Commit 7dcdd20f authored by Frédéric Karamé's avatar Frédéric Karamé
Browse files

gaussian mixture nonlinear filters: uses gaussian-mixture approximations for...

gaussian mixture nonlinear filters: uses gaussian-mixture approximations for particles. Fix bugs and normalize the way likelihood is calculated.
parent b0788fae
function [LIK,lik] = gaussian_mixture_filter(ReducedForm,Y,start,DynareOptions)
% 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 Smolyak quadrature à la Kronrod & Paterson (Heiss & Winschel 2010, Winschel & Kratzig 2010).
% - 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) 2009-2012 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 Gprime Gsecond
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 = DynareOptions.particle.mixture_state_variables; % number of GM components in state
I = DynareOptions.particle.mixture_structural_shocks ; % number of GM components in structural noise
J = DynareOptions.particle.mixture_measurement_shocks ; % number of GM components in observation noise
Gprime = G*I ;
Gsecond = G*I*J ;
number_of_particles = DynareOptions.particle.number_of_particles;
init_flag = 1;
end
SampleWeights = ones(Gsecond,1)/Gsecond ;
% compute gaussian quadrature nodes and weights on states and shocks
if isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'quadrature')
[nodes,weights] = nwspgr('GQN',number_of_state_variables,DynareOptions.particle.smolyak_accuracy) ;
weights_c = weights ;
elseif isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'cubature')
[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables) ;
weights_c = weights ;
elseif isempty(nodes) && strcmpi(DynareOptions.particle.approximation_method,'unscented')
[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables,DynareOptions) ;
else
% Set seed for randn().
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 all matrices
StateWeights = ones(1,G)/G ;
StateMu = ReducedForm.StateVectorMean*ones(1,G) ;
StateSqrtP = zeros(number_of_state_variables,number_of_state_variables,G) ;
for g=1:G
StateSqrtP(:,:,g) = reduced_rank_cholesky(ReducedForm.StateVectorVariance)' ;
end
StructuralShocksWeights = ones(1,I)/I ;
StructuralShocksMu = zeros(number_of_structural_innovations,I) ;
StructuralShocksSqrtP = zeros(number_of_structural_innovations,number_of_structural_innovations,I) ;
for i=1:I
StructuralShocksSqrtP(:,:,i) = Q_lower_triangular_cholesky ;
end
ObservationShocksWeights = ones(1,J)/J ;
ObservationShocksMu = zeros(number_of_observed_variables,J) ;
ObservationShocksSqrtP = zeros(number_of_observed_variables,number_of_observed_variables,J) ;
for j=1:J
ObservationShocksSqrtP(:,:,j) = H_lower_triangular_cholesky ;
end
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) ;
%estimate = zeros(sample_size,number_of_state_variables,3) ;
const_lik = (2*pi)^(.5*number_of_observed_variables) ;
ks = 0 ;
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 ;
a = g + (j-1)*G ;
b = a + (i-1)*Gprime ;
[StateMuPrior(:,a),StateSqrtPPrior(:,:,a),StateWeightsPrior(1,a),...
StateMuPost(:,b),StateSqrtPPost(:,:,b),StateWeightsPost(1,b)] =...
gaussian_mixture_filter_bank(ReducedForm,Y(:,t),StateMu(:,g),StateSqrtP(:,:,g),StateWeights(1,g),...
StructuralShocksMu(:,i),StructuralShocksSqrtP(:,:,i),StructuralShocksWeights(1,i),...
ObservationShocksMu(:,j),ObservationShocksSqrtP(:,:,j),ObservationShocksWeights(1,j),...
H,H_lower_triangular_cholesky,const_lik,DynareOptions) ;
end
end
end
% Normalize weights
StateWeightsPrior = StateWeightsPrior/sum(StateWeightsPrior,2) ;
StateWeightsPost = StateWeightsPost/sum(StateWeightsPost,2) ;
if strcmpi(DynareOptions.particle.approximation_method,'quadrature') || ... % sparse grids approximations
strcmpi(DynareOptions.particle.approximation_method,'cubature') || ...
strcmpi(DynareOptions.particle.approximation_method,'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,DynareOptions) ;
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(SampleWeights,DynareOptions.particle.resampling.method1,DynareOptions.particle.resampling.method2) ;
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) ;
% Compute prior, proposal and likelihood 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,DynareOptions) ;
% calculate importance weights of particles
SampleWeights = SampleWeights.*IncrementalWeights ;
SumSampleWeights = sum(SampleWeights,1) ;
SampleWeights = SampleWeights./SumSampleWeights ;
lik(t) = log(SumSampleWeights) ;
% First possible state point estimates
%estimate(t,:,1) = SampleWeights*StateParticles' ;
% Resampling if needed of required
Neff = 1/sum(bsxfun(@power,SampleWeights,2)) ;
if (Neff<.5*sample_size && strcmpi(DynareOptions.particle.resampling.status,'generic')) || strcmpi(DynareOptions.particle.resampling.status,'systematic')
ks = ks + 1 ;
StateParticles = StateParticles(:,resample(SampleWeights',DynareOptions.particle.resampling.method1,DynareOptions.particle.resampling.method2)) ;
StateVectorMean = mean(StateParticles,2) ;
StateVectorVarianceSquareRoot = reduced_rank_cholesky( (StateParticles*StateParticles')/number_of_particles - StateVectorMean*(StateVectorMean') )';
SampleWeights = 1/number_of_particles ;
elseif strcmpi(DynareOptions.particle.resampling.status,'none')
StateVectorMean = StateParticles*sampleWeights ;
temp = sqrt(SampleWeights').*StateParticles ;
StateVectorVarianceSquareRoot = reduced_rank_cholesky( temp*temp' - StateVectorMean*(StateVectorMean') )';
end
% Use the information from particles to update the gaussian mixture on state variables
[StateMu,StateSqrtP,StateWeights] = fit_gaussian_mixture(StateParticles,StateMu,StateSqrtP,StateWeights,0.001,10,1) ;
%estimate(t,:,3) = StateWeights*StateMu' ;
end
end
LIK = -sum(lik(start:end)) ;
\ No newline at end of file
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