.. module:: statsmodels.tsa.statespace
   :synopsis: Statespace models for time-series analysis

.. currentmodule:: statsmodels.tsa.statespace


.. _statespace:


Time Series Analysis by State Space Methods :mod:`statespace`
=============================================================

:mod:`statsmodels.tsa.statespace` contains classes and functions that are
useful for time series analysis using state space methods.

A general state space model is of the form

.. math::

  y_t & = Z_t \alpha_t + d_t + \varepsilon_t \\
  \alpha_t & = T_t \alpha_{t-1} + c_t + R_t \eta_t \\

where :math:`y_t` refers to the observation vector at time :math:`t`,
:math:`\alpha_t` refers to the (unobserved) state vector at time
:math:`t`, and where the irregular components are defined as

.. math::

  \varepsilon_t \sim N(0, H_t) \\
  \eta_t \sim N(0, Q_t) \\

The remaining variables (:math:`Z_t, d_t, H_t, T_t, c_t, R_t, Q_t`) in the
equations are matrices describing the process. Their variable names and
dimensions are as follows

Z : `design`          :math:`(k\_endog \times k\_states \times nobs)`

d : `obs_intercept`   :math:`(k\_endog \times nobs)`

H : `obs_cov`         :math:`(k\_endog \times k\_endog \times nobs)`

T : `transition`      :math:`(k\_states \times k\_states \times nobs)`

c : `state_intercept` :math:`(k\_states \times nobs)`

R : `selection`       :math:`(k\_states \times k\_posdef \times nobs)`

Q : `state_cov`       :math:`(k\_posdef \times k\_posdef \times nobs)`

In the case that one of the matrices is time-invariant (so that, for
example, :math:`Z_t = Z_{t+1} ~ \forall ~ t`), its last dimension may
be of size :math:`1` rather than size `nobs`.

This generic form encapsulates many of the most popular linear time series
models (see below) and is very flexible, allowing estimation with missing
observations, forecasting, impulse response functions, and much more.

**Example: AR(2) model**

An autoregressive model is a good introductory example to putting models in
state space form. Recall that an AR(2) model is often written as:

.. math::

   y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t

This can be put into state space form in the following way:

.. math::

   y_t & = \begin{bmatrix} 1 & 0 \end{bmatrix} \alpha_t \\
   \alpha_t & = \begin{bmatrix}
      \phi_1 & \phi_2 \\
           1 &      0
   \end{bmatrix} \alpha_{t-1} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \eta_t

Where

.. math::

   Z_t \equiv Z = \begin{bmatrix} 1 & 0 \end{bmatrix}

and

.. math::

   T_t \equiv T & = \begin{bmatrix}
      \phi_1 & \phi_2 \\
           1 &      0
   \end{bmatrix} \\
   R_t \equiv R & = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
   \eta_t & \sim N(0, \sigma^2)

There are three unknown parameters in this model:
:math:`\phi_1, \phi_2, \sigma^2`.

Models and Estimation
---------------------

The following are the main estimation classes, which can be accessed through
`statsmodels.tsa.statespace.api` and their result classes.

Seasonal Autoregressive Integrated Moving-Average with eXogenous regressors (SARIMAX)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The `SARIMAX` class is an example of a fully fledged model created using the
statespace backend for estimation. `SARIMAX` can be used very similarly to
:ref:`tsa <tsa>` models, but works on a wider range of models by adding the
estimation of additive and multiplicative seasonal effects, as well as
arbitrary trend polynomials.

.. autosummary::
   :toctree: generated/

   sarimax.SARIMAX
   sarimax.SARIMAXResults

For an example of the use of this model, see the
`SARIMAX example notebook <examples/notebooks/generated/statespace_sarimax_stata.html>`__
or the very brief code snippet below:


.. code-block:: python

   # Load the statsmodels api
   import statsmodels.api as sm

   # Load your dataset
   endog = pd.read_csv('your/dataset/here.csv')

   # We could fit an AR(2) model, described above
   mod_ar2 = sm.tsa.SARIMAX(endog, order=(2,0,0))
   # Note that mod_ar2 is an instance of the SARIMAX class

