"""
ARIMA model class.
Author: Chad Fulton
License: BSD-3
"""
from statsmodels.compat.pandas import Appender
import warnings
import numpy as np
from statsmodels.tools.data import _is_using_pandas
from statsmodels.tsa.statespace import sarimax
from statsmodels.tsa.statespace.kalman_filter import MEMORY_CONSERVE
from statsmodels.tsa.statespace.tools import diff
import statsmodels.base.wrapper as wrap
from statsmodels.tsa.arima.estimators.yule_walker import yule_walker
from statsmodels.tsa.arima.estimators.burg import burg
from statsmodels.tsa.arima.estimators.hannan_rissanen import hannan_rissanen
from statsmodels.tsa.arima.estimators.innovations import (
innovations, innovations_mle)
from statsmodels.tsa.arima.estimators.gls import gls as estimate_gls
from statsmodels.tsa.arima.specification import SARIMAXSpecification
[docs]class ARIMA(sarimax.SARIMAX):
r"""
Autoregressive Integrated Moving Average (ARIMA) model, and extensions
This model is the basic interface for ARIMA-type models, including those
with exogenous regressors and those with seasonal components. The most
general form of the model is SARIMAX(p, d, q)x(P, D, Q, s). It also allows
all specialized cases, including
- autoregressive models: AR(p)
- moving average models: MA(q)
- mixed autoregressive moving average models: ARMA(p, q)
- integration models: ARIMA(p, d, q)
- seasonal models: SARIMA(P, D, Q, s)
- regression with errors that follow one of the above ARIMA-type models
Parameters
----------
endog : array_like, optional
The observed time-series process :math:`y`.
exog : array_like, optional
Array of exogenous regressors.
order : tuple, optional
The (p,d,q) order of the model for the autoregressive, differences, and
moving average components. d is always an integer, while p and q may
either be integers or lists of integers.
seasonal_order : tuple, optional
The (P,D,Q,s) order of the seasonal component of the model for the
AR parameters, differences, MA parameters, and periodicity. Default
is (0, 0, 0, 0). D and s are always integers, while P and Q
may either be integers or lists of positive integers.
trend : str{'n','c','t','ct'} or iterable, optional
Parameter controlling the deterministic trend. Can be specified as a
string where 'c' indicates a constant term, 't' indicates a
linear trend in time, and 'ct' includes both. Can also be specified as
an iterable defining a polynomial, as in `numpy.poly1d`, where
`[1,1,0,1]` would denote :math:`a + bt + ct^3`. Default is 'c' for
models without integration, and no trend for models with integration.
Note that all trend terms are included in the model as exogenous
regressors, which differs from how trends are included in ``SARIMAX``
models. See the Notes section for a precise definition of the
treatment of trend terms.
enforce_stationarity : bool, optional
Whether or not to require the autoregressive parameters to correspond
to a stationarity process.
enforce_invertibility : bool, optional
Whether or not to require the moving average parameters to correspond
to an invertible process.
concentrate_scale : bool, optional
Whether or not to concentrate the scale (variance of the error term)
out of the likelihood. This reduces the number of parameters by one.
This is only applicable when considering estimation by numerical
maximum likelihood.
trend_offset : int, optional
The offset at which to start time trend values. Default is 1, so that
if `trend='t'` the trend is equal to 1, 2, ..., nobs. Typically is only
set when the model created by extending a previous dataset.
dates : array_like of datetime, optional
If no index is given by `endog` or `exog`, an array-like object of
datetime objects can be provided.
freq : str, optional
If no index is given by `endog` or `exog`, the frequency of the
time-series may be specified here as a Pandas offset or offset string.
missing : str
Available options are 'none', 'drop', and 'raise'. If 'none', no nan
checking is done. If 'drop', any observations with nans are dropped.
If 'raise', an error is raised. Default is 'none'.
Notes
-----
This model incorporates both exogenous regressors and trend components
through "regression with ARIMA errors". This differs from the
specification estimated using ``SARIMAX`` which treats the trend
components separately from any included exogenous regressors. The full
specification of the model estimated here is:
.. math::
Y_{t}-\delta_{0}-\delta_{1}t-\ldots-\delta_{k}t^{k}-X_{t}\beta
& =\epsilon_{t} \\
\left(1-L\right)^{d}\left(1-L^{s}\right)^{D}\Phi\left(L\right)
\Phi_{s}\left(L\right)\epsilon_{t}
& =\Theta\left(L\right)\Theta_{s}\left(L\right)\eta_{t}
where :math:`\eta_t \sim WN(0,\sigma^2)` is a white noise process, L
is the lag operator, and :math:`G(L)` are lag polynomials corresponding
to the autoregressive (:math:`\Phi`), seasonal autoregressive
(:math:`\Phi_s`), moving average (:math:`\Theta`), and seasonal moving
average components (:math:`\Theta_s`).
