seasonal_ma = (
self.polynomial_seasonal_ma.real.astype(params.dtype)
)
polynomial_seasonal_ma[idx] = params_seasonal_ma
self.polynomial_seasonal_ma = polynomial_seasonal_ma

# Get the reduced form lag polynomial terms by multiplying the regular
# and seasonal lag polynomials
# Note: that although the numpy np.polymul examples assume that they
# are ordered from highest degree to lowest, whereas our are from
# lowest to highest, it does not matter.
if self.k_seasonal_ar > 0:
reduced_polynomial_ar = -np.polymul(
self.polynomial_ar, self.polynomial_seasonal_ar
)
else:
reduced_polynomial_ar = -self.polynomial_ar
if self.k_seasonal_ma > 0:
reduced_polynomial_ma = np.polymul(
self.polynomial_ma, self.polynomial_seasonal_ma
)
else:
reduced_polynomial_ma = self.polynomial_ma

# Observation intercept
# Exogenous data with MLE estimation of parameters enters through a
# time-varying observation intercept (is equivalent to simply
# subtracting it out of the endogenous variable first)
if self.mle_regression:
self.ssm['obs_intercept'] = np.dot(self.exog, params_exog)[None, :]

# State intercept (Harvey) or additional observation intercept
# (Hamilton)
# SARIMA trend enters through the a time-varying state intercept,
# associated with the first row of the stationary component of the
# state vector (i.e. the first element of the state vector following
# any differencing elements)
if self.k_trend > 0:
data = np.dot(self._trend_data, params_trend).astype(params.dtype)
if not self.hamilton_representation:
self.ssm['state_intercept', self._k_states_diff, :] = data
else:
# The way the trend enters in the Hamilton representation means
# that the parameter is not an intercept'' but instead the
# mean of the process. The trend values in data are meant for
# an intercept, and so must be transformed to represent the
if self.hamilton_representation:
data /= np.sum(-reduced_polynomial_ar)

# If we already set the observation intercept for MLE
# regression, just add to it
if self.mle_regression:
self.ssm.obs_intercept += data[None, :]
# Otherwise set it directly
else:
self.ssm.obs_intercept = data[None, :]

# Observation covariance matrix
if self.measurement_error:
self.ssm['obs_cov', 0, 0] = params_measurement_variance

# Transition matrix
if self.k_ar > 0 or self.k_seasonal_ar > 0:
self.ssm[self.transition_ar_params_idx] = reduced_polynomial_ar[1:]
elif not self.ssm.transition.dtype == params.dtype:
# This is required if the transition matrix is not really in use
# (e.g. for an MA(q) process) so that it's dtype never changes as
# the parameters' dtype changes. This changes the dtype manually.
self.ssm.transition = self.ssm.transition.real.astype(params.dtype)

# Selection matrix (Harvey) or Design matrix (Hamilton)
if self.k_ma > 0 or self.k_seasonal_ma > 0:
if not self.hamilton_representation:
self.ssm[self.selection_ma_params_idx] = (
reduced_polynomial_ma[1:]
)
else:
self.ssm[self.design_ma_params_idx] = reduced_polynomial_ma[1:]

# State covariance matrix
if self.k_posdef > 0:
self.ssm['state_cov', 0, 0] = params_variance
if self.state_regression and self.time_varying_regression:
self.ssm[self._exog_variance_idx] = params_exog_variance

# Initialize
if not self._manual_initialization:
self.initialize_state(complex_step=complex_step)

return params

[docs]class SARIMAXResults(MLEResults):
"""
Class to hold results from fitting an SARIMAX model.

Parameters
----------
model : SARIMAX instance
The fitted model instance

Attributes
----------
specification : dictionary
Dictionary including all attributes from the SARIMAX model instance.
polynomial_ar : array
Array containing autoregressive lag polynomial coefficients,
ordered from lowest degree to highest. Initialized with ones, unless
a coefficient is constrained to be zero (in which case it is zero).
polynomial_ma : array
Array containing moving average lag polynomial coefficients,
ordered from lowest degree to highest. Initialized with ones, unless
a coefficient is constrained to be zero (in which case it is zero).
polynomial_seasonal_ar : array
Array containing seasonal autoregressive lag polynomial coefficients,
ordered from lowest degree to highest. Initialized with ones, unless
a coefficient is constrained to be zero (in which case it is zero).
polynomial_seasonal_ma : array
Array containing seasonal moving average lag polynomial coefficients,
ordered from lowest degree to highest. Initialized with ones, unless
a coefficient is constrained to be zero (in which case it is zero).
polynomial_trend : array
Array containing trend polynomial coefficients, ordered from lowest
degree to highest. Initialized with ones, unless a coefficient is
constrained to be zero (in which case it is zero).
model_orders : list of int
The orders of each of the polynomials in the model.
param_terms : list of str
List of parameters actually included in the model, in sorted order.

