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|
function pruned_state_space = pruned_state_space_system(M, options, dr, indy, nlags, useautocorr, compute_derivs)
% Set up the pruned state space ABCD representation:
% z = c + A*z(-1) + B*inov
% y = ys + d + C*z(-1) + D*inov
% References:
% - Andreasen, Martin M., Jesús Fernández-Villaverde and Juan F. Rubio-Ramírez (2018):
% "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications",
% Review of Economic Studies, Volume 85, Issue 1, Pages 1–49.
% - Mutschler, Willi (2018): "Higher-order statistics for DSGE models",
% Econometrics and Statistics, Volume 6, Pages 44-56.
% =========================================================================
% INPUTS
% M: [structure] storing the model information
% options: [structure] storing the options
% dr: [structure] storing the results from perturbation approximation
% indy: [vector] index of control variables in DR order
% nlags: [integer] number of lags in autocovariances and autocorrelations
% useautocorr: [boolean] true: compute autocorrelations
% -------------------------------------------------------------------------
% OUTPUTS
% pruned_state_space: [structure] with the following fields:
% indx: [x_nbr by 1]
% index of state variables
% indy: [y_nbr by 1]
% index of control variables
% A: [z_nbr by z_nbr]
% state space transition matrix A mapping previous states to current states
% B: [z_nbr by inov_nbr]
% state space transition matrix B mapping current inovations to current states
% c: [z_nbr by 1]
% state space transition matrix c mapping constants to current states
% C: [y_nbr by z_nbr]
% state space measurement matrix C mapping previous states to current controls
% D: [y_nbr by inov_nbr]
% state space measurement matrix D mapping current inovations to current controls
% d: [y_nbr by 1]
% state space measurement matrix d mapping constants to current controls
% Var_inov [inov_nbr by inov_nbr]
% contemporenous covariance matrix of innovations, i.e. E[inov*inov']
% Var_z [z_nbr by z_nbr]
% contemporenous covariance matrix of states z
% Var_y [y_nbr by y_nbr]
% contemporenous covariance matrix of controls y
% Var_yi [y_nbr by y_nbr by nlags]
% autocovariance matrix of controls y
% Corr_y [y_nbr by y_nbr]
% contemporenous correlation matrix of controls y
% Corr_yi [y_nbr by y_nbr by nlags]
% autocorrelation matrix of controls y
% E_y [y_nbr by 1]
% unconditional theoretical mean of control variables y
%
% if compute_derivs == 1, then the following additional fields are outputed:
% dA: [z_nbr by z_nbr by totparam_nbr]
% parameter Jacobian of A
% dB: [z_nbr by inov_nbr by totparam_nbr]
% parameter Jacobian of B
% dc: [z_nbr by totparam_nbr]
% parameter Jacobian of c
% dC: [y_nbr by z_nbr by totparam_nbr]
% parameter Jacobian of C
% dD: [y_nbr by inov_nbr by totparam_nbr]
% parameter Jacobian of D
% dd: [y_nbr by totparam_nbr]
% parameter Jacobian of d
% dVar_inov [inov_nbr by inov_nbr by totparam_nbr]
% parameter Jacobian of Var_inov
% dVar_z [z_nbr by z_nbr by totparam_nbr]
% parameter Jacobian of Var_z
% dVar_y [y_nbr by y_nbr by totparam_nbr]
% parameter Jacobian of Var_y
% dVar_yi [y_nbr by y_nbr by nlags by totparam_nbr]
% parameter Jacobian of Var_yi
% dCorr_y [y_nbr by y_nbr by totparam_nbr]
% parameter Jacobian of Corr_y
% dCorr_yi [y_nbr by y_nbr by nlags by totparam_nbr]
% parameter Jacobian of Corr_yi
% dE_y [y_nbr by totparam_nbr]
% parameter Jacobian of E_y
% -------------------------------------------------------------------------
% This function is called by
% * get_identification_jacobians.m
% * identification_numerical_objective.m
% -------------------------------------------------------------------------
% This function calls
% * allVL1.m
% * commutation.m
% * disclyap_fast.m
% * duplication.m
% * lyapunov_symm.m
% * prodmom
% * prodmom_deriv
% * Q6_plication
% * quadruplication.m
% * vec.m
% =========================================================================
% Copyright (C) 2019-2020 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/>.
% =========================================================================
%% MAIN IDEA:
% Decompose the state vector x into first-order effects xf, second-order
% effects xs, and third-order effects xrd, i.e. x=xf+xs+xrd. Then, Dynare's
% perturbation approximation for the state vector up to third order
% (with Gaussian innovations u, i.e. no odd moments, hxxs=huus=hxus=hsss=0) is:
% x = hx*( xf(-1)+xs(-1)+xrd(-1) )
% + hu*u
% + 1/2*hxx*kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) )
% + hxu*kron( xf(-1)+xs(-1)+xrd(-1) , u )
% + 1/2*huu*kron( u , u )
% + 1/2*hss*sig^2
% + 1/6*hxxx*kron( kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) ) , xf(-1)+xs(-1)+xrd(-1) )
% + 1/6*huuu*kron( kron( u , u ) , u )
% + 3/6*hxxu*kron( kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) ) , u )
% + 3/6*hxuu*kron( kron( xf(-1)+xs(-1)+xrd(-1) , u ) , u)
% + 3/6*hxss*( xf(-1)+xs(-1)+xrd(-1) )*sig^2
% + 3/6*huss*u*sig^2
% where:
% hx = dr.ghx(indx,:); hu = dr.ghu(indx,:);
% hxx = dr.ghxx(indx,:); hxu = dr.ghxu(indx,:); huu = dr.ghuu(indx,:); hss = dr.ghs2(indx,:);
% hxxx = dr.ghxxx(indx,:); huuu = dr.ghuuu(indx,:); hxxu = dr.ghxxu(indx,:); hxuu = dr.ghxxu(indx,:); hxss = dr.ghxss(indx,:); huss = dr.ghuss(indx,:);
% and similarly for control variables:
% y = gx*( xf(-1)+xs(-1)+xrd(-1) )
% + gu*u
% + 1/2*gxx*kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) )
% + gxu*kron( xf(-1)+xs(-1)+xrd(-1) , u )
% + 1/2*guu*kron( u , u )
% + 1/2*gss*sig^2
% + 1/6*gxxx*kron( kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) ) , xf(-1)+xs(-1)+xrd(-1) )
% + 1/6*guuu*kron( kron( u , u ) , u )
% + 3/6*gxxu*kron( kron( xf(-1)+xs(-1)+xrd(-1) , xf(-1)+xs(-1)+xrd(-1) ) , u )
% + 3/6*gxuu*kron( kron( xf(-1)+xs(-1)+xrd(-1) , u ) , u)
% + 3/6*gxss*( xf(-1)+xs(-1)+xrd(-1) )*sig^2
% + 3/6*guss*u*sig^2
% where:
% gx = dr.ghx(indy,:); gu = dr.ghu(indy,:);
% gxx = dr.ghxx(indy,:); gxu = dr.ghxu(indy,:); guu = dr.ghuu(indy,:); gss = dr.ghs2(indy,:);
% gxxx = dr.ghxxx(indy,:); guuu = dr.ghuuu(indy,:); gxxu = dr.ghxxu(indy,:); gxuu = dr.ghxxu(indy,:); gxss = dr.ghxss(indy,:); guss = dr.ghuss(indy,:);
%
% PRUNING means getting rid of terms higher than the approximation order, i.e.
