325 lines
16 KiB
Matlab
325 lines
16 KiB
Matlab
function [LIK,lik] = gaussian_mixture_filter(ReducedForm,Y,start,ParticleOptions,ThreadsOptions)
|
|
% Evaluates the likelihood of a non-linear model approximating the state
|
|
% variables distributions with gaussian mixtures. Gaussian Mixture allows reproducing
|
|
% a wide variety of generalized distributions (when multimodal for instance).
|
|
% Each gaussian distribution is obtained whether
|
|
% - with a radial-spherical cubature
|
|
% - with scaled unscented sigma-points
|
|
% A Sparse grid Kalman Filter is implemented on each component of the mixture,
|
|
% which confers it a weight about current information.
|
|
% Information on the current observables is then embodied in the proposal
|
|
% distribution in which we draw particles, which allows
|
|
% - reaching a greater precision relatively to a standard particle filter,
|
|
% - reducing the number of particles needed,
|
|
% - still being faster.
|
|
%
|
|
%
|
|
% 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.
|
|
% start [integer] scalar, likelihood evaluation starts at 'start'.
|
|
%
|
|
% OUTPUTS
|
|
% LIK [double] scalar, likelihood
|
|
% lik [double] vector, density of observations in each period.
|
|
%
|
|
% REFERENCES
|
|
%
|
|
% Van der Meerwe & Wan, Gaussian Mixture Sigma-Point Particle Filters for Sequential
|
|
% Probabilistic Inference in Dynamic State-Space Models.
|
|
% Heiss & Winschel, 2010, Journal of Applied Economics.
|
|
% Winschel & Kratzig, 2010, Econometrica.
|
|
%
|
|
% NOTES
|
|
% The vector "lik" is used to evaluate the jacobian of the likelihood.
|
|
% Copyright (C) 2009-2013 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 <http://www.gnu.org/licenses/>.
|
|
|
|
persistent init_flag mf0 mf1
|
|
persistent nodes weights weights_c I J G number_of_particles
|
|
persistent sample_size number_of_state_variables number_of_observed_variables number_of_structural_innovations
|
|
|
|
% Set default
|
|
if isempty(start)
|
|
start = 1;
|
|
end
|
|
|
|
% Set persistent variables.
|
|
if isempty(init_flag)
|
|
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_structural_innovations = length(ReducedForm.Q);
|
|
G = ParticleOptions.mixture_state_variables; % number of GM components in state
|
|
number_of_particles = ParticleOptions.number_of_particles;
|
|
init_flag = 1;
|
|
end
|
|
|
|
% compute gaussian quadrature nodes and weights on states and shocks
|
|
if isempty(nodes)
|
|
if ParticleOptions.distribution_approximation.cubature
|
|
[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables);
|
|
weights_c = weights;
|
|
elseif ParticleOptions.distribution_approximation.unscented
|
|
[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables,ParticleOptions);
|
|
else
|
|
if ~ParticleOptions.distribution_approximation.montecarlo
|
|
error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
end
|
|
end
|
|
end
|
|
|
|
if ParticleOptions.distribution_approximation.montecarlo
|
|
set_dynare_seed('default');
|
|
SampleWeights = 1/number_of_particles ;
|
|
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
|
|
Q_lower_triangular_cholesky = reduced_rank_cholesky(Q)';
|
|
|
|
% Initialize mixtures
|
|
StateWeights = ones(1,G)/G ;
|
|
StateMu = ReducedForm.StateVectorMean ;
|
|
StateSqrtP = zeros(number_of_state_variables,number_of_state_variables,G) ;
|
|
temp = reduced_rank_cholesky(ReducedForm.StateVectorVariance)' ;
|
|
StateMu = bsxfun(@plus,StateMu,bsxfun(@times,diag(temp),(-(G-1)/2:1:(G-1)/2))/10) ;
|
|
for g=1:G
|
|
StateSqrtP(:,:,g) = temp/sqrt(G) ;
|
|
end
|
|
|
|
% if ParticleOptions.mixture_structural_shocks==1
|
|
% StructuralShocksMu = zeros(1,number_of_structural_innovations) ;
|
|
% StructuralShocksWeights = 1 ;
|
|
% else
|
|
% if ParticleOptions.proposal_approximation.cubature
|
|
% [StructuralShocksMu,StructuralShocksWeights] = spherical_radial_sigma_points(number_of_structural_innovations);