   # Fit the model via maximum likelihood
   res_ar2 = mod_ar2.fit()
   # Note that res_ar2 is an instance of the SARIMAXResults class

   # Show the summary of results
   print(res_ar2.summary())

   # We could also fit a more complicated model with seasonal components.
   # As an example, here is an SARIMA(1,1,1) x (0,1,1,4):
   mod_sarimax = sm.tsa.SARIMAX(endog, order=(1,1,1),
                                seasonal_order=(0,1,1,4))
   res_sarimax = mod_sarimax.fit()

   # Show the summary of results
   print(res_sarimax.summary())

The results object has many of the attributes and methods you would expect from
other Statsmodels results objects, including standard errors, z-statistics,
and prediction / forecasting.

Behind the scenes, the `SARIMAX` model creates the design and transition
matrices (and sometimes some of the other matrices) based on the model
specification.

Unobserved Components
^^^^^^^^^^^^^^^^^^^^^

The `UnobservedComponents` class is another example of a statespace model.

.. autosummary::
   :toctree: generated/

   structural.UnobservedComponents
   structural.UnobservedComponentsResults

For examples of the use of this model, see the `example notebook <examples/notebooks/generated/statespace_structural_harvey_jaeger.html>`__ or a notebook on using the unobserved components model to `decompose a time series into a trend and cycle <examples/notebooks/generated/statespace_cycles.html>`__ or the very brief code snippet below:

.. code-block:: python

   # Load the statsmodels api
   import statsmodels.api as sm

   # Load your dataset
   endog = pd.read_csv('your/dataset/here.csv')

   # Fit a local level model
   mod_ll = sm.tsa.UnobservedComponents(endog, 'local level')
   # Note that mod_ll is an instance of the UnobservedComponents class

   # Fit the model via maximum likelihood
   res_ll = mod_ll.fit()
   # Note that res_ll is an instance of the UnobservedComponentsResults class

   # Show the summary of results
   print(res_ll.summary())

   # Show a plot of the estimated level and trend component series
   fig_ll = res_ll.plot_components()

   # We could further add a damped stochastic cycle as follows
   mod_cycle = sm.tsa.UnobservedComponents(endog, 'local level', cycle=True,
                                           damped_cycle=true,
                                           stochastic_cycle=True)
   res_cycle = mod_cycle.fit()

   # Show the summary of results
   print(res_cycle.summary())

   # Show a plot of the estimated level, trend, and cycle component series
   fig_cycle = res_cycle.plot_components()

Vector Autoregressive Moving-Average with eXogenous regressors (VARMAX)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The `VARMAX` class is an example of a multivariate statespace model.

.. autosummary::
   :toctree: generated/

   varmax.VARMAX
   varmax.VARMAXResults

For an example of the use of this model, see the `VARMAX example notebook <examples/notebooks/generated/statespace_varmax.html>`__ or the very brief code snippet below:

.. code-block:: python

   # Load the statsmodels api
   import statsmodels.api as sm

   # Load your (multivariate) dataset
   endog = pd.read_csv('your/dataset/here.csv')

   # Fit a local level model
   mod_var1 = sm.tsa.VARMAX(endog, order=(1,0))
   # Note that mod_var1 is an instance of the VARMAX class

   # Fit the model via maximum likelihood
   res_var1 = mod_var1.fit()
   # Note that res_var1 is an instance of the VARMAXResults class

   # Show the summary of results
   print(res_var1.summary())

   # Construct impulse responses
   irfs = res_ll.impulse_responses(steps=10)

Dynamic Factor Models
^^^^^^^^^^^^^^^^^^^^^

The `DynamicFactor` class is another example of a multivariate statespace
model.

