""" SARIMAX Model Author: Chad Fulton License: Simplified-BSD """ from warnings import warn import numpy as np import pandas as pd from statsmodels.compat.pandas import Appender from statsmodels.tools.tools import Bunch from statsmodels.tools.data import _is_using_pandas from statsmodels.tools.decorators import cache_readonly import statsmodels.base.wrapper as wrap from statsmodels.tsa.arima.specification import SARIMAXSpecification from statsmodels.tsa.arima.params import SARIMAXParams from statsmodels.tsa.tsatools import lagmat from .initialization import Initialization from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper from .tools import ( companion_matrix, diff, is_invertible, constrain_stationary_univariate, unconstrain_stationary_univariate, prepare_exog, prepare_trend_spec, prepare_trend_data) 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 non-zero polynomial exponents to include, in increasing order. For example, `[1,1,0,1]` denotes :math:`a + bt + ct^3`. Default is to not include a trend component. measurement_error : bool, optional Whether or not to assume the endogenous observations `endog` were measured with error. Default is False. time_varying_regression : bool, optional Used when an explanatory variables, `exog`, are provided to select whether or not coefficients on the exogenous regressors are allowed to vary over time. Default is False. mle_regression : bool, 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 : bool, 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 results in a smaller state-space formulation. See the Notes section for important details about interpreting results when this option is used. 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 : bool, optional Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True. enforce_invertibility : bool, 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 : bool, optional Whether or not to use the Hamilton representation of an ARMA process (if True) or the Harvey representation (if False). Default is False. 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 estimated by maximum likelihood by one, but standard errors will then not be available for the scale parameter. 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. use_exact_diffuse : bool, optional Whether or not to use exact diffuse initialization for non-stationary states. Default is False (in which case approximate diffuse initialization is used). **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 : bool Whether or not to assume the endogenous observations `endog` were measured with error. state_error : bool Whether or not the transition equation has an error component. mle_regression : bool Whether or not the regression coefficients for the exogenous variables were estimated via maximum likelihood estimation. state_regression : bool 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 : bool Whether or not coefficients on the exogenous regressors are allowed to vary over time. simple_differencing : bool Whether or not to use partially conditional maximum likelihood estimation. enforce_stationarity : bool Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. enforce_invertibility : bool Whether or not to transform the MA parameters to enforce invertibility in the moving average component of the model. hamilton_representation : bool 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 parameter documentation for more information. polynomial_ar : ndarray Array containing autoregressive lag polynomial lags, ordered from lowest degree to highest. The polynomial begins with lag 0. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero). polynomial_ma : ndarray Array containing moving average lag polynomial lags, 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 : ndarray Array containing seasonal moving average lag polynomial lags, 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 : ndarray Array containing seasonal moving average lag polynomial lags, 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 : ndarray 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 integration. 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 integration. 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 accommodate 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). If `filter_concentrated = True` is used, then the scale of the model is concentrated out of the likelihood. A benefit of this is that there the dimension of the parameter vector is reduced so that numerical maximization of the log-likelihood function may be faster and more stable. If this option in a model with measurement error, it is important to note that the estimated measurement error parameter will be relative to the scale, and is named "snr.measurement_error" instead of "var.measurement_error". To compute the variance of the measurement error in this case one would multiply `snr.measurement_error` parameter by the scale. If `simple_differencing = True` is used, then the `endog` and `exog` data are differenced prior to putting the model in state-space form. This has the same effect as if the user differenced the data prior to constructing the model, which has implications for using the results: - Forecasts and predictions will be about the *differenced* data, not about the original data. (while if `simple_differencing = False` is used, then forecasts and predictions will be about the original data). - If the original data has an Int64Index, a new RangeIndex will be created for the differenced data that starts from one, and forecasts and predictions will use this new index. 