|
|
% StructuralShocksWeights = ones(size(StructuralShocksMu,1),1)*StructuralShocksWeights ;
|
|
% elseif ParticleOptions.proposal_approximation.unscented
|
|
% [StructuralShocksMu,StructuralShocksWeights,raf] = unscented_sigma_points(number_of_structural_innovations,ParticleOptions);
|
|
% else
|
|
% if ~ParticleOptions.distribution_approximation.montecarlo
|
|
% error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
% end
|
|
% end
|
|
% end
|
|
% I = size(StructuralShocksWeights,1) ;
|
|
% StructuralShocksMu = Q_lower_triangular_cholesky*(StructuralShocksMu') ;
|
|
% StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
|
|
% for i=1:I
|
|
% StructuralShocksSqrtP(:,:,i) = Q_lower_triangular_cholesky/sqrt(StructuralShocksWeights(i)) ;
|
|
% end
|
|
%
|
|
% if ParticleOptions.mixture_measurement_shocks==1
|
|
% ObservationShocksMu = zeros(1,number_of_observed_variables) ;
|
|
% ObservationShocksWeights = 1 ;
|
|
% else
|
|
% if ParticleOptions.proposal_approximation.cubature
|
|
% [ObservationShocksMu,ObservationShocksWeights] = spherical_radial_sigma_points(number_of_observed_variables);
|
|
% ObservationShocksWeights = ones(size(ObservationShocksMu,1),1)*ObservationShocksWeights;
|
|
% elseif ParticleOptions.proposal_approximation.unscented
|
|
% [ObservationShocksMu,ObservationShocksWeights,raf] = unscented_sigma_points(number_of_observed_variables,ParticleOptions);
|
|
% else
|
|
% if ~ParticleOptions.distribution_approximation.montecarlo
|
|
% error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
% end
|
|
% end
|
|
% end
|
|
% J = size(ObservationShocksWeights,1) ;
|
|
% ObservationShocksMu = H_lower_triangular_cholesky*(ObservationShocksMu') ;
|
|
% ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
|
|
% for j=1:J
|
|
% ObservationShocksSqrtP(:,:,j) = H_lower_triangular_cholesky/sqrt(ObservationShocksWeights(j)) ;
|
|
% end
|
|
|
|
if ParticleOptions.mixture_structural_shocks==0
|
|
StructuralShocksMu = zeros(1,number_of_structural_innovations) ;
|
|
StructuralShocksWeights = 1 ;
|
|
I = 1 ;
|
|
StructuralShocksMu = Q_lower_triangular_cholesky*(StructuralShocksMu') ;
|
|
StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
|
|
StructuralShocksSqrtP(:,:,1) = Q_lower_triangular_cholesky ;
|
|
elseif ParticleOptions.mixture_structural_shocks==1
|
|
if ParticleOptions.proposal_approximation.cubature
|
|
[StructuralShocksMu,StructuralShocksWeights] = spherical_radial_sigma_points(number_of_structural_innovations);
|
|
StructuralShocksWeights = ones(size(StructuralShocksMu,1),1)*StructuralShocksWeights ;
|
|
elseif ParticleOptions.proposal_approximation.unscented
|
|
[StructuralShocksMu,StructuralShocksWeights,raf] = unscented_sigma_points(number_of_structural_innovations,ParticleOptions);
|
|
else
|
|
if ~ParticleOptions.distribution_approximation.montecarlo
|
|
error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
end
|
|
end
|
|
I = size(StructuralShocksWeights,1) ;
|
|
StructuralShocksMu = Q_lower_triangular_cholesky*(StructuralShocksMu') ;
|
|
StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
|
|
for i=1:I
|
|
StructuralShocksSqrtP(:,:,i) = Q_lower_triangular_cholesky ;
|
|
end
|
|
else
|
|
if ParticleOptions.proposal_approximation.cubature
|
|
[StructuralShocksMu,StructuralShocksWeights] = spherical_radial_sigma_points(number_of_structural_innovations);
|
|
StructuralShocksWeights = ones(size(StructuralShocksMu,1),1)*StructuralShocksWeights ;
|
|
elseif ParticleOptions.proposal_approximation.unscented
|
|
[StructuralShocksMu,StructuralShocksWeights,raf] = unscented_sigma_points(number_of_structural_innovations,ParticleOptions);
|
|
else
|
|
if ~ParticleOptions.distribution_approximation.montecarlo
|
|
error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
end
|
|
end
|
|
I = size(StructuralShocksWeights,1) ;
|
|