.. autosummary::
   :toctree: generated/

   dynamic_factor.DynamicFactor
   dynamic_factor.DynamicFactorResults

For an example of the use of this model, see the `Dynamic Factor example notebook <examples/notebooks/generated/statespace_dfm_coincident.html>`__ or the very brief code snippet below:

.. code-block:: python

   # Load the statsmodels api
   import statsmodels.api as sm

   # Load your dataset
   endog = pd.read_csv('your/dataset/here.csv')

   # Fit a local level model
   mod_dfm = sm.tsa.DynamicFactor(endog, k_factors=1, factor_order=2)
   # Note that mod_dfm is an instance of the DynamicFactor class

   # Fit the model via maximum likelihood
   res_dfm = mod_dfm.fit()
   # Note that res_dfm is an instance of the DynamicFactorResults class

   # Show the summary of results
   print(res_ll.summary())

   # Show a plot of the r^2 values from regressions of
   # individual estimated factors on endogenous variables.
   fig_dfm = res_ll.plot_coefficients_of_determination()

Custom state space models
^^^^^^^^^^^^^^^^^^^^^^^^^

The true power of the state space model is to allow the creation and estimation
of custom models. Usually that is done by extending the following two classes,
which bundle all of state space representation, Kalman filtering, and maximum
likelihood fitting functionality for estimation and results output.

.. autosummary::
   :toctree: generated/

   mlemodel.MLEModel
   mlemodel.MLEResults

For a basic example demonstrating creating and estimating a custom state space
model, see the `Local Linear Trend example notebook <examples/notebooks/generated/statespace_local_linear_trend.html>`__.
For a more sophisticated example, see the source code for the `SARIMAX` and
`SARIMAXResults` classes, which are built by extending `MLEModel` and
`MLEResults`.

In simple cases, the model can be constructed entirely using the MLEModel
class. For example, the AR(2) model from above could be constructed and
estimated using only the following code:

.. code-block:: python

   import numpy as np
   from scipy.signal import lfilter
   import statsmodels.api as sm

   # True model parameters
   nobs = int(1e3)
   true_phi = np.r_[0.5, -0.2]
   true_sigma = 1**0.5

   # Simulate a time series
   np.random.seed(1234)
   disturbances = np.random.normal(0, true_sigma, size=(nobs,))
   endog = lfilter([1], np.r_[1, -true_phi], disturbances)

   # Construct the model
   class AR2(sm.tsa.statespace.MLEModel):
       def __init__(self, endog):
           # Initialize the state space model
           super(AR2, self).__init__(endog, k_states=2, k_posdef=1,
                                     initialization='stationary')

           # Setup the fixed components of the state space representation
           self['design'] = [1, 0]
           self['transition'] = [[0, 0],
                                     [1, 0]]
           self['selection', 0, 0] = 1

       # Describe how parameters enter the model
       def update(self, params, transformed=True, **kwargs):
           params = super(AR2, self).update(params, transformed, **kwargs)

           self['transition', 0, :] = params[:2]
           self['state_cov', 0, 0] = params[2]

       # Specify start parameters and parameter names
       @property
       def start_params(self):
           return [0,0,1]  # these are very simple

   # Create and fit the model
   mod = AR2(endog)
   res = mod.fit()
   print(res.summary())

This results in the following summary table::

                              Statespace Model Results                           
   ==============================================================================
   Dep. Variable:                      y   No. Observations:                 1000
   Model:                            AR2   Log Likelihood               -1389.437
   Date:                Wed, 26 Oct 2016   AIC                           2784.874
   Time:                        00:42:03   BIC                           2799.598
   Sample:                             0   HQIC                          2790.470
                                  - 1000                                         
   Covariance Type:                  opg                                         
   ==============================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
   ------------------------------------------------------------------------------
   param.0        0.4395      0.030     14.730      0.000       0.381       0.498
   param.1       -0.2055      0.032     -6.523      0.000      -0.267      -0.144
   param.2        0.9425      0.042     22.413      0.000       0.860       1.025
   ===================================================================================
   Ljung-Box (Q):                       24.25   Jarque-Bera (JB):                 0.22
   Prob(Q):                              0.98   Prob(JB):                         0.90
   Heteroskedasticity (H):               1.05   Skew:                            -0.04
   Prob(H) (two-sided):                  0.66   Kurtosis:                         3.02
   ===================================================================================
   
   Warnings:
   [1] Covariance matrix calculated using the outer product of gradients (complex-step).

The results object has many of the attributes and methods you would expect from
other Statsmodels results objects, including standard errors, z-statistics,
and prediction / forecasting.