`enforce_stationarity` and `enforce_invertibility` are specified in the
constructor because they affect loglikelihood computations, and so should
not be changed on the fly. This is why they are not instead included as
arguments to the `fit` method.
.. todo:: should concentrate_scale=True by default
Examples
--------
>>> mod = sm.tsa.arima.ARIMA(endog, order=(1, 0, 0))
>>> res = mod.fit()
>>> print(res.summary())
"""
def __init__(self, endog, exog=None, order=(0, 0, 0),
seasonal_order=(0, 0, 0, 0), trend=None,
enforce_stationarity=True, enforce_invertibility=True,
concentrate_scale=False, trend_offset=1, dates=None,
freq=None, missing='none', validate_specification=True):
# Default for trend
# 'c' if there is no integration and 'n' otherwise
# TODO: if trend='c', then we could alternatively use `demean=True` in
# the estimation methods rather than setting up `exog` and using GLS.
# Not sure if it's worth the trouble though.
integrated = order[1] > 0 or seasonal_order[1] > 0
if trend is None and not integrated:
trend = 'c'
elif trend is None:
trend = 'n'
# Construct the specification
# (don't pass specific values of enforce stationarity/invertibility,
# because we don't actually want to restrict the estimators based on
# this criteria. Instead, we'll just make sure that the parameter
# estimates from those methods satisfy the criteria.)
self._spec_arima = SARIMAXSpecification(
endog, exog=exog, order=order, seasonal_order=seasonal_order,
trend=trend, enforce_stationarity=None, enforce_invertibility=None,
concentrate_scale=concentrate_scale, trend_offset=trend_offset,
dates=dates, freq=freq, missing=missing,
validate_specification=validate_specification)
exog = self._spec_arima._model.data.orig_exog
# Raise an error if we have a constant in an integrated model
has_trend = len(self._spec_arima.trend_terms) > 0
if has_trend:
lowest_trend = np.min(self._spec_arima.trend_terms)
if lowest_trend < order[1] + seasonal_order[1]:
raise ValueError(
'In models with integration (`d > 0`) or seasonal'
' integration (`D > 0`), trend terms of lower order than'
' `d + D` cannot be (as they would be eliminated due to'
' the differencing operation). For example, a constant'
' cannot be included in an ARIMA(1, 1, 1) model, but'
' including a linear trend, which would have the same'
' effect as fitting a constant to the differenced data,'
' is allowed.')
# Keep the given `exog` by removing the prepended trend variables
input_exog = None
if exog is not None:
if _is_using_pandas(exog, None):
input_exog = exog.iloc[:, self._spec_arima.k_trend:]
else:
input_exog = exog[:, self._spec_arima.k_trend:]
# Initialize the base SARIMAX class
# Note: we don't pass in a trend value to the base class, since ARIMA
# standardizes the trend to always be part of exog, while the base
# SARIMAX class puts it in the transition equation.
super().__init__(
endog, exog, trend=None, order=order,
seasonal_order=seasonal_order,
enforce_stationarity=enforce_stationarity,
enforce_invertibility=enforce_invertibility,
concentrate_scale=concentrate_scale, dates=dates, freq=freq,
missing=missing, validate_specification=validate_specification)
self.trend = trend
# Save the input exog and input exog names, so that we can refer to
# them later (see especially `ARIMAResults.append`)
self._input_exog = input_exog
if exog is not None:
self._input_exog_names = self.exog_names[self._spec_arima.k_trend:]
else:
self._input_exog_names = None
# Override the public attributes for k_exog and k_trend to reflect the
# distinction here (for the purpose of the superclass, these are both
# combined as `k_exog`)
self.k_exog = self._spec_arima.k_exog
self.k_trend = self._spec_arima.k_trend
# Remove some init kwargs that aren't used in this model
unused = ['measurement_error', 'time_varying_regression',
'mle_regression', 'simple_differencing',
'hamilton_representation']
self._init_keys = [key for key in self._init_keys if key not in unused]
@property
def _res_classes(self):
return {'fit': (ARIMAResults, ARIMAResultsWrapper)}
[docs] def fit(self, start_params=None, transformed=True, includes_fixed=False,
method=None, method_kwargs=None, gls=None, gls_kwargs=None,
cov_type=None, cov_kwds=None, return_params=False,
low_memory=False):
"""
Fit (estimate) the parameters of the model.