StructuralShocksMu = Q_lower_triangular_cholesky*(StructuralShocksMu') ;
|
|
StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
|
|
for i=1:I
|
|
StructuralShocksSqrtP(:,:,i) = Q_lower_triangular_cholesky/sqrt(StructuralShocksWeights(i)) ;
|
|
end
|
|
end
|
|
|
|
ObservationShocksMu = zeros(1,number_of_observed_variables) ;
|
|
ObservationShocksWeights = 1 ;
|
|
J = 1 ;
|
|
ObservationShocksMu = H_lower_triangular_cholesky*(ObservationShocksMu') ;
|
|
ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
|
|
ObservationShocksSqrtP(:,:,1) = H_lower_triangular_cholesky ;
|
|
|
|
% if ParticleOptions.mixture_measurement_shocks==0
|
|
% ObservationShocksMu = zeros(1,number_of_observed_variables) ;
|
|
% ObservationShocksWeights = 1 ;
|
|
% J = 1 ;
|
|
% ObservationShocksMu = H_lower_triangular_cholesky*(ObservationShocksMu') ;
|
|
% ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
|
|
% ObservationShocksSqrtP(:,:,1) = H_lower_triangular_cholesky ;
|
|
% elseif ParticleOptions.mixture_measurement_shocks==1
|
|
% if ParticleOptions.proposal_approximation.cubature
|
|
% [ObservationShocksMu,ObservationShocksWeights] = spherical_radial_sigma_points(number_of_observed_variables);
|
|
% ObservationShocksWeights = ones(size(ObservationShocksMu,1),1)*ObservationShocksWeights;
|
|
% elseif ParticleOptions.proposal_approximation.unscented
|
|
% [ObservationShocksMu,ObservationShocksWeights,raf] = unscented_sigma_points(number_of_observed_variables,ParticleOptions);
|
|
% else
|
|
% if ~ParticleOptions.distribution_approximation.montecarlo
|
|
% error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
% end
|
|
% end
|
|
% J = size(ObservationShocksWeights,1) ;
|
|
% ObservationShocksMu = H_lower_triangular_cholesky*(ObservationShocksMu') ;
|
|
% ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
|
|
% for j=1:J
|
|
% ObservationShocksSqrtP(:,:,j) = H_lower_triangular_cholesky ;
|
|
% end
|
|
% else
|
|
% if ParticleOptions.proposal_approximation.cubature
|
|
% [ObservationShocksMu,ObservationShocksWeights] = spherical_radial_sigma_points(number_of_observed_variables);
|
|
% ObservationShocksWeights = ones(size(ObservationShocksMu,1),1)*ObservationShocksWeights;
|
|
% elseif ParticleOptions.proposal_approximation.unscented
|
|
% [ObservationShocksMu,ObservationShocksWeights,raf] = unscented_sigma_points(number_of_observed_variables,ParticleOptions);
|
|
% else
|
|
% if ~ParticleOptions.distribution_approximation.montecarlo
|
|
% error('Estimation: This approximation for the proposal is not implemented or unknown!')
|
|
% end
|
|
% end
|
|
% J = size(ObservationShocksWeights,1) ;
|
|
% ObservationShocksMu = H_lower_triangular_cholesky*(ObservationShocksMu') ;
|
|
% ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
|
|
% for j=1:J
|
|
% ObservationShocksSqrtP(:,:,j) = H_lower_triangular_cholesky/sqrt(ObservationShocksWeights(j)) ;
|
|
% end
|
|
% end
|
|
|
|
Gprime = G*I ;
|
|
Gsecond = G*I*J ;
|
|
SampleWeights = ones(Gsecond,1)/Gsecond ;
|
|
|
|
StateWeightsPrior = zeros(1,Gprime) ;
|
|
StateMuPrior = zeros(number_of_state_variables,Gprime) ;
|
|
StateSqrtPPrior = zeros(number_of_state_variables,number_of_state_variables,Gprime) ;
|
|
|
|
StateWeightsPost = zeros(1,Gsecond) ;
|
|
StateMuPost = zeros(number_of_state_variables,Gsecond) ;
|
|
StateSqrtPPost = zeros(number_of_state_variables,number_of_state_variables,Gsecond) ;
|
|
|
|
const_lik = (2*pi)^(.5*number_of_observed_variables) ;
|
|
|
|
lik = NaN(sample_size,1);
|
|
LIK = NaN;
|
|
for t=1:sample_size
|
|
% Build the proposal joint quadratures of Gaussian on states, structural
|
|
% shocks and observation shocks based on each combination of mixtures
|
|
for i=1:I
|
|
for j=1:J
|
|
for g=1:G ;
|
|
gprime = g + (i-1)*G ;
|
|
gsecond = gprime + (j-1)*Gprime ;
|
|
[StateMuPrior(:,gprime),StateSqrtPPrior(:,:,gprime),StateWeightsPrior(1,gprime),...