More advanced usage is possible, including specifying parameter
transformations, and specifing names for parameters for a more informative
output summary.

State space representation and Kalman filtering
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

While creation of custom models will almost always be done by extending
`MLEModel` and `MLEResults`, it can be useful to understand the superstructure
behind those classes.

Maximum likelihood estimation requires evaluating the likelihood function of
the model, and for models in state space form the likelihood function is
evaluted as a byproduct of running the Kalman filter.

There are two classes used by `MLEModel` that facilitate specification of the
state space model and Kalman filtering: `Representation` and `KalmanFilter`.

The `Representation` class is the piece where the state space model
representation is defined. In simple terms, it holds the state space matrices
(`design`, `obs_intercept`, etc.; see the introduction to state space models,
above) and allows their manipulation.

`FrozenRepresentation` is the most basic results-type class, in that it takes a
"snapshot" of the state space representation at any given time. See the class
documentation for the full list of available attributes.

.. autosummary::
   :toctree: generated/

   representation.Representation
   representation.FrozenRepresentation

The `KalmanFilter` class is a subclass of Representation that provides
filtering capabilities. Once the state space representation matrices have been
constructed, the :py:meth:`filter <kalman_filter.KalmanFilter.filter>`
method can be called, producing a `FilterResults` instance; `FilterResults` is
a subclass of `FrozenRepresentation`.

The `FilterResults` class not only holds a frozen representation of the state
space model (the design, transition, etc. matrices, as well as model
dimensions, etc.) but it also holds the filtering output, including the
:py:attr:`filtered state <kalman_filter.FilterResults.filtered_state>` and
loglikelihood (see the class documentation for the full list of available
results). It also provides a
:py:meth:`predict <kalman_filter.FilterResults.predict>` method, which allows
in-sample prediction or out-of-sample forecasting. A similar method,
:py:meth:`predict <kalman_filter.FilterResults.get_prediction>`, provides
additional prediction or forecasting results, including confidence intervals.

.. autosummary::
   :toctree: generated/

   kalman_filter.KalmanFilter
   kalman_filter.FilterResults
   kalman_filter.PredictionResults

The `KalmanSmoother` class is a subclass of `KalmanFilter` that provides
smoothing capabilities. Once the state space representation matrices have been
constructed, the :py:meth:`filter <kalman_filter.KalmanSmoother.smooth>`
method can be called, producing a `SmootherResults` instance; `SmootherResults`
is a subclass of `FilterResults`.

The `SmootherResults` class holds all the output from `FilterResults`, but
also includes smoothing output, including the
:py:attr:`smoothed state <kalman_filter.SmootherResults.smoothed_state>` and
loglikelihood (see the class documentation for the full list of available
results). Whereas "filtered" output at time `t` refers to estimates conditional
on observations up through time `t`, "smoothed" output refers to estimates
conditional on the entire set of observations in the dataset.

.. autosummary::
   :toctree: generated/

   kalman_smoother.KalmanSmoother
   kalman_smoother.SmootherResults

Statespace diagnostics
----------------------

Three diagnostic tests are available after estimation of any statespace model,
whether built in or custom, to help assess whether the model conforms to the
underlying statistical assumptions. These tests are:

- :py:meth:`test_normality <mlemodel.MLEResults.test_normality>`
- :py:meth:`test_heteroskedasticity <mlemodel.MLEResults.test_heteroskedasticity>`
- :py:meth:`test_serial_correlation <mlemodel.MLEResults.test_serial_correlation>`

A number of standard plots of regression residuals are available for the same
purpose. These can be produced using the command
:py:meth:`plot_diagnostics <mlemodel.MLEResults.plot_diagnostics>`.

Statespace Tools
----------------

There are a variety of tools used for state space modeling or by the SARIMAX
class:

.. autosummary::
   :toctree: generated/

   tools.companion_matrix
   tools.diff
   tools.is_invertible
   tools.constrain_stationary_univariate
   tools.unconstrain_stationary_univariate
   tools.constrain_stationary_multivariate
   tools.unconstrain_stationary_multivariate
   tools.validate_matrix_shape
   tools.validate_vector_shape
