From 5922f88182f3cf94fbf15878d569976f24ab37d7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fr=C3=A9d=C3=A9ric=20Karam=C3=A9?= Date: Fri, 9 Oct 2015 14:42:53 +0200 Subject: [PATCH] Add the possibility to use an sparse-grid Kalman filter with approximation at order 2. --- src/Kalman_filter.m | 161 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 161 insertions(+) create mode 100644 src/Kalman_filter.m diff --git a/src/Kalman_filter.m b/src/Kalman_filter.m new file mode 100644 index 000000000..e5d36ee5e --- /dev/null +++ b/src/Kalman_filter.m @@ -0,0 +1,161 @@ +function [LIK,lik] = Kalman_filter(ReducedForm, Y, start, ParticleOptions, ThreadsOptions) +% 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 (C) 2009-2015 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 . + +persistent init_flag mf0 mf1 nodes weights weights_c +persistent sample_size number_of_state_variables number_of_observed_variables number_of_structural_innovations + +% Set default +if isempty(start) + start = 1; +end + +% 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; + +if any(any(isnan(ghx))) || any(any(isnan(ghu))) || any(any(isnan(ghxx))) || any(any(isnan(ghuu))) || any(any(isnan(ghxu))) || ... + any(any(isinf(ghx))) || any(any(isinf(ghu))) || any(any(isinf(ghxx))) || any(any(isinf(ghuu))) || any(any(isinf(ghxu))) ... + any(any(abs(ghx)>1e4)) || any(any(abs(ghu)>1e4)) || any(any(abs(ghxx)>1e4)) || any(any(abs(ghuu)>1e4)) || any(any(abs(ghxu)>1e4)) + ghx + ghu + ghxx + ghuu + ghxu +end + +constant = ReducedForm.constant; +state_variables_steady_state = ReducedForm.state_variables_steady_state; + +% 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); + init_flag = 1; +end + +% compute gaussian quadrature nodes and weights on states and shocks + +if ParticleOptions.proposal_approximation.cubature || ParticleOptions.proposal_approximation.montecarlo + [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 + set_dynare_seed('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); + tmp = local_state_space_iteration_2(yhat,epsilon,ghx,ghu,constant,ghxx,ghuu,ghxu,ThreadsOptions.local_state_space_iteration_2); + 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)] ]; + [mat1,mat] = qr2(big_mat,0); + mat = mat'; + clear('mat1'); + 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 = chol(StateVectorVariance)'; + PredictedObservedVarianceSquareRoot = chol(PredictedObservedVariance)' ; + end + lik(t) = log( probability2(0,PredictedObservedVarianceSquareRoot,PredictionError) ) ; +end + +LIK = -sum(lik(start:end));