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function [LIK,lik] = nonlinear_kalman_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions, options_, M_)
% Evaluates the likelihood of a non-linear model approximating the
% predictive (prior) and filtered (posterior) densities for state variables
% by a Kalman filter.
% Gaussian distribution approximation is done by:
% - a spherical-radial cubature (ref: Arasaratnam & Haykin, 2009).
% - a scaled unscented transform cubature (ref: Julier & Uhlmann 1995)
% - Monte-Carlo draws from a multivariate gaussian distribution.
% First and second moments of prior and posterior state densities are computed
% from the resulting nodes/particles and allows to generate new distributions at the
% following observation.
% Pros: The use of nodes is much faster than Monte-Carlo Gaussian particle and standard particles
% filters since it treats a lesser number of particles.
% Cons: 1. Application a linear projection formulae in a nonlinear context.
% 2. Parameter estimations may be biaised if the model is truly non-gaussian since predictive and
% filtered densities are unimodal.
%
% INPUTS
% Reduced_Form [structure] Matlab's structure describing the reduced form model.
% Y [double] matrix of original observed variables.
% start [double] structural parameters.
% ParticleOptions [structure] Matlab's structure describing options concerning particle filtering.
% ThreadsOptions [structure] Matlab's structure.
%
% OUTPUTS
% LIK [double] scalar, likelihood
% lik [double] vector, density of observations in each period.
%
% REFERENCES
%
% NOTES
% The vector "lik" is used to evaluate the jacobian of the likelihood.
% Copyright © 2009-2022 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 <https://www.gnu.org/licenses/>.
% Set default
if isempty(start)
start = 1;
end
order = options_.order;
if ReducedForm.use_k_order_solver
dr = ReducedForm.dr;
udr = ReducedForm.udr;
else
% 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)
% Set local state space model (third order approximation).
ghxxx = ReducedForm.ghxxx;
ghuuu = ReducedForm.ghuuu;
ghxxu = ReducedForm.ghxxu;
ghxuu = ReducedForm.ghxuu;
ghxss = ReducedForm.ghxss;
ghuss = ReducedForm.ghuss;
end
end
constant = ReducedForm.constant;
steadystate = ReducedForm.steadystate;
state_variables_steady_state = ReducedForm.state_variables_steady_state;
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);
% compute gaussian quadrature nodes and weights on states and shocks
if ParticleOptions.proposal_approximation.montecarlo
nodes = randn(ParticleOptions.number_of_particles,number_of_state_variables+number_of_structural_innovations);
weights = 1/ParticleOptions.number_of_particles;
weights_c = weights;
elseif ParticleOptions.proposal_approximation.cubature
[nodes,weights] = spherical_radial_sigma_points(number_of_state_variables+number_of_structural_innovations);
weights_c = weights;
elseif ParticleOptions.proposal_approximation.unscented
[nodes,weights,weights_c] = unscented_sigma_points(number_of_state_variables+number_of_structural_innovations,ParticleOptions);
else
error('Estimation: This approximation for the proposal is not implemented or unknown!')
end
if ParticleOptions.distribution_approximation.montecarlo
options_=set_dynare_seed_local_options(options_,'default');
end
% Get covariance matrices
H = ReducedForm.H;
H_lower_triangular_cholesky = chol(H)' ;
Q_lower_triangular_cholesky = chol(ReducedForm.Q)';
% Get initial condition for the state vector.
StateVectorMean = ReducedForm.StateVectorMean;
StateVectorVarianceSquareRoot = chol(ReducedForm.StateVectorVariance)';
% Initialization of the likelihood.
lik = NaN(sample_size,1);
LIK = NaN;
for t=1:sample_size
xbar = [StateVectorMean ; zeros(number_of_structural_innovations,1) ] ;
sqr_Px = [StateVectorVarianceSquareRoot zeros(number_of_state_variables,number_of_structural_innovations);
zeros(number_of_structural_innovations,number_of_state_variables) Q_lower_triangular_cholesky];
sigma_points = bsxfun(@plus,xbar,sqr_Px*(nodes'));
StateVectors = sigma_points(1:number_of_state_variables,:);
epsilon = sigma_points(number_of_state_variables+1:number_of_state_variables+number_of_structural_innovations,:);
yhat = bsxfun(@minus,StateVectors,state_variables_steady_state);
if ReducedForm.use_k_order_solver
tmp = local_state_space_iteration_k(yhat, epsilon, dr, M_, options_, udr);
else
if order == 2
tmp = local_state_space_iteration_2(yhat, epsilon, ghx, ghu, constant, ghxx, ghuu, ghxu, ThreadsOptions.local_state_space_iteration_2);
elseif order == 3
tmp = local_state_space_iteration_3(yhat, epsilon, ghx, ghu, ghxx, ghuu, ghxu, ghs2, ghxxx, ghuuu, ghxxu, ghxuu, ghxss, ghuss, steadystate, ThreadsOptions.local_state_space_iteration_3, false);
end
end
PredictedStateMean = tmp(mf0,:)*weights ;
PredictedObservedMean = tmp(mf1,:)*weights;
if ParticleOptions.proposal_approximation.cubature || ParticleOptions.proposal_approximation.montecarlo
PredictedStateMean = sum(PredictedStateMean,2);
PredictedObservedMean = sum(PredictedObservedMean,2);
dState = bsxfun(@minus,tmp(mf0,:),PredictedStateMean)'.*sqrt(weights);
dObserved = bsxfun(@minus,tmp(mf1,:),PredictedObservedMean)'.*sqrt(weights);
big_mat = [dObserved dState ; [H_lower_triangular_cholesky zeros(number_of_observed_variables,number_of_state_variables)] ];
[~, mat] = qr2(big_mat,0);
mat = mat';
PredictedObservedVarianceSquareRoot = mat(1:number_of_observed_variables,1:number_of_observed_variables);
CovarianceObservedStateSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),1:number_of_observed_variables);
StateVectorVarianceSquareRoot = mat(number_of_observed_variables+(1:number_of_state_variables),number_of_observed_variables+(1:number_of_state_variables));
PredictionError = Y(:,t) - PredictedObservedMean;
StateVectorMean = PredictedStateMean + (CovarianceObservedStateSquareRoot/PredictedObservedVarianceSquareRoot)*PredictionError;
else
dState = bsxfun(@minus,tmp(mf0,:),PredictedStateMean);
dObserved = bsxfun(@minus,tmp(mf1,:),PredictedObservedMean);
PredictedStateVariance = dState*diag(weights_c)*dState';
PredictedObservedVariance = dObserved*diag(weights_c)*dObserved' + H;
PredictedStateAndObservedCovariance = dState*diag(weights_c)*dObserved';
PredictionError = Y(:,t) - PredictedObservedMean;
KalmanFilterGain = PredictedStateAndObservedCovariance/PredictedObservedVariance;
StateVectorMean = PredictedStateMean + KalmanFilterGain*PredictionError;
StateVectorVariance = PredictedStateVariance - KalmanFilterGain*PredictedObservedVariance*KalmanFilterGain';
[StateVectorVarianceSquareRoot, p]= chol(StateVectorVariance,'lower');
if p
LIK=-Inf;
lik(t)=-Inf;
return
end
[~, p]= chol(PredictedObservedVariance,'lower');
if p
LIK=-Inf;
lik(t)=-Inf;
return
end
end
lik(t) = log( sum(probability2(Y(:,t),H_lower_triangular_cholesky,tmp(mf1,:)).*weights,1) ) ;
end
LIK = -sum(lik(start:end));
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