222 lines
11 KiB
Matlab
222 lines
11 KiB
Matlab
function [LIK,lik] = gaussian_mixture_filter(ReducedForm,Y,start,DynareOptions)
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% Evaluates the likelihood of a non-linear model approximating the state
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% variables distributions with gaussian mixtures. Gaussian Mixture allows reproducing
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% a wide variety of generalized distributions (when multimodal for instance).
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% Each gaussian distribution is obtained whether
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% - with a Smolyak quadrature à la Kronrod & Paterson (Heiss & Winschel 2010, Winschel & Kratzig 2010).
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% - with a radial-spherical cubature
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% - with scaled unscented sigma-points
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% A Sparse grid Kalman Filter is implemented on each component of the mixture,
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% which confers it a weight about current information.
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% Information on the current observables is then embodied in the proposal
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% distribution in which we draw particles, which allows
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% - reaching a greater precision relatively to a standard particle filter,
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% - reducing the number of particles needed,
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% - still being faster.
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%
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%
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% INPUTS
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% reduced_form_model [structure] Matlab's structure describing the reduced form model.
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% reduced_form_model.measurement.H [double] (pp x pp) variance matrix of measurement errors.
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% reduced_form_model.state.Q [double] (qq x qq) variance matrix of state errors.
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% reduced_form_model.state.dr [structure] output of resol.m.
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% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
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% start [integer] scalar, likelihood evaluation starts at 'start'.
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%
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% OUTPUTS
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% LIK [double] scalar, likelihood
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% lik [double] vector, density of observations in each period.
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%
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% REFERENCES
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%
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% Van der Meerwe & Wan, Gaussian Mixture Sigma-Point Particle Filters for Sequential
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% Probabilistic Inference in Dynamic State-Space Models.
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% Heiss & Winschel, 2010, Journal of Applied Economics.
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% Winschel & Kratzig, 2010, Econometrica.
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%
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% NOTES
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% The vector "lik" is used to evaluate the jacobian of the likelihood.
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% Copyright (C) 2009-2013 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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persistent init_flag mf0 mf1 Gprime Gsecond
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persistent nodes weights weights_c I J G number_of_particles
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persistent sample_size number_of_state_variables number_of_observed_variables number_of_structural_innovations
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% Set default
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if isempty(start)
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start = 1;
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end
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% Set persistent variables.
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if isempty(init_flag)
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mf0 = ReducedForm.mf0;
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mf1 = ReducedForm.mf1;
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sample_size = size(Y,2);
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number_of_state_variables = length(mf0);
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number_of_observed_variables = length(mf1);
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number_of_structural_innovations = length(ReducedForm.Q);
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G = DynareOptions.particle.mixture_state_variables; % number of GM components in state
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I = DynareOptions.particle.mixture_structural_shocks ; % number of GM components in structural noise
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J = DynareOptions.particle.mixture_measurement_shocks ; % number of GM components in observation noise
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Gprime = G*I ;
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Gsecond = G*I*J ;
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number_of_particles = DynareOptions.particle.number_of_particles;
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init_flag = 1;
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end
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SampleWeights = ones(Gsecond,1)/Gsecond ;
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% compute gaussian quadrature nodes and weights on states and shocks
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if isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'quadrature')
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[nodes,weights] = nwspgr('GQN',number_of_state_variables,DynareOptions.particle.smolyak_accuracy) ;
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weights_c = weights ;
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elseif isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'cubature')
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[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables) ;
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weights_c = weights ;
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elseif isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'unscented')
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[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables,DynareOptions) ;
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else
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% Set seed for randn().
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set_dynare_seed('default');
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SampleWeights = 1/number_of_particles ;
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end
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% Get covariance matrices
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Q = ReducedForm.Q;
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H = ReducedForm.H;
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if isempty(H)
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H = 0;
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H_lower_triangular_cholesky = 0;
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else
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H_lower_triangular_cholesky = reduced_rank_cholesky(H)';
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end
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Q_lower_triangular_cholesky = reduced_rank_cholesky(Q)';
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% Initialize all matrices
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StateWeights = ones(1,G)/G ;
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StateMu = ReducedForm.StateVectorMean*ones(1,G) ;
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StateSqrtP = zeros(number_of_state_variables,number_of_state_variables,G) ;
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for g=1:G
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StateSqrtP(:,:,g) = reduced_rank_cholesky(ReducedForm.StateVectorVariance)' ;
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end
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StructuralShocksWeights = ones(1,I)/I ;
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StructuralShocksMu = zeros(number_of_structural_innovations,I) ;
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StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
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for i=1:I
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StructuralShocksSqrtP(:,:,i) = Q_lower_triangular_cholesky ;
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end
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ObservationShocksWeights = ones(1,J)/J ;
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ObservationShocksMu = zeros(number_of_observed_variables,J) ;
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ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
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for j=1:J
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ObservationShocksSqrtP(:,:,j) = H_lower_triangular_cholesky ;
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end
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StateWeightsPrior = zeros(1,Gprime) ;
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StateMuPrior = zeros(number_of_state_variables,Gprime) ;
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StateSqrtPPrior = zeros(number_of_state_variables,number_of_state_variables,Gprime) ;
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StateWeightsPost = zeros(1,Gsecond) ;
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StateMuPost = zeros(number_of_state_variables,Gsecond) ;
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StateSqrtPPost = zeros(number_of_state_variables,number_of_state_variables,Gsecond) ;
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%estimate = zeros(sample_size,number_of_state_variables,3) ;
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const_lik = (2*pi)^(.5*number_of_observed_variables) ;
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ks = 0 ;
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lik = NaN(sample_size,1);
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LIK = NaN;
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for t=1:sample_size
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% Build the proposal joint quadratures of Gaussian on states, structural
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% shocks and observation shocks based on each combination of mixtures
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for i=1:I
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for j=1:J
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for g=1:G ;
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a = g + (j-1)*G ;
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b = a + (i-1)*Gprime ;
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[StateMuPrior(:,a),StateSqrtPPrior(:,:,a),StateWeightsPrior(1,a),...
