class_docstr: "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.
code: 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
  
file_path: statsmodels/statsmodels/tsa/statespace/cfa_simulation_smoother.py
imports: import numpy
  from .None import tools
label_desc: Use image node_id 1 to create a new CFASimulationSmoother object  with example: obj = CFASimulationSmoother(model, cfa_simulation_smoother_classes)