226 lines
11 KiB
Matlab
226 lines
11 KiB
Matlab
function [LIK, lik] = gaussian_mixture_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, DynareOptions, Model)
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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 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 © 2009-2017 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 <https://www.gnu.org/licenses/>.
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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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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 = ParticleOptions.mixture_state_variables; % number of GM components in state
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number_of_particles = ParticleOptions.number_of_particles;
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% compute gaussian quadrature nodes and weights on states and shocks
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if ParticleOptions.distribution_approximation.cubature
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[nodes, weights] = spherical_radial_sigma_points(number_of_state_variables);
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elseif ParticleOptions.distribution_approximation.unscented
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[nodes, weights] = unscented_sigma_points(number_of_state_variables, ParticleOptions);
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else
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if ~ParticleOptions.distribution_approximation.montecarlo
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error('This approximation for the proposal is unknown!')
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end
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end
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if ParticleOptions.distribution_approximation.montecarlo
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DynareOptions=set_dynare_seed_local_options(DynareOptions,'default');
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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 mixtures
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StateWeights = ones(1, G)/G;
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StateMu = ReducedForm.StateVectorMean;
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StateSqrtP = zeros(number_of_state_variables, number_of_state_variables, G);
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temp = reduced_rank_cholesky(ReducedForm.StateVectorVariance)';
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StateMu = bsxfun(@plus, StateMu, bsxfun(@times,diag(temp), (-(G-1)/2:1:(G-1)/2))/10);
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for g=1:G
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StateSqrtP(:,:,g) = temp/sqrt(G) ;
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end
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if ~ParticleOptions.mixture_structural_shocks
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StructuralShocksMu = zeros(1, number_of_structural_innovations);
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StructuralShocksWeights = 1;
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I = 1;
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StructuralShocksMu = Q_lower_triangular_cholesky*StructuralShocksMu';
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StructuralShocksSqrtP = zeros(number_of_structural_innovations, number_of_structural_innovations, I);
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StructuralShocksSqrtP(:,:,1) = Q_lower_triangular_cholesky;
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elseif ParticleOptions.mixture_structural_shocks==1
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if ParticleOptions.proposal_approximation.cubature
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[StructuralShocksMu, StructuralShocksWeights] = spherical_radial_sigma_points(number_of_structural_innovations);
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StructuralShocksWeights = ones(size(StructuralShocksMu, 1), 1)*StructuralShocksWeights;
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elseif ParticleOptions.proposal_approximation.unscented
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[StructuralShocksMu, StructuralShocksWeights] = unscented_sigma_points(number_of_structural_innovations, ParticleOptions);
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else
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if ~ParticleOptions.distribution_approximation.montecarlo
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error('This approximation for the proposal is unknown!')
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end
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end
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I = size(StructuralShocksWeights, 1);
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StructuralShocksMu = Q_lower_triangular_cholesky*StructuralShocksMu';
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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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else
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if ParticleOptions.proposal_approximation.cubature
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[StructuralShocksMu, StructuralShocksWeights] = spherical_radial_sigma_points(number_of_structural_innovations);
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StructuralShocksWeights = ones(size(StructuralShocksMu, 1), 1)*StructuralShocksWeights ;
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elseif ParticleOptions.proposal_approximation.unscented
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[StructuralShocksMu, StructuralShocksWeights] = unscented_sigma_points(number_of_structural_innovations, ParticleOptions);
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else
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if ~ParticleOptions.distribution_approximation.montecarlo
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error('This approximation for the proposal is unknown!')
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end
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end
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I = size(StructuralShocksWeights, 1);
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StructuralShocksMu = Q_lower_triangular_cholesky*StructuralShocksMu';
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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/sqrt(StructuralShocksWeights(i));
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end
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end
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ObservationShocksWeights = 1;
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J = 1 ;
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Gprime = G*I;
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Gsecond = G*I*J;
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SampleWeights = ones(Gsecond, 1)/Gsecond;
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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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const_lik = (2*pi)^(.5*number_of_observed_variables);
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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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gprime = g + (i-1)*G;
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gsecond = gprime + (j-1)*Gprime;
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[StateMuPrior(:,gprime), StateSqrtPPrior(:,:,gprime), StateWeightsPrior(1,gprime), ...
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StateMuPost(:,gsecond), StateSqrtPPost(:,:,gsecond), StateWeightsPost(1,gsecond)] = ...
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gaussian_mixture_filter_bank(ReducedForm,Y(:,t), StateMu(:,g), StateSqrtP(:,:,g), StateWeights(g),...
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StructuralShocksMu(:,i), StructuralShocksSqrtP(:,:,i), StructuralShocksWeights(i),...
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ObservationShocksWeights(j), H, H_lower_triangular_cholesky, const_lik, ...
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ParticleOptions, ThreadsOptions, DynareOptions, Model);
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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 ParticleOptions.distribution_approximation.cubature || ParticleOptions.distribution_approximation.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, StateParticles, H, ...
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ReducedForm, ThreadsOptions, DynareOptions, Model);
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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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[~, SortedRandomIndx] = sort(rand(1,Gsecond));
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SortedRandomIndx = SortedRandomIndx(1:G);
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indx = resample(0,SampleWeights,ParticleOptions);
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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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IncrementalWeights = gaussian_mixture_densities(Y(:,t), StateMuPrior, StateSqrtPPrior, StateWeightsPrior, ...
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StateMuPost, StateSqrtPPost, StateWeightsPost, StateParticles, H, ...
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ReducedForm, ThreadsOptions, DynareOptions, Model);
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SampleWeights = IncrementalWeights/number_of_particles;
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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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if (ParticleOptions.resampling.status.generic && neff(SampleWeights)<ParticleOptions.resampling.threshold*sample_size) || ParticleOptions.resampling.status.systematic
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StateParticles = resample(StateParticles',SampleWeights',ParticleOptions)';
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SampleWeights = ones(number_of_particles,1)/number_of_particles;
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end
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[StateMu, StateSqrtP, StateWeights] = fit_gaussian_mixture(StateParticles, SampleWeights', StateMu, StateSqrtP, StateWeights, 0.001, 10, 1);
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end
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end
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LIK = -sum(lik(start:end)); |