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.
method : str, optional
The method used for estimating the parameters of the model. Valid
options include 'statespace', 'innovations_mle', 'hannan_rissanen',
'burg', 'innovations', and 'yule_walker'. Not all options are
available for every specification (for example 'yule_walker' can
only be used with AR(p) models).
method_kwargs : dict, optional
Arguments to pass to the fit function for the parameter estimator
described by the `method` argument.
gls : bool, optional
Whether or not to use generalized least squares (GLS) to estimate
regression effects. The default is False if `method='statespace'`
and is True otherwise.
gls_kwargs : dict, optional
Arguments to pass to the GLS estimation fit method. Only applicable
if GLS estimation is used (see `gls` argument for details).
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 'oim'.
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.
return_params : bool, optional
Whether or not to return only the array of maximizing parameters.
Default is False.
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.
Returns
-------
ARIMAResults
Examples
--------
>>> mod = sm.tsa.arima.ARIMA(endog, order=(1, 0, 0))
>>> res = mod.fit()
>>> print(res.summary())
"""
# Determine which method to use
# 1. If method is specified, make sure it is valid
if method is not None:
self._spec_arima.validate_estimator(method)
# 2. Otherwise, use state space
# TODO: may want to consider using innovations (MLE) if possible here,
# (since in some cases it may be faster than state space), but it is
# less tested.
else:
method = 'statespace'
# Can only use fixed parameters with the following methods
methods_with_fixed_params = ['statespace', 'hannan_rissanen']
if self._has_fixed_params and method not in methods_with_fixed_params:
raise ValueError(
f"When parameters have been fixed, only the methods "
f"{methods_with_fixed_params} can be used; got '{method}'."
)
# Handle kwargs related to the fit method
if method_kwargs is None:
method_kwargs = {}
required_kwargs = []
if method == 'statespace':
required_kwargs = ['enforce_stationarity', 'enforce_invertibility',
'concentrate_scale']
elif method == 'innovations_mle':
required_kwargs = ['enforce_invertibility']
for name in required_kwargs:
if name in method_kwargs:
raise ValueError('Cannot override model level value for "%s"'
' when method="%s".' % (name, method))
method_kwargs[name] = getattr(self, name)
# Handle kwargs related to GLS estimation
if gls_kwargs is None:
gls_kwargs = {}
# Handle starting parameters
# TODO: maybe should have standard way of computing starting
# parameters in this class?
if start_params is not None:
if method not in ['statespace', 'innovations_mle']:
raise ValueError('Estimation method "%s" does not use starting'
' parameters, but `start_params` argument was'
' given.' % method)
method_kwargs['start_params'] = start_params
method_kwargs['transformed'] = transformed
method_kwargs['includes_fixed'] = includes_fixed
# Perform estimation, depending on whether we have exog or not
p = None
fit_details = None
has_exog = self._spec_arima.exog is not None
if has_exog or method == 'statespace':
# Use GLS if it was explicitly requested (`gls = True`) or if it
# was left at the default (`gls = None`) and the ARMA estimator is
# anything but statespace.
# Note: both GLS and statespace are able to handle models with
# integration, so we don't need to difference endog or exog here.
if has_exog and (gls or (gls is None and method != 'statespace')):
if self._has_fixed_params:
raise NotImplementedError(
'GLS estimation is not yet implemented for the case '
'with fixed parameters.'
)
p, fit_details = estimate_gls(
self.endog, exog=self.exog, order=self.order,
seasonal_order=self.seasonal_order, include_constant=False,
arma_estimator=method, arma_estimator_kwargs=method_kwargs,
**gls_kwargs)
elif method != 'statespace':
raise ValueError('If `exog` is given and GLS is disabled'
' (`gls=False`), then the only valid'
" method is 'statespace'. Got '%s'."
