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function initial_distribution = auxiliary_initialization(ReducedForm,Y,start,ParticleOptions,ThreadsOptions)
% Evaluates the likelihood of a nonlinear model with a particle filter allowing eventually resampling.
% Copyright © 2011-2022 Dynare Team
%
% This file is part of Dynare (particles module).
%
% 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 particles module 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/>.
persistent init_flag mf1 number_of_particles
persistent number_of_observed_variables number_of_structural_innovations
% Set default
if isempty(start)
start = 1;
end
% Set flag for prunning
%pruning = ParticleOptions.pruning;
% Get steady state and mean.
%steadystate = ReducedForm.steadystate;
constant = ReducedForm.constant;
ss = ReducedForm.ys;
state_variables_steady_state = ReducedForm.state_variables_steady_state;
% Set persistent variables.
if isempty(init_flag)
mf1 = ReducedForm.mf1;
number_of_observed_variables = length(mf1);
number_of_structural_innovations = length(ReducedForm.Q);
number_of_particles = ParticleOptions.number_of_particles;
init_flag = 1;
end
order = options_.order;
% Set local state space model (first order approximation).
ghx = ReducedForm.ghx;
ghu = ReducedForm.ghu;
% Set local state space model (second order approximation).
ghxx = ReducedForm.ghxx;
ghuu = ReducedForm.ghuu;
ghxu = ReducedForm.ghxu;
ghs2 = ReducedForm.ghs2;
if (order == 3)
ghxxx = ReducedForm.ghxxx;
ghuuu = ReducedForm.ghuuu;
ghxxu = ReducedForm.ghxxu;
ghxuu = ReducedForm.ghxuu;
ghxss = ReducedForm.ghxss;
ghuss = ReducedForm.ghuss;
end
% Get covariance matrices
Q = ReducedForm.Q;
H = ReducedForm.H;
if isempty(H)
H = 0;
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 = chol(Q)';
%if pruning
% StateVectorMean_ = StateVectorMean;
% StateVectorVarianceSquareRoot_ = StateVectorVarianceSquareRoot;
%end
% Set seed for randn().
options_=set_dynare_seed_local_options(options_,'default');
% Initialization of the likelihood.
const_lik = log(2*pi)*number_of_observed_variables;
% Initialization of the weights across particles.
weights = ones(1,number_of_particles)/number_of_particles ;
StateVectors = bsxfun(@plus,StateVectorVarianceSquareRoot*randn(state_variance_rank,number_of_particles),StateVectorMean);
%if pruning
% StateVectors_ = StateVectors;
%end
yhat = bsxfun(@minus,StateVectors,state_variables_steady_state);
%if pruning
% yhat_ = bsxfun(@minus,StateVectors_,state_variables_steady_state);
% [tmp, tmp_] = local_state_space_iteration_2(yhat,zeros(number_of_structural_innovations,number_of_particles),ghx,ghu,constant,ghxx,ghuu,ghxu,yhat_,steadystate,ThreadsOptions.local_state_space_iteration_2);
%else
if (order == 2)
tmp = local_state_space_iteration_2(yhat,zeros(number_of_structural_innovations,number_of_particles),ghx,ghu,constant,ghxx,ghuu,ghxu,ThreadsOptions.local_state_space_iteration_2);
elseif (order == 3)
tmp = local_state_space_iteration_3(yhat, zeros(number_of_structural_innovations,number_of_particles), ghx, ghu, ghxx, ghuu, ghxu, ghs2, ghxxx, ghuuu, ghxxu, ghxuu, ghxss, ghuss, ss, options_.threads.local_state_space_iteration_3, false);
else
error('Orders > 3 not allowed');
end
%end
PredictedObservedMean = weights*(tmp(mf1,:)');
PredictionError = bsxfun(@minus,Y(:,t),tmp(mf1,:));
dPredictedObservedMean = bsxfun(@minus,tmp(mf1,:),PredictedObservedMean');
PredictedObservedVariance = bsxfun(@times,weights,dPredictedObservedMean)*dPredictedObservedMean' + H;
wtilde = exp(-.5*(const_lik+log(det(PredictedObservedVariance))+sum(PredictionError.*(PredictedObservedVariance\PredictionError),1))) ;
tau_tilde = weights.*wtilde ;
tau_tilde = tau_tilde/sum(tau_tilde);
initial_distribution = resample(StateVectors',tau_tilde',ParticleOptions)' ;
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