% - involving fourth-order effects: kron(xf,xrd), kron(xs,xs), kron(xrd,xf), kron(xrd,u),
% kron(kron(xf,xf),xs), kron(kron(xf,xs),xf), kron(kron(xs,xf),xf)
% kron(kron(xf,xs),u), kron(kron(xs,xf),u)
% kron(kron(xs,u),u)
% xs*sig^2
% - involving fifth-order effects: kron(xs,xrd), kron(xrd,xs),
% kron(kron(xf,xf),xrd), kron(kron(xf,xs),xs), kron(kron(xf,xrd),xf), kron(kron(xs,xf),xs), kron(kron(xs,xs),xf), kron(kron(xrd,xf),xf)
% kron(kron(xf,xrd),u), kron(kron(xs,xs),u), kron(kron(xrd,xf),u)
% kron(kron(xrd,u),u)
% xrd*sig^2
% - involving sixth-order effects: kron(xrd,xrd),
% kron(kron(xf,xs),xrd), kron(kron(xf,xrd),xs), kron(kron(xs,xrd),xrd), kron(kron(xs,xs),xs), kron(kron(xs,xrd),xf), kron(kron(xf,xf),xs), kron(kron(xrd,xs),xf)
% kron(kron(xs,xrd),u), kron(kron(xrd,xs),u)
% - involving seventh-order effects: kron(kron(xf,xrd),xrd), kron(kron(xs,xs),xrd), kron(kron(xs,xrd),xs), kron(kron(xrd,xf),xrd), kron(kron(xrd,xs),xs), kron(kron(xrd,xrd),xf)
% kron(kron(xrd,xrd),u)
% - involving eighth-order effects: kron(kron(xs,xrd),xrd), kron(kron(xrd,xs),xrd), kron(kron(xrd,xrd),xs)
% - involving ninth-order effects: kron(kron(xrd,xrd),xrd)
% Note that u is treated as a first-order effect and the perturbation parameter sig as a variable.
%
% SUMMARY OF LAW OF MOTIONS: Set up the law of motions for the individual effects, but keep only effects of same order
% Notation: I_n=eye(n) and K_m_n=commutation(m,n)
%
% First-order effects: keep xf and u
% xf = hx*xf(-1) + hu*u
% Note that we
%
% Second-order effects: keep xs, kron(xf,xf), kron(u,u), kron(xf,u), and sig^2
% xs = hx*xs(-1) + 1/2*hxx*kron(xf(-1),xf(-1)) + 1/2*huu*(kron(u,u)-Sigma_e(:)+Sigma_e(:)) + hxu*kron(xf(-1),u) + 1/2*hss*sig^2%
%
% Third-order effects: keep xrd, kron(xf,xs), kron(xs,xf), kron(xs,u), kron(kron(xf,xf),xf), kron(kron(u,u),u), kron(kron(xf,xf),u), kron(kron(xf,u),u), xf*sig^2, u*sig^2
% xrd = hx*xrd(-1) + 1/2*hxx*(kron(xf(-1),xs(-1))+kron(xs(-1),xf(-1))) + hxu*kron(xs(-1),u) + 1/6*hxxx*kron(xf(-1),kron(xf(-1),xf(-1))) + 1/6*huuu*kron(u,kron(u,u)) + 3/6*hxxu*kron(xf(-1),kron(xf(-1),u)) + 3/6*hxuu*kron(xf(-1),kron(u,u)) + 3/6*hxss*xf(-1)*sig^2 + 3/6*huss*u*sig^2
% Simplified (due to symmetry in hxx):
% xrd = hx*xrd(-1) + hxx*(kron(xf(-1),xs(-1)) + hxu*kron(xs(-1),u) + 1/6*hxxx*kron(xf(-1),kron(xf(-1),xf(-1))) + 1/6*huuu*kron(u,kron(u,u)) + 3/6*hxxu*kron(xf(-1),kron(xf(-1),u)) + 3/6*hxuu*kron(xf(-1),kron(u,u)) + 3/6*hxss*xf(-1)*sig^2 + 3/6*huss*u*sig^2%
%
% Auxiliary equation kron(xf,xf) to set up the VAR(1) pruned state space system
% kron(xf,xf) = kron(hx,hx)*kron(xf(-1),xf(-1)) + kron(hu,hu)*(kron(u,u)-Sigma_e(:)+Sigma_e(:)) + kron(hx,u)*kron(xf(-1),u) + kron(u,hx)*kron(u,xf(-1))
% Simplified using commutation matrix:
% kron(xf,xf) = kron(hx,hx)*kron(xf(-1),xf(-1)) + (I_xx+K_x_x)*kron(hx,hu)*kron(xf(-1),u) + kron(hu,hu)*kron(u,u)
%
% Auxiliary equation kron(xf,xs) to set up the VAR(1) pruned state space system
% kron(xf,xs) = kron(hx,hx)*kron(xf(-1),xs(-1)) + kron(hu,hx)*kron(u,xs(-1))
% + kron(hx,1/2*hxx)*kron(kron(xf(-1),xf(-1)),xf(-1)) + kron(hu,1/2*hxx)*kron(kron(u,xf(-1)),xf(-1))
% + kron(hx,1/2*huu)*kron(kron(xf(-1),u),u) + kron(hu,1/2*huu)*kron(kron(u,u),u)
% + kron(hx,hxu)*kron(kron(xf(-1),xf(-1)),u) + kron(hu,hxu)*kron(kron(u,xf(-1)),u)
% + kron(hx,1/2*hss)*xf(-1)*sig^2 + kron(hu,1/2*hss)*u*sig^2
% Simplified using commutation matrix:
% kron(xf,xs) = kron(hx,hx)*kron(xf(-1),xs(-1))
% + K_x_x*kron(hx,hu)*kron(xs(-1),u)
% + kron(hx,1/2*hxx)*kron(kron(xf(-1),xf(-1)),xf(-1))
% + ( kron(hx,hxu) + K_x_x*kron(1/2*hxx,hu) )*kron(kron(xf(-1),xf(-1)),u)
% + ( kron(hx,1/2*huu) + K_x_x*kron(hxu,hu) )*kron(kron(xf(-1),u),u)
% + kron(hu,1/2*huu)*kron(kron(u,u),u)
% + kron(hx,1/2*hss)*xf(-1)*sig^2
% + kron(hu,1/2*hss)*u*sig^2
%
% Auxiliary equation kron(kron(xf,xf),xf) to set up the VAR(1) pruned state space system
% kron(kron(xf,xf),xf) = kron(kron(hx,hx),hx)*kron(kron(xf(-1),xf(-1)),xf(-1))
% + kron(kron(hx,hu),hx)*kron(kron(xf(-1),u),xf(-1))
% + kron(kron(hx,hx),hu)*kron(kron(xf(-1),xf(-1)),u)
% + kron(kron(hx,hu),hu)*kron(kron(xf(-1),u),u)
% + kron(kron(hu,hx),hx)*kron(kron(u,xf(-1)),xf(-1))
% + kron(kron(hu,hu),hx)*kron(kron(u,u),xf(-1))
% + kron(kron(hu,hx),hu)*kron(kron(u,xf(-1)),u)
% + kron(kron(hu,hu),hu)*kron(kron(u,u),u)
% Simplified using commutation matrix:
% kron(kron(xf,xf),xf) = kron(kron(hx,hx),hx)*kron(kron(xf(-1),xf(-1)),xf(-1))
% + ( kron(kron(hx,hx),hu) + K_xx_x*kron(hx,(I_xx+K_x_x)*kron(hx,hu)) )*kron(kron(xf(-1),xf(-1)),u)
% + ( kron((I_xx+K_x_x)*kron(hx,hu),hu) + K_xx_x*kron(kron(hx,hu),hu) )*kron(kron(xf(-1),u),u)
% + kron(kron(hu,hu),hu)*kron(kron(u,u),u)
%
% Law of motion for control variables y (either VAROBS variables or if no VAROBS statement is given then for all endogenous variables)
% y = steady_state(y)
% + gx*( xf(-1)+xs(-1)+xrd(-1) )
% + gu*u
% + 1/2*gxx*kron(xf(-1),xf(-1)) + gxx*kron(xf(-1),xs(-1))
% + gxu*kron(xf(-1),u) + gxu*kron(xs(-1),u)
% + 1/2*guu*(kron(u,u)-Sigma_e+Sigma_e)
% + 1/2*gss*sig^2
% + 1/6*gxxx*kron(kron(xf(-1),xf(-1)),xf(-1))
% + 1/6*guuu*kron(kron(u,u),u)
