# Source code for statsmodels.tsa.statespace.cfa_simulation_smoother

"""
"Cholesky Factor Algorithm" (CFA) simulation smoothing for state space models

"""

import numpy as np

from . import tools

[docs]class CFASimulationSmoother(object):
r"""
"Cholesky Factor Algorithm" (CFA) simulation smoother

Parameters
----------
model : Representation
The state space model.

Notes
-----
This class allows simulation smoothing by the "Cholesky Factor Algorithm"
(CFA) described in [1]_ and [2]_, which essentially takes advantage of the
existence of an efficient sparse Cholesky factor algorithm for banded
matrices that are held in a sparse matrix format.

In particular, this simulation smoother computes the joint posterior mean
and covariance matrix for the unobserved state vector all at once, rather
than using the recursive computations of the Kalman filter and smoother. It
then uses these posterior moments to sample directly from this joint
posterior. For some models, it can be more computationally efficient than
the simulation smoother based on the Kalman filter and smoother.

**Important caveat**:

However, this simulation smoother cannot be used with all state space
models, including several of the most popular. In particular, the CFA
algorithm cannot support degenerate distributions (i.e. positive
semi-definite covariance matrices) for the initial state (which is the
prior for the first state) or the observation or state innovations.

One practical problem with this algorithm is that an autoregressive term
with order higher than one is typically put into state space form by
augmenting the states using identities. As identities, these augmenting
terms will not be subject to random innovations, and so the state
innovation will be degenerate. It is possible to take these higher order
terms directly into account when constructing the posterior covariance
matrix, but this has not yet been implemented.

Similarly, some state space forms of SARIMA and VARMA models make
the observation equation an identity, which is not compatible with the CFA
simulation smoothing approach.

This simulation smoother has so-far found most of its use with dynamic
factor and stochastic volatility models, which satisfy the restrictions
described above.

**Not-yet-implemented**:

There are several features that are not yet available with this simulation
smoother:

- It does not yet allow diffuse initialization of the state vector.
- It produces simulated states only for exactly the observations in the
model (i.e. it cannot produce simulations for a subset of the model
observations or for observations outside the model).

References
----------
.. [1] McCausland, William J., Shirley Miller, and Denis Pelletier.
"Simulation smoothing for state-space models: A computational
efficiency analysis."
Computational Statistics & Data Analysis 55, no. 1 (2011): 199-212.
.. [2] Chan, Joshua CC, and Ivan Jeliazkov.
"Efficient simulation and integrated likelihood estimation in
state space models."
International Journal of Mathematical Modelling and Numerical
Optimisation 1, no. 1-2 (2009): 101-120.
"""

def __init__(self, model, cfa_simulation_smoother_classes=None):
self.model = model

# Get the simulation smoother classes
self.prefix_simulation_smoother_map = (
cfa_simulation_smoother_classes
if cfa_simulation_smoother_classes is not None
else tools.prefix_cfa_simulation_smoother_map.copy())

self._simulation_smoothers = {}

self._posterior_mean = None
self._posterior_cov_inv_chol = None
self._posterior_cov = None
self._simulated_state = None

@property
def _simulation_smoother(self):
prefix = self.model.prefix
if prefix in self._simulation_smoothers:
return self._simulation_smoothers[prefix]
return None

@property
def posterior_mean(self):
r"""
Posterior mean of the states conditional on the data

Notes
-----

.. math::

\hat \alpha_t = E[\alpha_t \mid Y^n ]

This posterior mean is identical to the smoothed_state computed by
the Kalman smoother.
"""
if self._posterior_mean is None:
self._posterior_mean = np.array(
self._simulation_smoother.posterior_mean, copy=True)
return self._posterior_mean

@property
def posterior_cov_inv_chol_sparse(self):
r"""
Sparse Cholesky factor of inverse posterior covariance matrix