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StateMuPost(:,b),StateSqrtPPost(:,:,b),StateWeightsPost(1,b)] =...
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gaussian_mixture_filter_bank(ReducedForm,Y(:,t),StateMu(:,g),StateSqrtP(:,:,g),StateWeights(1,g),...
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StructuralShocksMu(:,i),StructuralShocksSqrtP(:,:,i),StructuralShocksWeights(1,i),...
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ObservationShocksMu(:,j),ObservationShocksSqrtP(:,:,j),ObservationShocksWeights(1,j),...
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H,H_lower_triangular_cholesky,const_lik,DynareOptions) ;
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end
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end
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end
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% Normalize weights
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StateWeightsPrior = StateWeightsPrior/sum(StateWeightsPrior,2) ;
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StateWeightsPost = StateWeightsPost/sum(StateWeightsPost,2) ;
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if strcmpi(DynareOptions.particle.approximation_method,'quadrature') || ... % sparse grids approximations
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strcmpi(DynareOptions.particle.approximation_method,'cubature') || ...
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strcmpi(DynareOptions.particle.approximation_method,'unscented')
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for i=1:Gsecond
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StateParticles = bsxfun(@plus,StateMuPost(:,i),StateSqrtPPost(:,:,i)*nodes') ;
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IncrementalWeights = gaussian_mixture_densities(Y(:,t),StateMuPrior,StateSqrtPPrior,StateWeightsPrior,...
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StateMuPost,StateSqrtPPost,StateWeightsPost,...
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StateParticles,H,const_lik,weights,weights_c,ReducedForm,DynareOptions) ;
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SampleWeights(i) = sum(StateWeightsPost(i)*weights.*IncrementalWeights) ;
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end
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SumSampleWeights = sum(SampleWeights) ;
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lik(t) = log(SumSampleWeights) ;
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SampleWeights = SampleWeights./SumSampleWeights ;
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[ras,SortedRandomIndx] = sort(rand(1,Gsecond));
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SortedRandomIndx = SortedRandomIndx(1:G);
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indx = index_resample(0,SampleWeights,DynareOptions) ;
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indx = indx(SortedRandomIndx) ;
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StateMu = StateMuPost(:,indx);
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StateSqrtP = StateSqrtPPost(:,:,indx);
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StateWeights = ones(1,G)/G ;
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else
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% Sample particle in the proposal distribution, ie the posterior state GM
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StateParticles = importance_sampling(StateMuPost,StateSqrtPPost,StateWeightsPost',number_of_particles) ;
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% Compute prior, proposal and likelihood of particles
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IncrementalWeights = gaussian_mixture_densities(Y(:,t),StateMuPrior,StateSqrtPPrior,StateWeightsPrior,...
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StateMuPost,StateSqrtPPost,StateWeightsPost,...
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StateParticles,H,const_lik,1/number_of_particles,...
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1/number_of_particles,ReducedForm,DynareOptions) ;
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% calculate importance weights of particles
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SampleWeights = SampleWeights.*IncrementalWeights ;
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SumSampleWeights = sum(SampleWeights,1) ;
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SampleWeights = SampleWeights./SumSampleWeights ;
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lik(t) = log(SumSampleWeights) ;
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% First possible state point estimates
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%estimate(t,:,1) = SampleWeights*StateParticles' ;
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% Resampling if needed of required
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Neff = 1/sum(bsxfun(@power,SampleWeights,2)) ;
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if (Neff<.5*sample_size && strcmpi(DynareOptions.particle.resampling.status,'generic')) || strcmpi(DynareOptions.particle.resampling.status,'systematic')
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ks = ks + 1 ;
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StateParticles = resample(StateParticles',SampleWeights,DynareOptions)' ;
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StateVectorMean = mean(StateParticles,2) ;
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StateVectorVarianceSquareRoot = reduced_rank_cholesky( (StateParticles*StateParticles')/number_of_particles - StateVectorMean*(StateVectorMean') )';
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SampleWeights = 1/number_of_particles ;
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elseif strcmpi(DynareOptions.particle.resampling.status,'none')
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StateVectorMean = StateParticles*sampleWeights ;
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temp = sqrt(SampleWeights').*StateParticles ;
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StateVectorVarianceSquareRoot = reduced_rank_cholesky( temp*temp' - StateVectorMean*(StateVectorMean') )';
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end
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% Use the information from particles to update the gaussian mixture on state variables
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[StateMu,StateSqrtP,StateWeights] = fit_gaussian_mixture(StateParticles,StateMu,StateSqrtP,StateWeights,0.001,10,1) ;
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%estimate(t,:,3) = StateWeights*StateMu' ;
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end
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end
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LIK = -sum(lik(start:end)) ; |