p7]c@`s`dZddlmZmZmZddlmZddlZddl m Z ddl m Z ddl mZddlmZdd lmZddljjZdd lmZd d lmZd d lmZmZmZd dlm Z m!Z!m"Z"m#Z#m$Z$m%Z%defdYZ&defdYZ'defdYZ(ej)e(e'dS(s< SARIMAX Model Author: Chad Fulton License: Simplified-BSD i(tdivisiontabsolute_importtprint_function(twarnN(tlong(tBunch(t_is_using_pandas(tcache_readonly(t ValueWarning(tlagmati(tInitialization(tMLEModelt MLEResultstMLEResultsWrapper(tcompanion_matrixtdifft is_invertibletconstrain_stationary_univariatet!unconstrain_stationary_univariatet prepare_exogtSARIMAXc B`seZdZedddfddddfeeeeeeeeed ZdZdZdZ edZ e dZ e d Z e d Ze d Ze d Zeded Ze dZe edZdddddddddg Ze dZe dZe dZe dZe dZedZdZd Zeed!ZRS("s<2 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 results 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. concentrate_scale : boolean, 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. **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 parameter documentation for more information. 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). 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. 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. iic K`s |d|_||_||_||_| |_| |_| |_| |_| |_||_ ||_ |jr|jrt dnt |dt ttjfrtjdtj|df|_ntjd|df|_|jj|_t |dt ttjfr>tjdtj|df|_ntjd|df|_|jj|_t |dt ttjfrtjddg|jddg|df|_nutjddg|jt|df|_xDtt|dD],}|d|j}|d||j||dstj?|}n|j0|_@|j1|_A|jr|j0dks|j1dkrd|_0d|_1n|j0|j|j1|_Bt||_C||_D||_E|j:d|tFtG|jH|d|d|d|||jrt8|jI_Jn|j6dkst|j dkrtK|jI_Ln|jM|jId<|jN|jId<|jO|jId<|jP|jId<|jr* d|jId$tj |j df}nitj |j |jf}|jdkr d|d 0.(RUR<RRAR R2tlinalgtpinvtdottemptyRFR/Rtmean(RtRGR6RKR9RTt trend_dataRtrt k_params_art k_params_mat residualstYtXt params_artcolstparamst params_trendt params_matparams_variancetoffset((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyt_conditional_sum_squares=sV ,,     ? / 3        c C`s|j}|j r|jdks1|jdkrt|j|j|j|j}|jdk rt|j|j|j|j}nd}|d|j dddf}n3|jj }|jdk r|jj nd}|j }t j t j|r]t j|j }||}|dk rA||}n|dk r]||}q]ng}|jdkrt jj|j|}|t j||}n|jrg}n|j||j|j|j|j|j|\}}}} |jdko$|jo$tt jd| f } | rDtd|d9}n|jdkos|jostt jd|f } | rtd|d9}n|j||j|j |j!|j"\} } }}|jdko|jott jd| f }|rtd| d9} n|j!dkoE|joEtt jd|f }|retd|d9}ng}|jr|j#rdg|j}n|j$r*t%| t&kr*t'| dkr*t%|t&kot'|dks|} q*|jdkrt j(||} q*t j(|||j)} n|j*r9dng}t j+t,t j-| d} |j.rrg} nt j||||| |||| f S( sG Starting parameters for maximum likelihood estimation iNis\Non-stationary starting autoregressive parameters found. 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Using zeros as starting parameters.sSNon-stationary starting seasonal autoregressive Using zeros as starting parameters.sSNon-invertible starting seasonal moving average Using zeros as starting parameters.g|=(/RR)RRRSRRtR.RRARFR7tsqueezeR2tanytisnanRWRRRRXRRGR6RKR9RTR*RR4RR+RMR;RPR?R'RZttypeRoR<tinnerRcR&t atleast_1dRURCR-(RsRRtRtmaskt params_exogRRRRt invalid_art invalid_mat_tparams_seasonal_artparams_seasonal_matparams_seasonal_variancetinvalid_seasonal_artinvalid_seasonal_matparams_exog_variancetparams_measurement_variance((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyt start_params|s  &$       !             !  cC`sd}|jdkrU|jdkr9|r0dnd}qU|rEdnd|j}nd}|jdkr|jdkr|rdnd |j}q|rd nd |j|jf}n|j}|r |jdkr |jdkr |rd nd |||jjfS|r?|jdkr?|r+dnd||jjfS|rt|jdkrt|r`dnd||jjfS|jjSdS(sNames of endogenous variablestiis\DeltatDs \Delta^%dsD%ds \Delta_%dsDS%ds \Delta_%d^%dsD%dS%ds%s%s %ss%s.%s.%ss%s %ss%s.%sN(RJROR.R)Rtynames(RstlatexRt seasonal_difft endog_diff((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyt endog_names s.  $R%Rtartmat seasonal_art seasonal_mat exog_variancetmeasurement_variancetvariancecC`s_|j}g|jD]}||dkr|^q}d|kr[|j r[|jdn|S(s List of parameters actually included in the model, in sorted order. TODO Make this an OrderedDict with slice or indices as the values. iR(t model_orderstparams_completeR(tremove(RsRR#R((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyt param_terms-s  cC`s:|j}|j}g|D]}||D] }|^q'qS(sq List of human readable parameter names (for parameters actually included in the model). (Rt model_names(Rstparams_sort_orderRtparamtname((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyt param_namesAs  cC`si |jd6|jd6|jd6|jd6|jd6|jd6|j|jd6|j|jd6|jr||jr||jnd d 6t|j d 6t|j o|j d 6S( sE The orders of each of the polynomials in the model. R%RRRRRt reduced_art reduced_maiRRR( RTRWRGRKRMRPRXR'R1R&RZR-(Rs((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyRMs      "cC`s|jdtS(sH The plain text names of all possible model parameters. R(t_get_model_namesRh(Rs((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyRascC`s|jdtS(sC The latex names of all possible model parameters. R(RRY(Rs((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pytmodel_latex_nameshscC`si dd6dd6dd6dd6dd6dd6dd6dd6dd 6dd 6dd 6}|jd kr|rnd nd}g|d|j$j+j|jks|j$d jj|j|j$d RPRR@R2RRRRfRR,RbRHRRR RRRR(RsRRRRRRRRRRRRRRR6R9RR;R?RRR((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyRs                                       /! "  %     (t__name__t __module__t__doc__RARhRYReRyRRRrtpropertyRjRkRlRmRt staticmethodRRRRRRRRRRR RR(((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyRsB  / 8  @=  r h hRcB`seZdZddZedZedZedZedZedZ edZ ed Z ed Z ddeddd Zd dd Zejje_RS(sg 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. See Also -------- statsmodels.tsa.statespace.kalman_filter.FilterResults statsmodels.tsa.statespace.mlemodel.MLEResults topgc K`stt|j|||||tj|_|jj|_t i|jj d6|jj d6|jj d6|jj d6|jjd6|jjd6|jjd6|jjd6|jjd 6|jjd 6|jjd 6|jjd 6|jjd 6|jjd6|jjd6|jjd6|jjd6|jjd6|jjd6|jjd6|jjd6|jjd6|jj d6|_!|jj"|_#|jj$|_%|jj&|_'|jj(|_)|jj*|_+tj,|j%|j)|_-tj,|j'|j+|_.|jj/|_/|jj0|_0d}}x|j0D]}|dkrN|jj} n^|dkri|jj} nC|dkr|jj1} n(|dkr|jj2} n |j/|} || 7}t3|d||j4||!|| 7}q-W|j5j6ddgdS( NR.R&R'R)R*R+R,R-R#R$RJRORGRKRMRPRIRLR%RTRWR(RXiRRRRs _params_%sR]R^(7RdRReR2tinftdf_residtmodelRyt _init_kwdsRR.R&R'R)R*R+R,R-R#R$RJRORGRKRMRPRIRLR%RTRWR(RXt specificationRERBR8R6R:R9R>R;R@R?Rtpolynomial_reduced_artpolynomial_reduced_maRRRNRQtsetattrRt_data_attr_modeltextend( RsRRtfilter_resultstcov_typeRuRRRR((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pyResl                               cC`stj|jdS(sQ (array) Roots of the reduced form autoregressive lag polynomial i(R2trootsR (Rs((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pytarrootsscC`stj|jdS(sQ (array) Roots of the reduced form moving average lag polynomial i(R2R'R!(Rs((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pytmarootsscC`s7|j}|jsdStj|j|jdtjS(sj (array) Frequency of the roots of the reduced form autoregressive lag polynomial Ni(R(tsizeR2tarctan2timagRtpi(Rstz((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pytarfreqs  cC`s7|j}|jsdStj|j|jdtjS(sj (array) Frequency of the roots of the reduced form moving average lag polynomial Ni(R)R*R2R+R,RR-(RsR.((sAlib/python2.7/site-packages/statsmodels/tsa/statespace/sarimax.pytmafreqs  cC`s|jS(s (array) Autoregressive parameters actually estimated in the model. 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