% + 3/6*gxxu*kron(kron(xf(-1),xf(-1),u)
% + 3/6*gxuu*kron(kron(xf(-1),u),u)
% + 3/6*gxss*xf(-1)*sig^2
% + 3/6*guss*u*sig^2
%
% See code below how z and inov are defined at first, second, and third order,
% and how to set up A, B, C, D and compute unconditional first and second moments of inov, z and y
%% Auxiliary indices and objects
order = options.order;
if isempty(options.qz_criterium)
% set default value for qz_criterium: if there are no unit roots one can use 1.0
% If they are possible, you may have have multiple unit roots and the accuracy
% decreases when computing the eigenvalues in lyapunov_symm. Hence, we normally use 1+1e-6
% Note that unit roots are only possible at first-order, at higher order we set it to 1
options.qz_criterium = 1+1e-6;
end
indx = [M.nstatic+(1:M.nspred) M.endo_nbr+(1:size(dr.ghx,2)-M.nspred)]';
if isempty(indy)
indy = (1:M.endo_nbr)'; %by default select all variables in DR order
end
u_nbr = M.exo_nbr;
x_nbr = length(indx);
y_nbr = length(indy);
Yss = dr.ys(dr.order_var);
hx = dr.ghx(indx,:);
gx = dr.ghx(indy,:);
hu = dr.ghu(indx,:);
gu = dr.ghu(indy,:);
E_uu = M.Sigma_e; %this is E[u*u']
if compute_derivs
stderrparam_nbr = length(dr.derivs.indpstderr);
corrparam_nbr = size(dr.derivs.indpcorr,1);
modparam_nbr = length(dr.derivs.indpmodel);
totparam_nbr = stderrparam_nbr+corrparam_nbr+modparam_nbr;
dYss = dr.derivs.dYss;
dhx = dr.derivs.dghx(indx,:,:);
dgx = dr.derivs.dghx(indy,:,:);
dhu = dr.derivs.dghu(indx,:,:);
dgu = dr.derivs.dghu(indy,:,:);
dE_uu = dr.derivs.dSigma_e;
end
% first-order approximation indices for extended state vector z and extended innovations vector inov
id_z1_xf = (1:x_nbr);
id_inov1_u = (1:u_nbr);
if order > 1
% second-order approximation indices for extended state vector z and extended innovations vector inov
id_z2_xs = id_z1_xf(end) + (1:x_nbr);
id_z3_xf_xf = id_z2_xs(end) + (1:x_nbr*x_nbr);
id_inov2_u_u = id_inov1_u(end) + (1:u_nbr*u_nbr);
id_inov3_xf_u = id_inov2_u_u(end) + (1:x_nbr*u_nbr);
hxx = dr.ghxx(indx,:);
gxx = dr.ghxx(indy,:);
hxu = dr.ghxu(indx,:);
gxu = dr.ghxu(indy,:);
huu = dr.ghuu(indx,:);
guu = dr.ghuu(indy,:);
hss = dr.ghs2(indx,:);
gss = dr.ghs2(indy,:);
if compute_derivs
dhxx = dr.derivs.dghxx(indx,:,:);
dgxx = dr.derivs.dghxx(indy,:,:);
dhxu = dr.derivs.dghxu(indx,:,:);
dgxu = dr.derivs.dghxu(indy,:,:);
dhuu = dr.derivs.dghuu(indx,:,:);
dguu = dr.derivs.dghuu(indy,:,:);
dhss = dr.derivs.dghs2(indx,:);
dgss = dr.derivs.dghs2(indy,:);
end
end
if order > 2
% third-order approximation indices for extended state vector z and extended innovations vector inov
id_z4_xrd = id_z3_xf_xf(end) + (1:x_nbr);
id_z5_xf_xs = id_z4_xrd(end) + (1:x_nbr*x_nbr);
id_z6_xf_xf_xf = id_z5_xf_xs(end) + (1:x_nbr*x_nbr*x_nbr);
id_inov4_xs_u = id_inov3_xf_u(end) + (1:x_nbr*u_nbr);
id_inov5_xf_xf_u = id_inov4_xs_u(end) + (1:x_nbr*x_nbr*u_nbr);
id_inov6_xf_u_u = id_inov5_xf_xf_u(end) + (1:x_nbr*u_nbr*u_nbr);
id_inov7_u_u_u = id_inov6_xf_u_u(end) + (1:u_nbr*u_nbr*u_nbr);
hxxx = dr.ghxxx(indx,:);
gxxx = dr.ghxxx(indy,:);
hxxu = dr.ghxxu(indx,:);
gxxu = dr.ghxxu(indy,:);
hxuu = dr.ghxuu(indx,:);
gxuu = dr.ghxuu(indy,:);
huuu = dr.ghuuu(indx,:);
guuu = dr.ghuuu(indy,:);
hxss = dr.ghxss(indx,:);
gxss = dr.ghxss(indy,:);
huss = dr.ghuss(indx,:);
guss = dr.ghuss(indy,:);
if compute_derivs
dhxxx = dr.derivs.dghxxx(indx,:,:);
dgxxx = dr.derivs.dghxxx(indy,:,:);
dhxxu = dr.derivs.dghxxu(indx,:,:);
dgxxu = dr.derivs.dghxxu(indy,:,:);
dhxuu = dr.derivs.dghxuu(indx,:,:);
dgxuu = dr.derivs.dghxuu(indy,:,:);
dhuuu = dr.derivs.dghuuu(indx,:,:);
dguuu = dr.derivs.dghuuu(indy,:,:);
dhxss = dr.derivs.dghxss(indx,:,:);
dgxss = dr.derivs.dghxss(indy,:,:);
dhuss = dr.derivs.dghuss(indx,:,:);
dguss = dr.derivs.dghuss(indy,:,:);
end
end
%% First-order state space system
% Auxiliary state vector z is defined by: z = [xf]
% Auxiliary innovations vector inov is defined by: inov = [u]
z_nbr = x_nbr;
inov_nbr = M.exo_nbr;
A = hx;
B = hu;
c = zeros(x_nbr,1);
C = gx;
D = gu;
d = zeros(y_nbr,1);
Varinov = E_uu;
E_inovzlag1 = zeros(inov_nbr,z_nbr); %at first-order E[inov*z(-1)'] = 0
Om_z = B*Varinov*B';
E_xf = zeros(x_nbr,1);
lyapunov_symm_method = 1; %method=1 to initialize persistent variables
[Var_z,Schur_u] = lyapunov_symm(A, Om_z,... %at first-order this algorithm is well established and also used in th_autocovariances.m
options.lyapunov_fixed_point_tol, options.qz_criterium, options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug); %we use Schur_u to take care of (possible) nonstationary VAROBS variables in moment computations
%find stationary vars
stationary_vars = (1:y_nbr)';
if ~isempty(Schur_u)
%base this only on first order, because if first-order is stable so are the higher-order pruned systems
x = abs(gx*Schur_u);
stationary_vars = find(all(x < options.Schur_vec_tol,2));
end
if compute_derivs == 1
dA = dhx;
dB = dhu;
dc = zeros(x_nbr,totparam_nbr);
dC = dgx;
dD = dgu;
dd = zeros(y_nbr,totparam_nbr);
dVarinov = dE_uu;
dE_xf = zeros(x_nbr,totparam_nbr);
dE_inovzlag1 = zeros(z_nbr,inov_nbr,totparam_nbr);
dVar_z = zeros(z_nbr,z_nbr,totparam_nbr);