|
|
StateMuPost(:,gsecond),StateSqrtPPost(:,:,gsecond),StateWeightsPost(1,gsecond)] =...
|
|
gaussian_mixture_filter_bank(ReducedForm,Y(:,t),StateMu(:,g),StateSqrtP(:,:,g),StateWeights(g),...
|
|
StructuralShocksMu(:,i),StructuralShocksSqrtP(:,:,i),StructuralShocksWeights(i),...
|
|
ObservationShocksMu(:,j),ObservationShocksSqrtP(:,:,j),ObservationShocksWeights(j),...
|
|
H,H_lower_triangular_cholesky,const_lik,ParticleOptions,ThreadsOptions) ;
|
|
end
|
|
end
|
|
end
|
|
|
|
% Normalize weights
|
|
StateWeightsPrior = StateWeightsPrior/sum(StateWeightsPrior,2) ;
|
|
StateWeightsPost = StateWeightsPost/sum(StateWeightsPost,2) ;
|
|
|
|
if ParticleOptions.distribution_approximation.cubature || ParticleOptions.distribution_approximation.unscented
|
|
for i=1:Gsecond
|
|
StateParticles = bsxfun(@plus,StateMuPost(:,i),StateSqrtPPost(:,:,i)*nodes') ;
|
|
IncrementalWeights = gaussian_mixture_densities(Y(:,t),StateMuPrior,StateSqrtPPrior,StateWeightsPrior,...
|
|
StateMuPost,StateSqrtPPost,StateWeightsPost,...
|
|
StateParticles,H,const_lik,weights,weights_c,ReducedForm,ThreadsOptions) ;
|
|
SampleWeights(i) = sum(StateWeightsPost(i)*weights.*IncrementalWeights) ;
|
|
end
|
|
SumSampleWeights = sum(SampleWeights) ;
|
|
lik(t) = log(SumSampleWeights) ;
|
|
SampleWeights = SampleWeights./SumSampleWeights ;
|
|
[ras,SortedRandomIndx] = sort(rand(1,Gsecond));
|
|
SortedRandomIndx = SortedRandomIndx(1:G);
|
|
indx = resample(0,SampleWeights,ParticleOptions) ;
|
|
indx = indx(SortedRandomIndx) ;
|
|
StateMu = StateMuPost(:,indx);
|
|
StateSqrtP = StateSqrtPPost(:,:,indx);
|
|
StateWeights = ones(1,G)/G ;
|
|
else
|
|
% Sample particle in the proposal distribution, ie the posterior state GM
|
|
StateParticles = importance_sampling(StateMuPost,StateSqrtPPost,StateWeightsPost',number_of_particles) ;
|
|
IncrementalWeights = gaussian_mixture_densities(Y(:,t),StateMuPrior,StateSqrtPPrior,StateWeightsPrior,...
|
|
StateMuPost,StateSqrtPPost,StateWeightsPost,...
|
|
StateParticles,H,const_lik,1/number_of_particles,...
|
|
1/number_of_particles,ReducedForm,ThreadsOptions) ;
|
|
SampleWeights = IncrementalWeights/number_of_particles ;
|
|
SumSampleWeights = sum(SampleWeights,1) ;
|
|
SampleWeights = SampleWeights./SumSampleWeights ;
|
|
lik(t) = log(SumSampleWeights) ;
|
|
[StateMu,StateSqrtP,StateWeights] = fit_gaussian_mixture(StateParticles,StateMu,StateSqrtP,StateWeights,0.001,10,1) ;
|
|
end
|
|
end
|
|
|
|
LIK = -sum(lik(start:end)) ; |