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, concentrate_scale=False, trend_offset=1, use_exact_diffuse=False, dates=None, freq=None, missing='none', validate_specification=True, **kwargs): self._spec = SARIMAXSpecification( endog, exog=exog, order=order, seasonal_order=seasonal_order, trend=trend, enforce_stationarity=None, enforce_invertibility=None, concentrate_scale=concentrate_scale, dates=dates, freq=freq, missing=missing, validate_specification=validate_specification) self._params = SARIMAXParams(self._spec) # Save given orders order = self._spec.order seasonal_order = self._spec.seasonal_order self.order = order self.seasonal_order = seasonal_order # 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 self.concentrate_scale = concentrate_scale self.use_exact_diffuse = use_exact_diffuse # 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 self._params.ar_params = -1 self.polynomial_ar = self._params.ar_poly.coef self._polynomial_ar = self.polynomial_ar.copy() self._params.ma_params = 1 self.polynomial_ma = self._params.ma_poly.coef self._polynomial_ma = self.polynomial_ma.copy() self._params.seasonal_ar_params = -1 self.polynomial_seasonal_ar = self._params.seasonal_ar_poly.coef self._polynomial_seasonal_ar = self.polynomial_seasonal_ar.copy() self._params.seasonal_ma_params = 1 self.polynomial_seasonal_ma = self._params.seasonal_ma_poly.coef self._polynomial_seasonal_ma = self.polynomial_seasonal_ma.copy() # Deterministic trend polynomial self.trend = trend self.trend_offset = trend_offset self.polynomial_trend, self.k_trend = prepare_trend_spec(self.trend) self._polynomial_trend = self.polynomial_trend.copy() self._k_trend = self.k_trend # (we internally use _k_trend for mechanics so that the public # attribute can be overridden by subclasses) # 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 = self._spec.max_ar_order self.k_ar_params = self._spec.k_ar_params self.k_diff = int(order[1]) self.k_ma = self._spec.max_ma_order self.k_ma_params = self._spec.k_ma_params self.k_seasonal_ar = (self._spec.max_seasonal_ar_order * self._spec.seasonal_periods) self.k_seasonal_ar_params = self._spec.k_seasonal_ar_params self.k_seasonal_diff = int(seasonal_order[1]) self.k_seasonal_ma = (self._spec.max_seasonal_ma_order * self._spec.seasonal_periods) self.k_seasonal_ma_params = self._spec.k_seasonal_ma_params # 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') # 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: # Handle time-varying regression if self.time_varying_regression: self._k_order = 0 # Exogenous data (self._k_exog, exog) = prepare_exog(exog) # (we internally use _k_exog for mechanics so that the public attribute # can be overridden by subclasses) self.k_exog = self._k_exog # 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 estimated 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 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) # Handle non-default loglikelihood burn self._loglikelihood_burn = kwargs.get('loglikelihood_burn', None) # Number of parameters self.k_params = ( self.k_ar_params + self.k_ma_params + self.k_seasonal_ar_params + self.k_seasonal_ma_params + self._k_trend + self.measurement_error + int(not self.concentrate_scale) ) 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 # Initialize the statespace super().__init__( endog, exog=exog, k_states=k_states, k_posdef=k_posdef, **kwargs ) # Set the filter to concentrate out the scale if requested if self.concentrate_scale: self.ssm.filter_concentrated = True # 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 # 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 self.concentrate_scale: self.ssm['state_cov', 0, 0] = 1. # 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', 'concentrate_scale', 'trend_offset'] + list(kwargs.keys()) # TODO: I think the kwargs or not attached, need to recover from ??? # Initialize the state if self.ssm.initialization is None: self.initialize_default() def prepare_data(self): endog, exog = super().