lyapunov_symm_method = 2;%to spare a lot of computing time while not repeating Schur every time
for jp1 = 1:totparam_nbr
if jp1 <= stderrparam_nbr+corrparam_nbr
dOm_z_jp1 = B*dVarinov(:,:,jp1)*B';
dVar_z(:,:,jp1) = lyapunov_symm(A, dOm_z_jp1,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
else
dOm_z_jp1 = dB(:,:,jp1)*Varinov*B' + B*Varinov*dB(:,:,jp1)';
dVar_z(:,:,jp1) = lyapunov_symm(A, dA(:,:,jp1)*Var_z*A' + A*Var_z*dA(:,:,jp1)' + dOm_z_jp1,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
end
end
end
if order > 1
options.qz_criterium = 1; %pruned state space has no unit roots
% Some common and useful objects for order > 1
E_xfxf = Var_z;
if compute_derivs
dE_xfxf = dVar_z;
end
hx_hx = kron(hx,hx);
hx_hu = kron(hx,hu);
hu_hu = kron(hu,hu);
I_xx = eye(x_nbr^2);
K_x_x = commutation(x_nbr,x_nbr,1);
invIx_hx = (eye(x_nbr)-hx)\eye(x_nbr);
%Compute unique fourth order product moments of u, i.e. unique(E[kron(kron(kron(u,u),u),u)],'stable')
u_nbr4 = u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4;
QPu = quadruplication(u_nbr);
COMBOS4 = flipud(allVL1(u_nbr, 4)); %all possible (unique) combinations of powers that sum up to four
E_u_u_u_u = zeros(u_nbr4,1); %only unique entries
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u = zeros(u_nbr4,stderrparam_nbr+corrparam_nbr);
end
for j4 = 1:size(COMBOS4,1)
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
[E_u_u_u_u(j4), dE_u_u_u_u(j4,:)] = prodmom_deriv(E_uu, 1:u_nbr, COMBOS4(j4,:), dE_uu(:,:,1:(stderrparam_nbr+corrparam_nbr)), dr.derivs.dCorrelation_matrix(:,:,1:(stderrparam_nbr+corrparam_nbr)));
else
E_u_u_u_u(j4) = prodmom(E_uu, 1:u_nbr, COMBOS4(j4,:));
end
end
E_xfxf_uu = kron(E_xfxf,E_uu');
%% Second-order pruned state space system
% Auxiliary state vector z is defined by: z = [xf;xs;kron(xf,xf)]
% Auxiliary innovations vector inov is defined by: inov = [u;kron(u,u)-E_uu(:);kron(xf,u)]
z_nbr = x_nbr + x_nbr + x_nbr^2;
inov_nbr = u_nbr + u_nbr^2 + x_nbr*u_nbr;
A = zeros(z_nbr, z_nbr);
A(id_z1_xf , id_z1_xf ) = hx;
A(id_z2_xs , id_z2_xs ) = hx;
A(id_z2_xs , id_z3_xf_xf) = 1/2*hxx;
A(id_z3_xf_xf , id_z3_xf_xf) = hx_hx;
B = zeros(z_nbr, inov_nbr);
B(id_z1_xf , id_inov1_u ) = hu;
B(id_z2_xs , id_inov2_u_u ) = 1/2*huu;
B(id_z2_xs , id_inov3_xf_u) = hxu;
B(id_z3_xf_xf , id_inov2_u_u ) = hu_hu;
B(id_z3_xf_xf , id_inov3_xf_u) = (I_xx+K_x_x)*hx_hu;
c = zeros(z_nbr, 1);
c(id_z2_xs , 1) = 1/2*hss + 1/2*huu*E_uu(:);
c(id_z3_xf_xf , 1) = hu_hu*E_uu(:);
C = zeros(y_nbr, z_nbr);
C(: , id_z1_xf ) = gx;
C(: , id_z2_xs ) = gx;
C(: , id_z3_xf_xf) = 1/2*gxx;
D = zeros(y_nbr, inov_nbr);
D(: , id_inov1_u ) = gu;
D(: , id_inov2_u_u ) = 1/2*guu;
D(: , id_inov3_xf_u) = gxu;
d = 1/2*gss + 1/2*guu*E_uu(:);
Varinov = zeros(inov_nbr,inov_nbr);
Varinov(id_inov1_u , id_inov1_u) = E_uu;
%Varinov(id_inov1_u , id_inov2_u_u ) = zeros(u_nbr,u_nbr^2);
%Varinov(id_inov1_u , id_inov3_xf_u) = zeros(u_nbr,x_nbr*u_nbr);
%Varinov(id_inov2_u_u , id_inov1_u ) = zeros(u_nbr^2,u_nbr);
Varinov(id_inov2_u_u , id_inov2_u_u ) = reshape(QPu*E_u_u_u_u,u_nbr^2,u_nbr^2)-E_uu(:)*E_uu(:)';
%Varinov(id_inov2_u_u , id_inov3_xf_u) = zeros(u_nbr^2,x_nbr*u_nbr);
%Varinov(id_inov3_xf_u , id_inov1_u ) = zeros(x_nbr*u_nbr,u_nbr);
%Varinov(id_inov3_xf_u , id_inov2_u_u ) = zeros(x_nbr*u_nbr,u_nbr^2);
Varinov(id_inov3_xf_u , id_inov3_xf_u) = E_xfxf_uu;
E_xs = invIx_hx*(1/2*hxx*E_xfxf(:) + c(id_z2_xs,1));
E_inovzlag1 = zeros(inov_nbr,z_nbr); %at second-order E[z(-1)*inov'] = 0
Om_z = B*Varinov*transpose(B);
lyapunov_symm_method = 1; %method=1 to initialize persistent variables (if errorflag)
[Var_z, errorflag] = disclyap_fast(A,Om_z,options.lyapunov_doubling_tol);
if errorflag %use Schur-based method
fprintf('PRUNED_STATE_SPACE_SYSTEM: error flag in disclyap_fast at order=2, use lyapunov_symm\n');
Var_z = lyapunov_symm(A,Om_z,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
lyapunov_symm_method = 2; %in the following we can reuse persistent variables
end
% Make sure some stuff is zero due to Gaussianity
Var_z(id_z1_xf , id_z2_xs ) = zeros(x_nbr,x_nbr);
Var_z(id_z1_xf , id_z3_xf_xf) = zeros(x_nbr,x_nbr^2);
Var_z(id_z2_xs , id_z1_xf ) = zeros(x_nbr,x_nbr);
Var_z(id_z3_xf_xf , id_z1_xf ) = zeros(x_nbr^2,x_nbr);
if compute_derivs
dA = zeros(z_nbr,z_nbr,totparam_nbr);
dB = zeros(z_nbr,inov_nbr,totparam_nbr);
dc = zeros(z_nbr,totparam_nbr);
dC = zeros(y_nbr,z_nbr,totparam_nbr);
dD = zeros(y_nbr,inov_nbr,totparam_nbr);
dd = zeros(y_nbr,totparam_nbr);
dVarinov = zeros(inov_nbr,inov_nbr,totparam_nbr);
dE_xs = zeros(x_nbr,totparam_nbr);
dE_inovzlag1 = zeros(inov_nbr,z_nbr,totparam_nbr);
dVar_z = zeros(z_nbr,z_nbr,totparam_nbr);
for jp2 = 1:totparam_nbr
if jp2 <= (stderrparam_nbr+corrparam_nbr)
dE_uu_jp2 = dE_uu(:,:,jp2);
dE_u_u_u_u_jp2 = QPu*dE_u_u_u_u(:,jp2);
else
dE_uu_jp2 = zeros(u_nbr,u_nbr);
dE_u_u_u_u_jp2 = zeros(u_nbr^4,1);
end
dhx_jp2 = dhx(:,:,jp2);
dhu_jp2 = dhu(:,:,jp2);
dhxx_jp2 = dhxx(:,:,jp2);
dhxu_jp2 = dhxu(:,:,jp2);
dhuu_jp2 = dhuu(:,:,jp2);
dhss_jp2 = dhss(:,jp2);
dgx_jp2 = dgx(:,:,jp2);
dgu_jp2 = dgu(:,:,jp2);
dgxx_jp2 = dgxx(:,:,jp2);
dgxu_jp2 = dgxu(:,:,jp2);
dguu_jp2 = dguu(:,:,jp2);
dgss_jp2 = dgss(:,jp2);
