Filtering and decompositions
=============================

.. index:: filtering

Linear filtration
---------------------

Filtering with constant parameters can be done efficiently with algorithms presented in e.g. :cite:`DeJongCCL2003`,
:cite:`KoopmanDurbin2000`, :cite:`KoopmanDurbin2003`, :cite:`DurbinKoopman2012`. Unfortunately those procedures
cannot be used in a regime-switching context.

The main references in the economics literature for the filtration of Markov-switching state-space models
are :cite:`kim1999state` and :cite:`KimNelson2001`, which combine the Kalman and Hamilton filters. Similar algorithms can be found in the engineering
literature. See e.g. :cite:`BarShalomEtAl2001`,  :cite:`BarShalomEtAl2004`.

RISE uses an extension of the algorithms by :cite:`DurbinKoopman2012` except for the initialization process,
which in the case of nonstationarity, hasn't been extended to regime-switching models. In the case of
nonstationarity, RISE simply applies an arbitrary variance for the initialization process.

Also, for efficiency reasons, RISE deviates from :cite:`KimNelson2001` when it comes to the collapsing of the
regimes. While :cite:`KimNelson2001` collapse the updates, RISE collapse the forecasts. The two procedures
yield numerically similar results but the procedure in RISE is faster.

Historical Decomposition
-----------------------------

.. warning:: Exact only for constant-parameter linear models.

Observables decomposition
-------------------------------

.. important:: Available only for constant-parameter models

Conditional forecasting via filtering
-----------------------------------------

.. index:: filtering

Conditional forecasting can be implemented using filtering algorithms.

Suppose we want to condition on a variable X0.

Hard conditions
~~~~~~~~~~~~~~~~~~~~~

We simply smooth over an extended sample.

Soft conditions
~~~~~~~~~~~~~~~~~~~~~~

We relate X0 to some variable X in the model as :math:`X0_t=X_t+\sigma \varepsilon_t`.

And we let :math:`\sigma` control the softness of the condition :

	- If :math:`\sigma` tends to :math:`\infty` the condition is irrelevant
	- If :math:`\sigma` tends to 0 we have **hard conditions**
	- If :math:`\sigma` is greater than 0 but finite (small enough) we have **soft conditions**. 
	
User-defined filter
--------------------