--------
statsmodels.tsa.statespace.kalman_filter.FilterResults
statsmodels.tsa.statespace.mlemodel.MLEResults
"""
def __init__(self, model, params, filter_results, cov_type='opg', **kwargs):
super(SARIMAXResults, self).__init__(model, params, filter_results,
cov_type, **kwargs)

self.df_resid = np.inf  # attribute required for wald tests

# Save _init_kwds
self._init_kwds = self.model._get_init_kwds()

# Save model specification
self.specification = Bunch(**{
'seasonal_periods': self.model.seasonal_periods,
'measurement_error': self.model.measurement_error,
'time_varying_regression': self.model.time_varying_regression,
'simple_differencing': self.model.simple_differencing,
'enforce_stationarity': self.model.enforce_stationarity,
'enforce_invertibility': self.model.enforce_invertibility,
'hamilton_representation': self.model.hamilton_representation,

'order': self.model.order,
'seasonal_order': self.model.seasonal_order,

# Model order
'k_diff': self.model.k_diff,
'k_seasonal_diff': self.model.k_seasonal_diff,
'k_ar': self.model.k_ar,
'k_ma': self.model.k_ma,
'k_seasonal_ar': self.model.k_seasonal_ar,
'k_seasonal_ma': self.model.k_seasonal_ma,

# Param Numbers
'k_ar_params': self.model.k_ar_params,
'k_ma_params': self.model.k_ma_params,

# Trend / Regression
'trend': self.model.trend,
'k_trend': self.model.k_trend,
'k_exog': self.model.k_exog,

'mle_regression': self.model.mle_regression,
'state_regression': self.model.state_regression,
})

# Polynomials
self.polynomial_trend = self.model.polynomial_trend
self.polynomial_ar = self.model.polynomial_ar
self.polynomial_ma = self.model.polynomial_ma
self.polynomial_seasonal_ar = self.model.polynomial_seasonal_ar
self.polynomial_seasonal_ma = self.model.polynomial_seasonal_ma
self.polynomial_reduced_ar = np.polymul(
self.polynomial_ar, self.polynomial_seasonal_ar
)
self.polynomial_reduced_ma = np.polymul(
self.polynomial_ma, self.polynomial_seasonal_ma
)

# Distinguish parameters
self.model_orders = self.model.model_orders
self.param_terms = self.model.param_terms
start = end = 0
for name in self.param_terms:
end += self.model_orders[name]
setattr(self, '_params_%s' % name, self.params[start:end])
start += self.model_orders[name]

# Handle removing data
self._data_attr_model.extend(['orig_endog', 'orig_exog'])

[docs]    def arroots(self):
"""
(array) Roots of the reduced form autoregressive lag polynomial
"""
return np.roots(self.polynomial_reduced_ar)**-1

[docs]    def maroots(self):
"""
(array) Roots of the reduced form moving average lag polynomial
"""
return np.roots(self.polynomial_reduced_ma)**-1

[docs]    def arfreq(self):
"""
(array) Frequency of the roots of the reduced form autoregressive
lag polynomial
"""
z = self.arroots
if not z.size:
return
return np.arctan2(z.imag, z.real) / (2 * np.pi)

[docs]    def mafreq(self):
"""
(array) Frequency of the roots of the reduced form moving average
lag polynomial
"""
z = self.maroots
if not z.size:
return
return np.arctan2(z.imag, z.real) / (2 * np.pi)

[docs]    def arparams(self):
"""
(array) Autoregressive parameters actually estimated in the model.
Does not include parameters whose values are constrained to be zero.
"""
return self._params_ar

[docs]    def maparams(self):
"""
(array) Moving average parameters actually estimated in the model.
Does not include parameters whose values are constrained to be zero.
"""
return self._params_ma

[docs]    def get_prediction(self, start=None, end=None, dynamic=False, exog=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, ie.,
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, ie.,
the first forecast is start. 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.
exog : array_like, optional
If the model includes exogenous regressors, you must provide
exactly enough out-of-sample values for the exogenous variables if
end is beyond the last observation in the sample.
dynamic : boolean, 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
full_results : boolean, optional
If True, returns a FilterResults instance; if False returns a
tuple with forecasts, the forecast errors, and the forecast error
covariance matrices. Default is False.
**kwargs
Additional arguments may required for forecasting beyond the end
of the sample. See FilterResults.predict for more details.