dhx_hx_jp2 = kron(dhx_jp2,hx) + kron(hx,dhx_jp2);
dhu_hu_jp2 = kron(dhu_jp2,hu) + kron(hu,dhu_jp2);
dhx_hu_jp2 = kron(dhx_jp2,hu) + kron(hx,dhu_jp2);
dE_xfxf_jp2 = dE_xfxf(:,:,jp2);
dE_xfxf_uu_jp2 = kron(dE_xfxf_jp2,E_uu) + kron(E_xfxf,dE_uu_jp2);
dA(id_z1_xf , id_z1_xf , jp2) = dhx_jp2;
dA(id_z2_xs , id_z2_xs , jp2) = dhx_jp2;
dA(id_z2_xs , id_z3_xf_xf , jp2) = 1/2*dhxx_jp2;
dA(id_z3_xf_xf , id_z3_xf_xf , jp2) = dhx_hx_jp2;
dB(id_z1_xf , id_inov1_u , jp2) = dhu_jp2;
dB(id_z2_xs , id_inov2_u_u , jp2) = 1/2*dhuu_jp2;
dB(id_z2_xs , id_inov3_xf_u , jp2) = dhxu_jp2;
dB(id_z3_xf_xf , id_inov2_u_u , jp2) = dhu_hu_jp2;
dB(id_z3_xf_xf , id_inov3_xf_u , jp2) = (I_xx+K_x_x)*dhx_hu_jp2;
dc(id_z2_xs , jp2) = 1/2*dhss_jp2 + 1/2*dhuu_jp2*E_uu(:) + 1/2*huu*dE_uu_jp2(:);
dc(id_z3_xf_xf , jp2) = dhu_hu_jp2*E_uu(:) + hu_hu*dE_uu_jp2(:);
dC(: , id_z1_xf , jp2) = dgx_jp2;
dC(: , id_z2_xs , jp2) = dgx_jp2;
dC(: , id_z3_xf_xf , jp2) = 1/2*dgxx_jp2;
dD(: , id_inov1_u , jp2) = dgu_jp2;
dD(: , id_inov2_u_u , jp2) = 1/2*dguu_jp2;
dD(: , id_inov3_xf_u , jp2) = dgxu_jp2;
dd(:,jp2) = 1/2*dgss_jp2 + 1/2*guu*dE_uu_jp2(:) + 1/2*dguu_jp2*E_uu(:);
dVarinov(id_inov1_u , id_inov1_u , jp2) = dE_uu_jp2;
dVarinov(id_inov2_u_u , id_inov2_u_u , jp2) = reshape(dE_u_u_u_u_jp2,u_nbr^2,u_nbr^2) - dE_uu_jp2(:)*E_uu(:)' - E_uu(:)*dE_uu_jp2(:)';
dVarinov(id_inov3_xf_u , id_inov3_xf_u , jp2) = dE_xfxf_uu_jp2;
dE_xs(:,jp2) = invIx_hx*( dhx_jp2*E_xs + 1/2*dhxx_jp2*E_xfxf(:) + 1/2*hxx*dE_xfxf_jp2(:) + dc(id_z2_xs,jp2) );
dOm_z_jp2 = dB(:,:,jp2)*Varinov*B' + B*dVarinov(:,:,jp2)*B' + B*Varinov*dB(:,:,jp2)';
[dVar_z(:,:,jp2), errorflag] = disclyap_fast(A, dA(:,:,jp2)*Var_z*A' + A*Var_z*dA(:,:,jp2)' + dOm_z_jp2, options.lyapunov_doubling_tol);
if errorflag
dVar_z(:,:,jp2) = lyapunov_symm(A, dA(:,:,jp2)*Var_z*A' + A*Var_z*dA(:,:,jp2)' + dOm_z_jp2,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
if lyapunov_symm_method == 1
lyapunov_symm_method = 2; %now we can reuse persistent schur
end
end
% Make sure some stuff is zero due to Gaussianity
dVar_z(id_z1_xf , id_z2_xs , jp2) = zeros(x_nbr,x_nbr);
dVar_z(id_z1_xf , id_z3_xf_xf , jp2) = zeros(x_nbr,x_nbr^2);
dVar_z(id_z2_xs , id_z1_xf , jp2) = zeros(x_nbr,x_nbr);
dVar_z(id_z3_xf_xf , id_z1_xf , jp2) = zeros(x_nbr^2,x_nbr);
end
end
if order > 2
% Some common and useful objects for order > 2
K_u_xx = commutation(u_nbr,x_nbr^2,1);
K_u_ux = commutation(u_nbr,u_nbr*x_nbr,1);
hx_hss2 = kron(hx,1/2*hss);
hu_hss2 = kron(hu,1/2*hss);
hx_hxx2 = kron(hx,1/2*hxx);
hxx2_hu = kron(1/2*hxx,hu);
hx_hxu = kron(hx,hxu);
hxu_hu = kron(hxu,hu);
hx_huu2 = kron(hx,1/2*huu);
hu_huu2 = kron(hu,1/2*huu);
hx_hx_hx = kron(hx,hx_hx);
hx_hx_hu = kron(hx_hx,hu);
hu_hx_hx = kron(hu,hx_hx);
hu_hu_hu = kron(hu_hu,hu);
hx_hu_hu = kron(hx,hu_hu);
hu_hx_hu = kron(hu,hx_hu);
invIxx_hx_hx = (eye(x_nbr^2)-hx_hx)\eye(x_nbr^2);
% Reuse second-order results
%E_xfxf = Var_z(id_z1_xf, id_z1_xf ); %this is E[xf*xf'], we already have that
%E_xfxs = Var_z(id_z1_xf, id_z2_xs ); %this is E[xf*xs']=0 due to gaussianity
%E_xfxf_xf = Var_z(id_z1_xf, id_z3_xf_xf ); %this is E[xf*kron(xf_xf)']=0 due to gaussianity
%E_xsxf = Var_z(id_z2_xs, id_z1_xf ); %this is E[xs*xf']=0 due to gaussianity
E_xsxs = Var_z(id_z2_xs, id_z2_xs ) + E_xs*transpose(E_xs); %this is E[xs*xs']
E_xsxf_xf = Var_z(id_z2_xs, id_z3_xf_xf ) + E_xs*E_xfxf(:)'; %this is E[xs*kron(xf,xf)']
%E_xf_xfxf = Var_z(id_z3_xf_xf, id_z1_xf ); %this is E[kron(xf,xf)*xf']=0 due to gaussianity
E_xf_xfxs = Var_z(id_z3_xf_xf, id_z2_xs ) + E_xfxf(:)*E_xs'; %this is E[kron(xf,xf)*xs']
E_xf_xfxf_xf = Var_z(id_z3_xf_xf, id_z3_xf_xf) + E_xfxf(:)*E_xfxf(:)'; %this is E[kron(xf,xf)*kron(xf,xf)']
E_xrdxf = reshape(invIxx_hx_hx*vec(...
hxx*reshape( commutation(x_nbr^2,x_nbr,1)*E_xf_xfxs(:), x_nbr^2,x_nbr)*hx'...
+ hxu*kron(E_xs,E_uu)*hu'...
+ 1/6*hxxx*reshape(E_xf_xfxf_xf,x_nbr^3,x_nbr)*hx'...
+ 1/6*huuu*reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr)*hu'...
+ 3/6*hxxu*kron(E_xfxf(:),E_uu)*hu'...
+ 3/6*hxuu*kron(E_xfxf,E_uu(:))*hx'...
+ 3/6*hxss*E_xfxf*hx'...
+ 3/6*huss*E_uu*hu'...
),...
x_nbr,x_nbr); %this is E[xrd*xf']
if compute_derivs
dE_xsxs = zeros(x_nbr,x_nbr,totparam_nbr);
dE_xsxf_xf = zeros(x_nbr,x_nbr^2,totparam_nbr);
dE_xf_xfxs = zeros(x_nbr^2,x_nbr,totparam_nbr);
dE_xf_xfxf_xf = zeros(x_nbr^2,x_nbr^2,totparam_nbr);
dE_xrdxf = zeros(x_nbr,x_nbr,totparam_nbr);
for jp2 = 1:totparam_nbr
if jp2 < (stderrparam_nbr+corrparam_nbr)
dE_u_u_u_u_jp2 = QPu*dE_u_u_u_u(:,jp2);
else
dE_u_u_u_u_jp2 = zeros(u_nbr^4,1);
end
dE_xsxs(:,:,jp2) = dVar_z(id_z2_xs , id_z2_xs , jp2) + dE_xs(:,jp2)*transpose(E_xs) + E_xs*transpose(dE_xs(:,jp2));
dE_xsxf_xf(:,:,jp2) = dVar_z(id_z2_xs , id_z3_xf_xf , jp2) + dE_xs(:,jp2)*E_xfxf(:)' + E_xs*vec(dE_xfxf(:,:,jp2))';
dE_xf_xfxs(:,:,jp2) = dVar_z(id_z3_xf_xf , id_z2_xs , jp2) + vec(dE_xfxf(:,:,jp2))*E_xs' + E_xfxf(:)*dE_xs(:,jp2)';
dE_xf_xfxf_xf(:,:,jp2) = dVar_z(id_z3_xf_xf , id_z3_xf_xf , jp2) + vec(dE_xfxf(:,:,jp2))*E_xfxf(:)' + E_xfxf(:)*vec(dE_xfxf(:,:,jp2))';
dE_xrdxf(:,:,jp2) = reshape(invIxx_hx_hx*vec(...
dhx(:,:,jp2)*E_xrdxf*hx' + hx*E_xrdxf*dhx(:,:,jp2)'...
