135 lines
6.4 KiB
Matlab
135 lines
6.4 KiB
Matlab
function [LIK,lik] = gaussian_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, DynareOptions, Model)
|
|
|
|
% 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
|
|
set_dynare_seed('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, DynareOptions, Model);
|
|
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, ...
|
|
DynareOptions, Model);
|
|
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, ...
|
|
DynareOptions, Model);
|
|
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)); |