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 and cache self.data.endog, self.data.exog = ( self.data._convert_endog_exog(endog, exog)) # Reset indexes, if provided new_length = self.data.endog.shape[0] if self.data.row_labels is not None: self.data._cache['row_labels'] = ( self.data.row_labels[orig_length - new_length:]) if self._index is not None: if self._index_int64: self._index = pd.RangeIndex(start=1, stop=new_length + 1) elif self._index_generated: self._index = self._index[:-(orig_length - new_length)] else: self._index = self._index[orig_length - new_length:] # Reset the nobs self.nobs = endog.shape[0] # Cache the arrays for calculating the intercept from the trend # components self._trend_data = prepare_trend_data( self.polynomial_trend, self._k_trend, self.nobs, self.trend_offset) return endog, exog def initialize(self): """ Initialize the SARIMAX model. Notes ----- These initialization steps must occur following the parent class __init__ function calls. """ super().initialize() # 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 # do not 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:]) def initialize_default(self, approximate_diffuse_variance=None): """Initialize default""" if approximate_diffuse_variance is None: approximate_diffuse_variance = self.ssm.initial_variance if self.use_exact_diffuse: diffuse_type = 'diffuse' else: diffuse_type = 'approximate_diffuse' # Set the loglikelihood burn parameter, if not given in constructor if self._loglikelihood_burn is None: k_diffuse_states = self.k_states if self.enforce_stationarity: k_diffuse_states -= self._k_order self.loglikelihood_burn = k_diffuse_states init = Initialization( self.k_states, approximate_diffuse_variance=approximate_diffuse_variance) if self.enforce_stationarity: # Differencing operators are at the beginning init.set((0, self._k_states_diff), diffuse_type) # Stationary component in the middle init.set((self._k_states_diff, self._k_states_diff + self._k_order), 'stationary') # Regression components at the end init.set((self._k_states_diff + self._k_order, self._k_states_diff + self._k_order + self._k_exog), diffuse_type) # If we're not enforcing a stationarity, then we cannot initialize a # stationary component else: init.set(None, diffuse_type) self.ssm.initialization = init @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 len(design) == 0: design = np.r_[0] # 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 if self._k_order > 0: 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 if d < self._k_seasonal_diff - 1: 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] if len(selection) == 0: selection = np.zeros((self.k_states, self.k_posdef)) 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 def clone(self, endog, exog=None, **kwargs): return self._clone_from_init_kwds(endog, exog=exog, **kwargs) @property def _res_classes(self): return {'fit': (SARIMAXResults, SARIMAXResultsWrapper)} @staticmethod def _conditional_sum_squares(endog, k_ar, polynomial_ar, k_ma, polynomial_ma, k_trend=0, trend_data=None, warning_description=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: try: # 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) except ValueError: if warning_description is not None: warning_description = ' for %s' % warning_description else: warning_description = '' warn('Too few observations to estimate starting parameters%s.' ' All parameters except for variances will be set to' ' zeros.' % warning_description) # Typically this will be raised if there are not enough # observations for the `lagmat` calls. params = np.zeros(k_trend + k_ar + k_ma, dtype=endog.dtype) if len(endog) == 0: # This case usually happens when there are not even enough # observations for a complete set of differencing # operations (no hope of fitting, just set starting # variance to 1) residuals = np.ones(k_params_ma * 2 + 1, dtype=endog.dtype) else: residuals = np.r_[ np.zeros(k_params_ma * 2, dtype=endog.dtype), endog - np.mean(endog)] # 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: if len(residuals) > max(1, k_params_ma): params_variance = (residuals[k_params_ma:] ** 2).mean() else: params_variance = np.var(endog) 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)): mask = ~np.isnan(endog).squeeze() endog = endog[mask] if exog is not None: exog = exog[mask] if trend_data is not None: trend_data = trend_data[mask] # 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, warning_description='ARMA and trend') # If we have estimated non-stationary start parameters but enforce # stationarity is on, start with 0 parameters and warn invalid_ar = ( self.k_ar > 0 and self.enforce_stationarity and not is_invertible(np.r_[1, -params_ar]) ) if invalid_ar: warn('Non-stationary starting autoregressive parameters' ' found. Using zeros as starting parameters.') params_ar *= 0 # 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: warn('Non-invertible starting MA parameters found.' ' Using zeros as starting parameters.', UserWarning) params_ma *= 0 # 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, warning_description='seasonal ARMA')) # If we have estimated non-stationary start parameters but enforce # stationarity is on, warn and set start params to 0 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: warn('Non-stationary starting seasonal autoregressive' ' Using zeros as starting parameters.') params_seasonal_ar *= 0 # 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: warn('Non-invertible starting seasonal moving average' ' Using zeros as starting parameters.') params_seasonal_ma *= 0 # 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 type(params_variance) is list and len(params_variance) == 0): if not (type(params_seasonal_variance) is list and len(params_seasonal_variance) == 0): params_variance = params_seasonal_variance elif self._k_exog > 0: params_variance = np.inner(endog, endog) else: params_variance = np.inner(endog, endog) / self.nobs params_measurement_variance = 1 if self.measurement_error else [] # We want to bound the starting variance away from zero params_variance = np.atleast_1d(np.array(params_variance)) if params_variance.size: # Avoid comparisons with empty arrays due to changes in NumPy 2.2 params_variance = np.atleast_1d(max(params_variance[0], 1e-10)) # Remove state variance as parameter if scale is concentrated out if self.concentrate_scale: params_variance = [] # 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 = r'\Delta' if latex else 'D' else: diff = (r'\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 = ((r'\Delta_%d' if latex else 'DS%d') % (self.seasonal_periods)) else: seasonal_diff = ((r'\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 dict 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 state_names(self): # TODO: we may be able to revisit these states to get somewhat more # informative names, but ultimately probably not much better. # TODO: alternatively, we may be able to get better for certain models, # like pure AR models. k_ar_states = self._k_order if not self.simple_differencing: k_ar_states += (self.seasonal_periods * self._k_seasonal_diff + self._k_diff) names = ['state.%d' % i for i in range(k_ar_states)] if self._k_exog > 0 and self.state_regression: names += ['beta.%s' % self.exog_names[i] for i in range(self._k_exog)] return names @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 and not self.concentrate_scale), } @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: if not self.concentrate_scale: exog_var_template = ('$\\sigma_\\text{%s}^2$' if latex else 'var.%s') else: exog_var_template = ( '$\\sigma_\\text{%s}^2 / \\sigma_\\zeta^2$' if latex else 'snr.%s') names['exog_variance'] = [ exog_var_template % exog_name for exog_name in self.exog_names ] # Measurement error variance if self.measurement_error: if not self.concentrate_scale: meas_var_tpl = ( '$\\sigma_\\eta^2$' if latex else 'var.measurement_error') else: meas_var_tpl = ( '$\\sigma_\\eta^2 / \\sigma_\\zeta^2$' if latex else 'snr.measurement_error') names['measurement_variance'] = [meas_var_tpl] # State variance if self.state_error and not self.concentrate_scale: var_tpl = '$\\sigma_\\zeta^2$' if latex else 'sigma2' names['variance'] = [var_tpl] return names 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 and not self.concentrate_scale: constrained[start] = unconstrained[start]**2 # start += 1 # end += 1 return constrained 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 and not self.concentrate_scale: unconstrained[start] = constrained[start]**0.5 # start += 1 # end += 1 return unconstrained def _validate_can_fix_params(self, param_names): super()._validate_can_fix_params(param_names) model_names = self.model_names items = [ ('ar', 'autoregressive', self.enforce_stationarity, '`enforce_stationarity=True`'), ('seasonal_ar', 'seasonal autoregressive', self.enforce_stationarity, '`enforce_stationarity=True`'), ('ma', 'moving average', self.enforce_invertibility, '`enforce_invertibility=True`'), ('seasonal_ma', 'seasonal moving average', self.enforce_invertibility, '`enforce_invertibility=True`')] for name, title, condition, condition_desc in items: names = set(model_names[name] or []) fix_all = param_names.issuperset(names) fix_any = len(param_names.intersection(names)) > 0 if condition and fix_any and not fix_all: raise ValueError('Cannot fix individual %s parameters when' ' %s. Must either fix all %s parameters or' ' none.' % (title, condition_desc, title)) def update(self, params, transformed=True, includes_fixed=False, 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 : 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. """ params = self.handle_params(params, transformed=transformed, includes_fixed=includes_fixed) 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 and not self.concentrate_scale: 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 # mean instead 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: if not self.concentrate_scale: self['state_cov', 0, 0] = params_variance if self.state_regression and self.time_varying_regression: self.ssm[self._exog_variance_idx] = params_exog_variance return params def _get_extension_time_varying_matrices( self, params, exog, out_of_sample, extend_kwargs=None, transformed=True, includes_fixed=False, **kwargs): """ Get time-varying state space system matrices for extended model Notes ----- We need to override this method for SARIMAX because we need some special handling in the `simple_differencing=True` case. """ # Get the appropriate exog for the extended sample exog = self._validate_out_of_sample_exog(exog, out_of_sample) # Get the tmp endog, exog if self.simple_differencing: nobs = self.data.orig_endog.shape[0] + out_of_sample tmp_endog = np.zeros((nobs, self.k_endog)) if exog is not None: tmp_exog = np.c_[self.data.orig_exog.T, exog.T].T else: tmp_exog = None else: tmp_endog = np.zeros((out_of_sample, self.k_endog)) tmp_exog = exog # Create extended model if extend_kwargs is None: extend_kwargs = {} if not self.simple_differencing and self.k_trend > 0: extend_kwargs.setdefault( 'trend_offset', self.trend_offset + self.nobs) extend_kwargs.setdefault('validate_specification', False) mod_extend = self.clone( endog=tmp_endog, exog=tmp_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 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 : ndarray 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 : ndarray 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 : ndarray 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 : ndarray 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 : ndarray 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. See Also -------- statsmodels.tsa.statespace.kalman_filter.FilterResults statsmodels.tsa.statespace.mlemodel.MLEResults """ def __init__(self, model, params, filter_results, cov_type=None, **kwargs): super().__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(**{ # Set additional model parameters '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, 'concentrate_scale': self.model.concentrate_scale, 'trend_offset': self.model.trend_offset, '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: if name == 'ar': k = self.model.k_ar_params elif name == 'ma': k = self.model.k_ma_params elif name == 'seasonal_ar': k = self.model.k_seasonal_ar_params elif name == 'seasonal_ma': k = self.model.k_seasonal_ma_params else: k = self.model_orders[name] end += k setattr(self, '_params_%s' % name, self.params[start:end]) start += k # GH7527, all terms must be defined all_terms = ['ar', 'ma', 'seasonal_ar', 'seasonal_ma', 'variance'] for name in set(all_terms).difference(self.param_terms): setattr(self, '_params_%s' % name, np.empty(0)) # Handle removing data self._data_attr_model.extend(['orig_endog', 'orig_exog']) def extend(self, endog, exog=None, **kwargs): kwargs.setdefault('trend_offset', self.nobs + 1) return super().extend(endog, exog=exog, **kwargs) @cache_readonly def arroots(self): """ (array) Roots of the reduced form autoregressive lag polynomial """ return np.roots(self.polynomial_reduced_ar)**-1 @cache_readonly def maroots(self): """ (array) Roots of the reduced form moving average lag polynomial """ return np.roots(self.polynomial_reduced_ma)**-1 @cache_readonly 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) @cache_readonly 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) @cache_readonly def arparams(self): """ (array) Autoregressive parameters actually estimated in the model. Does not include seasonal autoregressive parameters (see `seasonalarparams`) or parameters whose values are constrained to be zero. """ return self._params_ar @cache_readonly def seasonalarparams(self): """ (array) Seasonal autoregressive parameters actually estimated in the model. Does not include nonseasonal autoregressive parameters (see `arparams`) or parameters whose values are constrained to be zero. """ return self._params_seasonal_ar @cache_readonly def maparams(self): """ (array) Moving average parameters actually estimated in the model. Does not include seasonal moving average parameters (see `seasonalmaparams`) or parameters whose values are constrained to be zero. """ return self._params_ma @cache_readonly def seasonalmaparams(self): """ (array) Seasonal moving average parameters actually estimated in the model. Does not include nonseasonal moving average parameters (see `maparams`) or parameters whose values are constrained to be zero. """ return self._params_seasonal_ma @Appender(MLEResults.summary.__doc__) 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 = list(self.model._spec.ar_lags) if self.model.k_ma == self.model.k_ma_params: order_ma = self.model.k_ma else: order_ma = list(self.model._spec.ma_lags) # 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: tmp = int(self.model.k_seasonal_ar / self.model.seasonal_periods) if tmp == self.model.k_seasonal_ar_params: order_seasonal_ar = ( int(self.model.k_seasonal_ar / self.model.seasonal_periods) ) else: order_seasonal_ar = list(self.model._spec.seasonal_ar_lags) tmp = int(self.model.k_seasonal_ma / self.model.seasonal_periods) if tmp == self.model.k_ma_params: order_seasonal_ma = tmp else: order_seasonal_ma = list(self.model._spec.seasonal_ma_lags) # 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 = f'{self.model.__class__.__name__}{order}{seasonal_order}' return super().summary( alpha=alpha, start=start, title='SARIMAX Results', model_name=model_name ) 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) # noqa:E305