+ dhxx(:,:,jp2)*reshape( commutation(x_nbr^2,x_nbr,1)*E_xf_xfxs(:), x_nbr^2,x_nbr)*hx' + hxx*reshape( commutation(x_nbr^2,x_nbr,1)*vec(dE_xf_xfxs(:,:,jp2)), x_nbr^2,x_nbr)*hx' + hxx*reshape( commutation(x_nbr^2,x_nbr,1)*E_xf_xfxs(:), x_nbr^2,x_nbr)*dhx(:,:,jp2)'...
+ dhxu(:,:,jp2)*kron(E_xs,E_uu)*hu' + hxu*kron(dE_xs(:,jp2),E_uu)*hu' + hxu*kron(E_xs,dE_uu(:,:,jp2))*hu' + hxu*kron(E_xs,E_uu)*dhu(:,:,jp2)'...
+ 1/6*dhxxx(:,:,jp2)*reshape(E_xf_xfxf_xf,x_nbr^3,x_nbr)*hx' + 1/6*hxxx*reshape(dE_xf_xfxf_xf(:,:,jp2),x_nbr^3,x_nbr)*hx' + 1/6*hxxx*reshape(E_xf_xfxf_xf,x_nbr^3,x_nbr)*dhx(:,:,jp2)'...
+ 1/6*dhuuu(:,:,jp2)*reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr)*hu' + 1/6*huuu*reshape(dE_u_u_u_u_jp2,u_nbr^3,u_nbr)*hu' + 1/6*huuu*reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr)*dhu(:,:,jp2)'...
+ 3/6*dhxxu(:,:,jp2)*kron(E_xfxf(:),E_uu)*hu' + 3/6*hxxu*kron(vec(dE_xfxf(:,:,jp2)),E_uu)*hu' + 3/6*hxxu*kron(E_xfxf(:),dE_uu(:,:,jp2))*hu' + 3/6*hxxu*kron(E_xfxf(:),E_uu)*dhu(:,:,jp2)'...
+ 3/6*dhxuu(:,:,jp2)*kron(E_xfxf,E_uu(:))*hx' + 3/6*hxuu*kron(dE_xfxf(:,:,jp2),E_uu(:))*hx' + 3/6*hxuu*kron(E_xfxf,vec(dE_uu(:,:,jp2)))*hx' + 3/6*hxuu*kron(E_xfxf,E_uu(:))*dhx(:,:,jp2)'...
+ 3/6*dhxss(:,:,jp2)*E_xfxf*hx' + 3/6*hxss*dE_xfxf(:,:,jp2)*hx' + 3/6*hxss*E_xfxf*dhx(:,:,jp2)'...
+ 3/6*dhuss(:,:,jp2)*E_uu*hu' + 3/6*huss*dE_uu(:,:,jp2)*hu' + 3/6*huss*E_uu*dhu(:,:,jp2)'...
), x_nbr, x_nbr);
end
end
% Compute unique sixth-order product moments of u, i.e. unique(E[kron(kron(kron(kron(kron(u,u),u),u),u),u)],'stable')
u_nbr6 = u_nbr*(u_nbr+1)/2*(u_nbr+2)/3*(u_nbr+3)/4*(u_nbr+4)/5*(u_nbr+5)/6;
Q6Pu = Q6_plication(u_nbr);
COMBOS6 = flipud(allVL1(u_nbr, 6)); %all possible (unique) combinations of powers that sum up to six
E_u_u_u_u_u_u = zeros(u_nbr6,1); %only unique entries
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
dE_u_u_u_u_u_u = zeros(u_nbr6,stderrparam_nbr+corrparam_nbr);
end
for j6 = 1:size(COMBOS6,1)
if compute_derivs && (stderrparam_nbr+corrparam_nbr>0)
[E_u_u_u_u_u_u(j6), dE_u_u_u_u_u_u(j6,:)] = prodmom_deriv(E_uu, 1:u_nbr, COMBOS6(j6,:), dE_uu(:,:,1:(stderrparam_nbr+corrparam_nbr)), dr.derivs.dCorrelation_matrix(:,:,1:(stderrparam_nbr+corrparam_nbr)));
else
E_u_u_u_u_u_u(j6) = prodmom(E_uu, 1:u_nbr, COMBOS6(j6,:));
end
end
%% Third-order pruned state space system
% Auxiliary state vector z is defined by: z = [xf; xs; kron(xf,xf); xrd; kron(xf,xs); kron(kron(xf,xf),xf)]
% Auxiliary innovations vector inov is defined by: inov = [u; kron(u,u)-E_uu(:); kron(xf,u); kron(xs,u); kron(kron(xf,xf),u); kron(kron(xf,u),u); kron(kron(u,u),u))]
z_nbr = x_nbr + x_nbr + x_nbr^2 + x_nbr + x_nbr^2 + x_nbr^3;
inov_nbr = u_nbr + u_nbr^2 + x_nbr*u_nbr + x_nbr*u_nbr + x_nbr^2*u_nbr + x_nbr*u_nbr^2 + u_nbr^3;
A = zeros(z_nbr,z_nbr);
A(id_z1_xf , id_z1_xf ) = hx;
A(id_z2_xs , id_z2_xs ) = hx;
A(id_z2_xs , id_z3_xf_xf ) = 1/2*hxx;
A(id_z3_xf_xf , id_z3_xf_xf ) = hx_hx;
A(id_z4_xrd , id_z1_xf ) = 3/6*hxss;
A(id_z4_xrd , id_z4_xrd ) = hx;
A(id_z4_xrd , id_z5_xf_xs ) = hxx;
A(id_z4_xrd , id_z6_xf_xf_xf) = 1/6*hxxx;
A(id_z5_xf_xs , id_z1_xf ) = hx_hss2;
A(id_z5_xf_xs , id_z5_xf_xs ) = hx_hx;
A(id_z5_xf_xs , id_z6_xf_xf_xf) = hx_hxx2;
A(id_z6_xf_xf_xf , id_z6_xf_xf_xf) = hx_hx_hx;
B = zeros(z_nbr,inov_nbr);
B(id_z1_xf , id_inov1_u ) = hu;
B(id_z2_xs , id_inov2_u_u ) = 1/2*huu;
B(id_z2_xs , id_inov3_xf_u ) = hxu;
B(id_z3_xf_xf , id_inov2_u_u ) = hu_hu;
B(id_z3_xf_xf , id_inov3_xf_u ) = (I_xx+K_x_x)*hx_hu;
B(id_z4_xrd , id_inov1_u ) = 3/6*huss;
B(id_z4_xrd , id_inov4_xs_u ) = hxu;
B(id_z4_xrd , id_inov5_xf_xf_u) = 3/6*hxxu;
B(id_z4_xrd , id_inov6_xf_u_u ) = 3/6*hxuu;
B(id_z4_xrd , id_inov7_u_u_u ) = 1/6*huuu;
B(id_z5_xf_xs , id_inov1_u ) = hu_hss2;
B(id_z5_xf_xs , id_inov4_xs_u ) = K_x_x*hx_hu;
B(id_z5_xf_xs , id_inov5_xf_xf_u) = hx_hxu + K_x_x*hxx2_hu;
B(id_z5_xf_xs , id_inov6_xf_u_u ) = hx_huu2 + K_x_x*hxu_hu;
B(id_z5_xf_xs , id_inov7_u_u_u ) = hu_huu2;
B(id_z6_xf_xf_xf , id_inov5_xf_xf_u) = hx_hx_hu + kron(hx,K_x_x*hx_hu) + hu_hx_hx*K_u_xx;
B(id_z6_xf_xf_xf , id_inov6_xf_u_u ) = hx_hu_hu + hu_hx_hu*K_u_ux + kron(hu,K_x_x*hx_hu)*K_u_ux;
B(id_z6_xf_xf_xf , id_inov7_u_u_u ) = hu_hu_hu;
c = zeros(z_nbr, 1);
c(id_z2_xs , 1) = 1/2*hss + 1/2*huu*E_uu(:);
c(id_z3_xf_xf , 1) = hu_hu*E_uu(:);
C = zeros(y_nbr,z_nbr);
C(: , id_z1_xf ) = gx + 3/6*gxss;
C(: , id_z2_xs ) = gx;
C(: , id_z3_xf_xf ) = 1/2*gxx;
C(: , id_z4_xrd ) = gx;
C(: , id_z5_xf_xs ) = gxx;
C(: , id_z6_xf_xf_xf) = 1/6*gxxx;