Returns
-------
forecast : array
Array of out of sample forecasts.
"""
if start is None:
start = 0

# Handle end (e.g. date)
_start = self.model._get_predict_start(start)
_end, _out_of_sample = self.model._get_predict_end(end)

# Handle exogenous parameters
if _out_of_sample and (self.model.k_exog + self.model.k_trend > 0):
# Create a new faux SARIMAX model for the extended dataset
nobs = self.model.data.orig_endog.shape[0] + _out_of_sample
endog = np.zeros((nobs, self.model.k_endog))

if self.model.k_exog > 0:
if exog is None:
raise ValueError('Out-of-sample forecasting in a model'
' with a regression component requires'
' additional exogenous values via the'
' exog argument.')
exog = np.array(exog)
required_exog_shape = (_out_of_sample, self.model.k_exog)
if not exog.shape == required_exog_shape:
raise ValueError('Provided exogenous values are not of the'
' appropriate shape. Required %s, got %s.'
% (str(required_exog_shape),
str(exog.shape)))
exog = np.c_[self.model.data.orig_exog.T, exog.T].T

model_kwargs = self._init_kwds.copy()
model_kwargs['exog'] = exog
model = SARIMAX(endog, **model_kwargs)
model.update(self.params)

# Set the kwargs with the update time-varying state space
# representation matrices
for name in self.filter_results.shapes.keys():
if name == 'obs':
continue
mat = getattr(model.ssm, name)
if mat.shape[-1] > 1:
if len(mat.shape) == 2:
kwargs[name] = mat[:, -_out_of_sample:]
else:
kwargs[name] = mat[:, :, -_out_of_sample:]
elif self.model.k_exog == 0 and exog is not None:
warn('Exogenous array provided to predict, but additional data not'
' required. exog argument ignored.', ValueWarning)

return super(SARIMAXResults, self).get_prediction(
start=start, end=end, dynamic=dynamic, exog=exog, **kwargs
)

[docs]    def summary(self, alpha=.05, start=None):
# Create the model name

# See if we have an ARIMA component
order = ''
if self.model.k_ar + self.model.k_diff + self.model.k_ma > 0:
if self.model.k_ar == self.model.k_ar_params:
order_ar = self.model.k_ar
else:
order_ar = tuple(self.polynomial_ar.nonzero()[0][1:])
if self.model.k_ma == self.model.k_ma_params:
order_ma = self.model.k_ma
else:
order_ma = tuple(self.polynomial_ma.nonzero()[0][1:])
# If there is simple differencing, then that is reflected in the
# dependent variable name
k_diff = 0 if self.model.simple_differencing else self.model.k_diff
order = '(%s, %d, %s)' % (order_ar, k_diff, order_ma)
# See if we have an SARIMA component
seasonal_order = ''
has_seasonal = (
self.model.k_seasonal_ar +
self.model.k_seasonal_diff +
self.model.k_seasonal_ma
) > 0
if has_seasonal:
if self.model.k_ar == self.model.k_ar_params:
order_seasonal_ar = (
int(self.model.k_seasonal_ar / self.model.seasonal_periods)
)
else:
order_seasonal_ar = (
tuple(self.polynomial_seasonal_ar.nonzero()[0][1:])
)
if self.model.k_ma == self.model.k_ma_params:
order_seasonal_ma = (
int(self.model.k_seasonal_ma / self.model.seasonal_periods)
)
else:
order_seasonal_ma = (
tuple(self.polynomial_seasonal_ma.nonzero()[0][1:])
)
# If there is simple differencing, then that is reflected in the
# dependent variable name
k_seasonal_diff = self.model.k_seasonal_diff
if self.model.simple_differencing:
k_seasonal_diff = 0
seasonal_order = ('(%s, %d, %s, %d)' %
(str(order_seasonal_ar), k_seasonal_diff,
str(order_seasonal_ma), self.model.seasonal_periods))
if not order == '':
order += 'x'
model_name = (
'%s%s%s' % (self.model.__class__.__name__, order, seasonal_order)
)
return super(SARIMAXResults, self).summary(
alpha=alpha, start=start, model_name=model_name
)
summary.__doc__ = MLEResults.summary.__doc__

class SARIMAXResultsWrapper(MLEResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(SARIMAXResultsWrapper, SARIMAXResults)