Notes
-----
This attribute holds in sparse diagonal banded storage the Cholesky
factor of the inverse of the posterior covariance matrix. If we denote
:math:P = Var[\alpha \mid Y^n ], then the this attribute holds the
lower Cholesky factor :math:L, defined from :math:L L' = P^{-1}.
This attribute uses the sparse diagonal banded storage described in the
documentation of, for example, the SciPy function
scipy.linalg.solveh_banded.
"""
if self._posterior_cov_inv_chol is None:
self._posterior_cov_inv_chol = np.array(
self._simulation_smoother.posterior_cov_inv_chol, copy=True)
return self._posterior_cov_inv_chol

@property
def posterior_cov(self):
r"""
Posterior covariance of the states conditional on the data

Notes
-----
**Warning**: the matrix computed when accessing this property can be
extremely large: it is shaped (nobs * k_states, nobs * k_states). In
most cases, it is better to use the posterior_cov_inv_chol_sparse
property if possible, which holds in sparse diagonal banded storage
the Cholesky factor of the inverse of the posterior covariance matrix.

.. math::

Var[\alpha \mid Y^n ]

This posterior covariance matrix is *not* identical to the
smoothed_state_cov attribute produced by the Kalman smoother, because
smoothed_state_cov contains the (k_states, k_states) block
diagonal entries of this posterior covariance matrix.
"""
if self._posterior_cov is None:
from scipy.linalg import cho_solve_banded
inv_chol = self.posterior_cov_inv_chol_sparse
self._posterior_cov = cho_solve_banded(
(inv_chol, True), np.eye(inv_chol.shape[1]))
return self._posterior_cov

[docs]    def simulate(self, variates=None, update_posterior=True):
r"""
Perform simulation smoothing (via Cholesky factor algorithm)

Does not return anything, but populates the object's simulated_state
attribute, and also makes available the attributes posterior_mean,
posterior_cov, and posterior_cov_inv_chol_sparse.

Parameters
----------
variates : array_like, optional
Random variates, distributed standard Normal. Usually only
specified if results are to be replicated (e.g. to enforce a seed)
or for testing. If not specified, random variates are drawn. Must
be shaped (nobs, k_states).

Notes
-----
The first step in simulating from the joint posterior of the state
vector conditional on the data is to compute the two relevant moments
of the joint posterior distribution:

.. math::

\alpha \mid Y_n \sim N(\hat \alpha, Var(\alpha \mid Y_n))

Let :math:L L' = Var(\alpha \mid Y_n)^{-1}. Then simulation proceeds
according to the following steps:

1. Draw :math:u \sim N(0, I)
2. Compute :math:x = \hat \alpha + (L')^{-1} u

And then :math:x is a draw from the joint posterior of the states.
The output of the function is as follows:

- The simulated draw :math:x is held in the simulated_state
attribute.
- The posterior mean :math:\hat \alpha is held in the
posterior_mean attribute.
- The (lower triangular) Cholesky factor of the inverse posterior
covariance matrix, :math:L, is held in sparse diagonal banded
storage in the posterior_cov_inv_chol attribute.
- The posterior covariance matrix :math:Var(\alpha \mid Y_n) can be
computed on demand by accessing the posterior_cov property. Note
that this matrix can be extremely large, so care must be taken when
accessing this property. In most cases, it will be preferred to make
use of the posterior_cov_inv_chol attribute rather than the
posterior_cov attribute.

"""
# (Re) initialize the _statespace representation
prefix, dtype, create = self.model._initialize_representation()

# Validate variates and get in required datatype
if variates is not None:
tools.validate_matrix_shape('variates', variates.shape,
self.model.k_states,
self.model.nobs, 1)
variates = np.ravel(variates, order='F').astype(dtype)

# (Re) initialize the state
self.model._initialize_state(prefix=prefix)

# Construct the Cython simulation smoother instance, if necessary
if create or prefix not in self._simulation_smoothers:
cls = self.prefix_simulation_smoother_map[prefix]
self._simulation_smoothers[prefix] = cls(
self.model._statespaces[prefix])
sim = self._simulation_smoothers[prefix]

# Update posterior moments, if requested
if update_posterior:
sim.update_sparse_posterior_moments()
self._posterior_mean = None
self._posterior_cov_inv_chol = None
self._posterior_cov = None

# Perform simulation smoothing
self.simulated_state = sim.simulate(variates=variates)