D = zeros(y_nbr,inov_nbr);
D(: , id_inov1_u ) = gu + 3/6*guss;
D(: , id_inov2_u_u ) = 1/2*guu;
D(: , id_inov3_xf_u ) = gxu;
D(: , id_inov4_xs_u ) = gxu;
D(: , id_inov5_xf_xf_u) = 3/6*gxxu;
D(: , id_inov6_xf_u_u) = 3/6*gxuu;
D(: , id_inov7_u_u_u ) = 1/6*guuu;
d = 1/2*gss + 1/2*guu*E_uu(:);
Varinov = zeros(inov_nbr,inov_nbr);
Varinov(id_inov1_u , id_inov1_u ) = E_uu;
%Varinov(id_inov1_u , id_inov2_u_u ) = zeros(u_nbr,u_nbr^2);
%Varinov(id_inov1_u , id_inov3_xf_u ) = zeros(u_nbr,x_nbr*u_nbr);
Varinov(id_inov1_u , id_inov4_xs_u ) = kron(E_xs',E_uu);
Varinov(id_inov1_u , id_inov5_xf_xf_u) = kron(E_xfxf(:)',E_uu);
%Varinov(id_inov1_u , id_inov6_xf_u_u ) = zeros(u_nbr,x_nbr*u_nbr^2);
Varinov(id_inov1_u , id_inov7_u_u_u ) = reshape(QPu*E_u_u_u_u,u_nbr,u_nbr^3);
%Varinov(id_inov2_u_u , id_inov1_u ) = zeros(u_nbr^2,u_nbr);
Varinov(id_inov2_u_u , id_inov2_u_u ) = reshape(QPu*E_u_u_u_u,u_nbr^2,u_nbr^2)-E_uu(:)*E_uu(:)';
%Varinov(id_inov2_u_u , id_inov3_xf_u ) = zeros(u_nbr^2,x_nbr*u_nbr);
%Varinov(id_inov2_u_u , id_inov4_xs_u ) = zeros(u_nbr^2,x_nbr*u_nbr);
%Varinov(id_inov2_u_u , id_inov5_xf_xf_u) = zeros(u_nbr^2,x_nbr^2,u_nbr);
%Varinov(id_inov2_u_u , id_inov6_xf_u_u ) = zeros(u_nbr^2,x_nbr*u_nbr^2);
%Varinov(id_inov2_u_u , id_inov7_u_u_u ) = zeros(u_nbr^2,u_nbr^3);
%Varinov(id_inov3_xf_u , id_inov1_u ) = zeros(x_nbr*u_nbr,u_nbr);
%Varinov(id_inov3_xf_u , id_inov2_u_u ) = zeros(x_nbr*u_nbr,u_nbr^2);
Varinov(id_inov3_xf_u , id_inov3_xf_u ) = E_xfxf_uu;
%Varinov(id_inov3_xf_u , id_inov4_xs_u ) = zeros(x_nbr*u_nbr,x_nbr*u_nbr);
%Varinov(id_inov3_xf_u , id_inov5_xf_xf_u) = zeros(x_nbr*u_nbr,x_nbr^2*u_nbr);
%Varinov(id_inov3_xf_u , id_inov6_xf_u_u ) = zeros(x_nbr*u_nbr,x_nbr*u_nbr^2);
%Varinov(id_inov3_xf_u , id_inov7_u_u_u ) = zeros(x_nbr*u_nbr,u_nbr^3);
Varinov(id_inov4_xs_u , id_inov1_u ) = kron(E_xs,E_uu);
%Varinov(id_inov4_xs_u , id_inov2_u_u ) = zeros(x_nbr*u_nbr,u_nbr^2);
%Varinov(id_inov4_xs_u , id_inov3_xf_u ) = zeros(x_nbr*u_nbr,x_nbr*u_nbr);
Varinov(id_inov4_xs_u , id_inov4_xs_u ) = kron(E_xsxs,E_uu);
Varinov(id_inov4_xs_u , id_inov5_xf_xf_u) = kron(E_xsxf_xf, E_uu);
%Varinov(id_inov4_xs_u , id_inov6_xf_u_u ) = zeros(x_nbr*u_nbr,x_nbr*u_nbr^2);
Varinov(id_inov4_xs_u , id_inov7_u_u_u ) = kron(E_xs,reshape(QPu*E_u_u_u_u,u_nbr,u_nbr^3));
Varinov(id_inov5_xf_xf_u , id_inov1_u ) = kron(E_xfxf(:),E_uu);
%Varinov(id_inov5_xf_xf_u , id_inov2_u_u ) = zeros(x_nbr^2*u_nbr,u_nbr^2);
%Varinov(id_inov5_xf_xf_u , id_inov3_xf_u ) = zeros(x_nbr^2*u_nbr,x_nbr*u_nbr);
Varinov(id_inov5_xf_xf_u , id_inov4_xs_u ) = kron(E_xf_xfxs,E_uu);
Varinov(id_inov5_xf_xf_u , id_inov5_xf_xf_u) = kron(E_xf_xfxf_xf,E_uu);
%Varinov(id_inov5_xf_xf_u , id_inov6_xf_u_u ) = zeros(x_nbr^2*u_nbr,x_nbr*u_nbr^2);
Varinov(id_inov5_xf_xf_u , id_inov7_u_u_u ) = kron(E_xfxf(:),reshape(QPu*E_u_u_u_u,u_nbr,u_nbr^3));
%Varinov(id_inov6_xf_u_u , id_inov1_u ) = zeros(x_nbr*u_nbr^2,u_nbr);
%Varinov(id_inov6_xf_u_u , id_inov2_u_u ) = zeros(x_nbr*u_nbr^2,u_nbr^2);
%Varinov(id_inov6_xf_u_u , id_inov3_xf_u ) = zeros(x_nbr*u_nbr^2,x_nbr*u_nbr);
%Varinov(id_inov6_xf_u_u , id_inov4_xs_u ) = zeros(x_nbr*u_nbr^2,x_nbr*u_nbr);
%Varinov(id_inov6_xf_u_u , id_inov5_xf_xf_u) = zeros(x_nbr*u_nbr^2,x_nbr^2*u_nbr);
Varinov(id_inov6_xf_u_u , id_inov6_xf_u_u ) = kron(E_xfxf,reshape(QPu*E_u_u_u_u,u_nbr^2,u_nbr^2));
%Varinov(id_inov6_xf_u_u , id_inov7_u_u_u ) = zeros(x_nbr*u_nbr^2,u_nbr^3);
Varinov(id_inov7_u_u_u , id_inov1_u ) = reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr);
%Varinov(id_inov7_u_u_u , id_inov2_u_u ) = zeros(u_nbr^3,u_nbr^2);
%Varinov(id_inov7_u_u_u , id_inov3_xf_u ) = zeros(u_nbr^3,x_nbr*u_nbr);
Varinov(id_inov7_u_u_u , id_inov4_xs_u ) = kron(E_xs',reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr));
Varinov(id_inov7_u_u_u , id_inov5_xf_xf_u) = kron(transpose(E_xfxf(:)),reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr));
%Varinov(id_inov7_u_u_u , id_inov6_xf_u_u ) = zeros(u_nbr^3,x_nbr*u_nbr^2);
Varinov(id_inov7_u_u_u , id_inov7_u_u_u ) = reshape(Q6Pu*E_u_u_u_u_u_u,u_nbr^3,u_nbr^3);
E_xrd = zeros(x_nbr,1);%due to gaussianity
E_inovzlag1 = zeros(inov_nbr,z_nbr); % Attention: E[inov*z(-1)'] is not equal to zero for a third-order approximation due to kron(kron(xf(-1),u),u)
E_inovzlag1(id_inov6_xf_u_u , id_z1_xf ) = kron(E_xfxf,E_uu(:));
E_inovzlag1(id_inov6_xf_u_u , id_z4_xrd ) = kron(E_xrdxf',E_uu(:));
E_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) ;
