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function [LIK,lik] = gaussian_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, options_, M_)
% [LIK,lik] = gaussian_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, options_, M_)
% Evaluates the likelihood of a non-linear model approximating the
% predictive (prior) and filtered (posterior) densities for state variables
% by gaussian distributions.
% Gaussian approximation is done by:
% - a spherical-radial cubature (ref: Arasaratnam & Haykin, 2009).
% - a scaled unscented transform cubature (ref: Julier & Uhlmann 1995)
% - Monte-Carlo draws from a multivariate gaussian distribution.
% First and second moments of prior and posterior state densities are computed
% from the resulting nodes/particles and allows to generate new distributions at the
% following observation.
% Pros: The use of nodes is much faster than Monte-Carlo Gaussian particle and standard particles
% filters since it treats a lesser number of particles. Furthermore, in all cases, there is no need
% of resampling.
% Cons: estimations may be biaised if the model is truly non-gaussian
% since predictive and filtered densities are unimodal.
%
% INPUTS
% Reduced_Form [structure] Matlab's structure describing the reduced form model.
% Y [double] matrix of original observed variables.
% start [double] structural parameters.
% ParticleOptions [structure] Matlab's structure describing options concerning particle filtering.
% ThreadsOptions [structure] Matlab's structure.
%
% 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 © 2009-2019 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/>.
% Set default
if isempty(start)
start = 1;
end
mf0 = ReducedForm.mf0;
mf1 = ReducedForm.mf1;
sample_size = size(Y,2);
number_of_state_variables = length(mf0);
number_of_observed_variables = length(mf1);
number_of_particles = ParticleOptions.number_of_particles;
% compute gaussian quadrature nodes and weights on states and shocks
if ParticleOptions.distribution_approximation.cubature
[nodes2, weights2] = spherical_radial_sigma_points(number_of_state_variables);
weights_c2 = weights2;
elseif ParticleOptions.distribution_approximation.unscented
[nodes2, weights2, weights_c2] = unscented_sigma_points(number_of_state_variables,ParticleOptions);
else
if ~ParticleOptions.distribution_approximation.montecarlo
error('This approximation for the proposal is unknown!')
end
end
if ParticleOptions.distribution_approximation.montecarlo
options_=set_dynare_seed_local_options(options_,'default');
end
% Get covariance matrices
Q = ReducedForm.Q;
H = ReducedForm.H;
if isempty(H)
H = 0;
H_lower_triangular_cholesky = 0;
else
H_lower_triangular_cholesky = reduced_rank_cholesky(H)';
end
% Get initial condition for the state vector.
StateVectorMean = ReducedForm.StateVectorMean;
StateVectorVarianceSquareRoot = reduced_rank_cholesky(ReducedForm.StateVectorVariance)';
state_variance_rank = size(StateVectorVarianceSquareRoot,2);
Q_lower_triangular_cholesky = reduced_rank_cholesky(Q)';
% Initialization of the likelihood.
const_lik = (2*pi)^(number_of_observed_variables/2) ;
lik = NaN(sample_size,1);
LIK = NaN;
for t=1:sample_size
[PredictedStateMean, PredictedStateVarianceSquareRoot, StateVectorMean, StateVectorVarianceSquareRoot] = ...
gaussian_filter_bank(ReducedForm, Y(:,t), StateVectorMean, StateVectorVarianceSquareRoot, Q_lower_triangular_cholesky, H_lower_triangular_cholesky, ...
H, ParticleOptions, ThreadsOptions, options_, M_);
if ParticleOptions.distribution_approximation.cubature || ParticleOptions.distribution_approximation.unscented
StateParticles = bsxfun(@plus, StateVectorMean, StateVectorVarianceSquareRoot*nodes2');
IncrementalWeights = gaussian_densities(Y(:,t), StateVectorMean, StateVectorVarianceSquareRoot, PredictedStateMean, ...
PredictedStateVarianceSquareRoot, StateParticles, H, const_lik, ...
weights2, weights_c2, ReducedForm, ThreadsOptions, ...
options_, M_);
SampleWeights = weights2.*IncrementalWeights;
else
StateParticles = bsxfun(@plus, StateVectorVarianceSquareRoot*randn(state_variance_rank, number_of_particles), StateVectorMean) ;
IncrementalWeights = gaussian_densities(Y(:,t), StateVectorMean, StateVectorVarianceSquareRoot, PredictedStateMean, ...
PredictedStateVarianceSquareRoot,StateParticles,H,const_lik, ...
1/number_of_particles,1/number_of_particles,ReducedForm,ThreadsOptions, ...
options_, M_);
SampleWeights = IncrementalWeights/number_of_particles;
end
SampleWeights = SampleWeights + 1e-6*ones(size(SampleWeights, 1), 1);
SumSampleWeights = sum(SampleWeights);
lik(t) = log(SumSampleWeights);
SampleWeights = SampleWeights./SumSampleWeights;
if not(ParticleOptions.distribution_approximation.cubature || ParticleOptions.distribution_approximation.unscented)
if (ParticleOptions.resampling.status.generic && neff(SampleWeights)<ParticleOptions.resampling.threshold*sample_size) || ParticleOptions.resampling.status.systematic
StateParticles = resample(StateParticles', SampleWeights, ParticleOptions)';
SampleWeights = ones(number_of_particles, 1)/number_of_particles;
end
end
StateVectorMean = StateParticles*SampleWeights;
temp = bsxfun(@minus, StateParticles, StateVectorMean);
StateVectorVarianceSquareRoot = reduced_rank_cholesky(bsxfun(@times,SampleWeights',temp)*temp')';
end
LIK = -sum(lik(start:end));
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