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function [PredictedStateMean, PredictedStateVarianceSquareRoot, StateVectorMean, StateVectorVarianceSquareRoot] = ...
gaussian_filter_bank(ReducedForm, obs, StateVectorMean, StateVectorVarianceSquareRoot, Q_lower_triangular_cholesky, H_lower_triangular_cholesky, H, ...
ParticleOptions, ThreadsOptions, options_, M_)
%
% Computes the proposal with a gaussian approximation for importance
% sampling
% This proposal is a gaussian distribution calculated à la Kalman
%
% INPUTS
% reduced_form_model [structure] Matlab's structure describing the reduced form model.
% reduced_form_model.measurement.H [double] (pp x pp) variance matrix of measurement errors.
% reduced_form_model.state.Q [double] (qq x qq) variance matrix of state errors.
% reduced_form_model.state.dr [structure] output of resol.m.
% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
%
% OUTPUTS
% LIK [double] scalar, likelihood
% lik [double] vector, density of observations in each period.
%
% REFERENCES
%
% NOTES
% The vector "lik" is used to evaluate the jacobian of the likelihood.
% Copyright © 2009-2022 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
order = options_.order;
if ReducedForm.use_k_order_solver
dr = ReducedForm.dr;
udr = ReducedForm.udr;
else
% Set local state space model (first-order approximation).
ghx = ReducedForm.ghx;
ghu = ReducedForm.ghu;
% Set local state space model (second-order approximation).
ghxx = ReducedForm.ghxx;
ghuu = ReducedForm.ghuu;
ghxu = ReducedForm.ghxu;
ghs2 = ReducedForm.ghs2;
if order == 3
% Set local state space model (third order approximation).
ghxxx = ReducedForm.ghxxx;
ghuuu = ReducedForm.ghuuu;
ghxxu = ReducedForm.ghxxu;
ghxuu = ReducedForm.ghxuu;
ghxss = ReducedForm.ghxss;
ghuss = ReducedForm.ghuss;
end
end
constant = ReducedForm.constant;
steadystate = ReducedForm.steadystate;
state_variables_steady_state = ReducedForm.state_variables_steady_state;
mf0 = ReducedForm.mf0;
mf1 = ReducedForm.mf1;
number_of_state_variables = length(mf0);
number_of_observed_variables = length(mf1);
number_of_structural_innovations = length(ReducedForm.Q);
if ParticleOptions.proposal_approximation.montecarlo
nodes = randn(ParticleOptions.number_of_particles, number_of_state_variables+number_of_structural_innovations) ;
weights = 1/ParticleOptions.number_of_particles ;
weights_c = weights ;
elseif ParticleOptions.proposal_approximation.cubature
[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables+number_of_structural_innovations) ;
weights_c = weights ;
elseif ParticleOptions.proposal_approximation.unscented
[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables+number_of_structural_innovations, ParticleOptions);
else
error('This approximation for the proposal is not implemented or unknown!')
end
xbar = [StateVectorMean ; zeros(number_of_structural_innovations,1)] ;
sqr_Px = [ StateVectorVarianceSquareRoot, zeros(number_of_state_variables, number_of_structural_innovations);
zeros(number_of_structural_innovations, number_of_state_variables) Q_lower_triangular_cholesky];
sigma_points = bsxfun(@plus, xbar, sqr_Px*(nodes'));
StateVectors = sigma_points(1:number_of_state_variables,:);
epsilon = sigma_points(number_of_state_variables+1:number_of_state_variables+number_of_structural_innovations,:);
yhat = bsxfun(@minus, StateVectors, state_variables_steady_state);
if ReducedForm.use_k_order_solver
tmp = local_state_space_iteration_k(yhat, epsilon, dr, M_, options_, udr);
else
if order == 2
tmp = local_state_space_iteration_2(yhat, epsilon, ghx, ghu, constant, ghxx, ghuu, ghxu, ThreadsOptions.local_state_space_iteration_2);
elseif order == 3
tmp = local_state_space_iteration_3(yhat, epsilon, ghx, ghu, ghxx, ghuu, ghxu, ghs2, ghxxx, ghuuu, ghxxu, ghxuu, ghxss, ghuss, steadystate, ThreadsOptions.local_state_space_iteration_3, false);
else
error('Order > 3: use_k_order_solver should be set to true');
end
end
PredictedStateMean = tmp(mf0,:)*weights;
PredictedObservedMean = tmp(mf1,:)*weights;
if ParticleOptions.proposal_approximation.cubature || ParticleOptions.proposal_approximation.montecarlo
PredictedStateMean = sum(PredictedStateMean, 2);
PredictedObservedMean = sum(PredictedObservedMean, 2);
dState = bsxfun(@minus,tmp(mf0,:), PredictedStateMean)'.*sqrt(weights);
dObserved = bsxfun(@minus, tmp(mf1,:), PredictedObservedMean)'.*sqrt(weights);
PredictedStateVarianceSquareRoot = chol(dState'*dState)';
big_mat = [dObserved, dState ; H_lower_triangular_cholesky, zeros(number_of_observed_variables,number_of_state_variables)];
[~, mat] = qr2(big_mat, 0);
mat = mat';
PredictedObservedVarianceSquareRoot = mat(1:number_of_observed_variables,1:number_of_observed_variables);
CovarianceObservedStateSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),1:number_of_observed_variables);
StateVectorVarianceSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),number_of_observed_variables+(1:number_of_state_variables));
PredictionError = obs - PredictedObservedMean;
StateVectorMean = PredictedStateMean + (CovarianceObservedStateSquareRoot/PredictedObservedVarianceSquareRoot)*PredictionError;
else
dState = bsxfun(@minus, tmp(mf0,:), PredictedStateMean);
dObserved = bsxfun(@minus, tmp(mf1,:), PredictedObservedMean);
PredictedStateVariance = dState*diag(weights_c)*dState';
PredictedObservedVariance = dObserved*diag(weights_c)*dObserved' + H;
PredictedStateAndObservedCovariance = dState*diag(weights_c)*dObserved';
PredictedStateVarianceSquareRoot = chol(PredictedStateVariance)';
PredictionError = obs - PredictedObservedMean;
KalmanFilterGain = PredictedStateAndObservedCovariance/PredictedObservedVariance;
StateVectorMean = PredictedStateMean + KalmanFilterGain*PredictionError;
StateVectorVariance = PredictedStateVariance - KalmanFilterGain*PredictedObservedVariance*KalmanFilterGain';
StateVectorVariance = .5*(StateVectorVariance+StateVectorVariance');
StateVectorVarianceSquareRoot = reduced_rank_cholesky(StateVectorVariance)';
end
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