""" "Cholesky Factor Algorithm" (CFA) simulation smoothing for state space models Author: Chad Fulton License: BSD-3 """ import numpy as np from . import tools class CFASimulationSmoother: 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 it additionally contains all cross-covariance terms. Instead, `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 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)