% method)
else:
method_kwargs.setdefault('disp', 0)
res = super().fit(
return_params=return_params, low_memory=low_memory,
cov_type=cov_type, cov_kwds=cov_kwds, **method_kwargs)
if not return_params:
res.fit_details = res.mlefit
else:
# Handle differencing if we have an integrated model
# (these methods do not support handling integration internally,
# so we need to manually do the differencing)
endog = self.endog
order = self._spec_arima.order
seasonal_order = self._spec_arima.seasonal_order
if self._spec_arima.is_integrated:
warnings.warn('Provided `endog` series has been differenced'
' to eliminate integration prior to parameter'
' estimation by method "%s".' % method)
endog = diff(
endog, k_diff=self._spec_arima.diff,
k_seasonal_diff=self._spec_arima.seasonal_diff,
seasonal_periods=self._spec_arima.seasonal_periods)
if order[1] > 0:
order = (order[0], 0, order[2])
if seasonal_order[1] > 0:
seasonal_order = (seasonal_order[0], 0, seasonal_order[2],
seasonal_order[3])
if self._has_fixed_params:
method_kwargs['fixed_params'] = self._fixed_params.copy()
# Now, estimate parameters
if method == 'yule_walker':
p, fit_details = yule_walker(
endog, ar_order=order[0], demean=False,
**method_kwargs)
elif method == 'burg':
p, fit_details = burg(endog, ar_order=order[0],
demean=False, **method_kwargs)
elif method == 'hannan_rissanen':
p, fit_details = hannan_rissanen(
endog, ar_order=order[0],
ma_order=order[2], demean=False, **method_kwargs)
elif method == 'innovations':
p, fit_details = innovations(
endog, ma_order=order[2], demean=False,
**method_kwargs)
# innovations computes estimates through the given order, so
# we want to take the estimate associated with the given order
p = p[-1]
elif method == 'innovations_mle':
p, fit_details = innovations_mle(
endog, order=order,
seasonal_order=seasonal_order,
demean=False, **method_kwargs)
# In all cases except method='statespace', we now need to extract the
# parameters and, optionally, create a new results object
if p is not None:
# Need to check that fitted parameters satisfy given restrictions
if (self.enforce_stationarity
and self._spec_arima.max_reduced_ar_order > 0
and not p.is_stationary):
raise ValueError('Non-stationary autoregressive parameters'
' found with `enforce_stationarity=True`.'
' Consider setting it to False or using a'
' different estimation method, such as'
' method="statespace".')
if (self.enforce_invertibility
and self._spec_arima.max_reduced_ma_order > 0
and not p.is_invertible):
raise ValueError('Non-invertible moving average parameters'
' found with `enforce_invertibility=True`.'
' Consider setting it to False or using a'
' different estimation method, such as'
' method="statespace".')
# Build the requested results
if return_params:
res = p.params
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(p.params, transformed=True, includes_fixed=True,
cov_type=cov_type, cov_kwds=cov_kwds)
# Save any details from the fit method
res.fit_details = fit_details
# Reset memory conservation
if low_memory:
self.ssm.set_conserve_memory(conserve_memory)
return res
[docs]@Appender(sarimax.SARIMAXResults.__doc__)
class ARIMAResults(sarimax.SARIMAXResults):
[docs] @Appender(sarimax.SARIMAXResults.append.__doc__)
def append(self, endog, exog=None, refit=False, fit_kwargs=None, **kwargs):
# MLEResults.append will concatenate the given `exog` here with
# `data.orig_exog`. However, `data.orig_exog` already has had any
# trend variables prepended to it, while the `exog` given here should
# not. Instead, we need to temporarily replace `orig_exog` and
# `exog_names` with the ones that correspond to those that were input
# by the user.
if exog is not None:
orig_exog = self.model.data.orig_exog
exog_names = self.model.exog_names
self.model.data.orig_exog = self.model._input_exog
self.model.exog_names = self.model._input_exog_names
# Perform the appending procedure
out = super().append(endog, exog=exog, refit=refit,
fit_kwargs=fit_kwargs, **kwargs)
# Now we reverse the temporary change made above
if exog is not None:
self.model.data.orig_exog = orig_exog
self.model.exog_names = exog_names
return out
class ARIMAResultsWrapper(sarimax.SARIMAXResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(
sarimax.SARIMAXResultsWrapper._wrap_attrs, _attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(
sarimax.SARIMAXResultsWrapper._wrap_methods, _methods)
wrap.populate_wrapper(ARIMAResultsWrapper, ARIMAResults) # noqa:E305