E_inovzlag1(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
Binovzlag1A= B*E_inovzlag1*transpose(A);
Om_z = B*Varinov*transpose(B) + Binovzlag1A + transpose(Binovzlag1A);
lyapunov_symm_method = 1; %method=1 to initialize persistent variables
[Var_z, errorflag] = disclyap_fast(A,Om_z,options.lyapunov_doubling_tol);
if errorflag %use Schur-based method
fprintf('PRUNED_STATE_SPACE_SYSTEM: error flag in disclyap_fast at order=3, use lyapunov_symm\n');
Var_z = lyapunov_symm(A,Om_z,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
lyapunov_symm_method = 2; %we can now make use of persistent variables from shur
end
%make sure some stuff is zero due to Gaussianity
Var_z(id_z1_xf , id_z2_xs) = zeros(x_nbr,x_nbr);
Var_z(id_z1_xf , id_z3_xf_xf) = zeros(x_nbr,x_nbr^2);
Var_z(id_z2_xs , id_z1_xf) = zeros(x_nbr,x_nbr);
Var_z(id_z2_xs , id_z4_xrd) = zeros(x_nbr,x_nbr);
Var_z(id_z2_xs , id_z5_xf_xs) = zeros(x_nbr,x_nbr^2);
Var_z(id_z2_xs , id_z6_xf_xf_xf) = zeros(x_nbr,x_nbr^3);
Var_z(id_z3_xf_xf , id_z1_xf) = zeros(x_nbr^2,x_nbr);
Var_z(id_z3_xf_xf , id_z4_xrd) = zeros(x_nbr^2,x_nbr);
Var_z(id_z3_xf_xf , id_z5_xf_xs) = zeros(x_nbr^2,x_nbr^2);
Var_z(id_z3_xf_xf , id_z6_xf_xf_xf) = zeros(x_nbr^2,x_nbr^3);
Var_z(id_z4_xrd , id_z2_xs) = zeros(x_nbr,x_nbr);
Var_z(id_z4_xrd , id_z3_xf_xf) = zeros(x_nbr,x_nbr^2);
Var_z(id_z5_xf_xs , id_z2_xs) = zeros(x_nbr^2,x_nbr);
Var_z(id_z5_xf_xs , id_z3_xf_xf) = zeros(x_nbr^2,x_nbr^2);
Var_z(id_z6_xf_xf_xf , id_z2_xs) = zeros(x_nbr^3,x_nbr);
Var_z(id_z6_xf_xf_xf , id_z3_xf_xf) = zeros(x_nbr^3,x_nbr^2);
if compute_derivs
dA = zeros(z_nbr,z_nbr,totparam_nbr);
dB = zeros(z_nbr,inov_nbr,totparam_nbr);
dc = zeros(z_nbr,totparam_nbr);
dC = zeros(y_nbr,z_nbr,totparam_nbr);
dD = zeros(y_nbr,inov_nbr,totparam_nbr);
dd = zeros(y_nbr,totparam_nbr);
dVarinov = zeros(inov_nbr,inov_nbr,totparam_nbr);
dE_xrd = zeros(x_nbr,totparam_nbr);
dE_inovzlag1 = zeros(inov_nbr,z_nbr,totparam_nbr);
dVar_z = zeros(z_nbr,z_nbr,totparam_nbr);
for jp3 = 1:totparam_nbr
if jp3 <= (stderrparam_nbr+corrparam_nbr)
dE_uu_jp3 = dE_uu(:,:,jp3);
dE_u_u_u_u_jp3 = QPu*dE_u_u_u_u(:,jp3);
dE_u_u_u_u_u_u_jp3 = Q6Pu*dE_u_u_u_u_u_u(:,jp3);
else
dE_uu_jp3 = zeros(u_nbr,u_nbr);
dE_u_u_u_u_jp3 = zeros(u_nbr^4,1);
dE_u_u_u_u_u_u_jp3 = zeros(u_nbr^6,1);
end
dhx_jp3 = dhx(:,:,jp3);
dhu_jp3 = dhu(:,:,jp3);
dhxx_jp3 = dhxx(:,:,jp3);
dhxu_jp3 = dhxu(:,:,jp3);
dhuu_jp3 = dhuu(:,:,jp3);
dhss_jp3 = dhss(:,jp3);
dhxxx_jp3 = dhxxx(:,:,jp3);
dhxxu_jp3 = dhxxu(:,:,jp3);
dhxuu_jp3 = dhxuu(:,:,jp3);
dhuuu_jp3 = dhuuu(:,:,jp3);
dhxss_jp3 = dhxss(:,:,jp3);
dhuss_jp3 = dhuss(:,:,jp3);
dgx_jp3 = dgx(:,:,jp3);
dgu_jp3 = dgu(:,:,jp3);
dgxx_jp3 = dgxx(:,:,jp3);
dgxu_jp3 = dgxu(:,:,jp3);
dguu_jp3 = dguu(:,:,jp3);
dgss_jp3 = dgss(:,jp3);
dgxxx_jp3 = dgxxx(:,:,jp3);
dgxxu_jp3 = dgxxu(:,:,jp3);
dgxuu_jp3 = dgxuu(:,:,jp3);
dguuu_jp3 = dguuu(:,:,jp3);
dgxss_jp3 = dgxss(:,:,jp3);
dguss_jp3 = dguss(:,:,jp3);
dhx_hx_jp3 = kron(dhx_jp3,hx) + kron(hx,dhx_jp3);
dhx_hu_jp3 = kron(dhx_jp3,hu) + kron(hx,dhu_jp3);
dhu_hu_jp3 = kron(dhu_jp3,hu) + kron(hu,dhu_jp3);
dhx_hss2_jp3 = kron(dhx_jp3,1/2*hss) + kron(hx,1/2*dhss_jp3);
dhu_hss2_jp3 = kron(dhu_jp3,1/2*hss) + kron(hu,1/2*dhss_jp3);
dhx_hxx2_jp3 = kron(dhx_jp3,1/2*hxx) + kron(hx,1/2*dhxx_jp3);
dhxx2_hu_jp3 = kron(1/2*dhxx_jp3,hu) + kron(1/2*hxx,dhu_jp3);
dhx_hxu_jp3 = kron(dhx_jp3,hxu) + kron(hx,dhxu_jp3);
dhxu_hu_jp3 = kron(dhxu_jp3,hu) + kron(hxu,dhu_jp3);
dhx_huu2_jp3 = kron(dhx_jp3,1/2*huu) + kron(hx,1/2*dhuu_jp3);
dhu_huu2_jp3 = kron(dhu_jp3,1/2*huu) + kron(hu,1/2*dhuu_jp3);
dhx_hx_hx_jp3 = kron(dhx_jp3,hx_hx) + kron(hx,dhx_hx_jp3);
dhx_hx_hu_jp3 = kron(dhx_hx_jp3,hu) + kron(hx_hx,dhu_jp3);
dhu_hx_hx_jp3 = kron(dhu_jp3,hx_hx) + kron(hu,dhx_hx_jp3);
dhu_hu_hu_jp3 = kron(dhu_hu_jp3,hu) + kron(hu_hu,dhu_jp3);
dhx_hu_hu_jp3 = kron(dhx_jp3,hu_hu) + kron(hx,dhu_hu_jp3);
dhu_hx_hu_jp3 = kron(dhu_jp3,hx_hu) + kron(hu,dhx_hu_jp3);
dE_xs_jp3 = dE_xs(:,jp3);
dE_xfxf_jp3 = dE_xfxf(:,:,jp3);
dE_xsxs_jp3 = dE_xsxs(:,:,jp3);
dE_xsxf_xf_jp3 = dE_xsxf_xf(:,:,jp3);
dE_xfxf_uu_jp3 = kron(dE_xfxf_jp3,E_uu) + kron(E_xfxf,dE_uu_jp3);
dE_xf_xfxs_jp3 = dE_xf_xfxs(:,:,jp3);
dE_xf_xfxf_xf_jp3 = dE_xf_xfxf_xf(:,:,jp3);
dE_xrdxf_jp3 = dE_xrdxf(:,:,jp3);
dA(id_z1_xf , id_z1_xf , jp3) = dhx_jp3;
dA(id_z2_xs , id_z2_xs , jp3) = dhx_jp3;
dA(id_z2_xs , id_z3_xf_xf , jp3) = 1/2*dhxx_jp3;
dA(id_z3_xf_xf , id_z3_xf_xf , jp3) = dhx_hx_jp3;
dA(id_z4_xrd , id_z1_xf , jp3) = 3/6*dhxss_jp3;
dA(id_z4_xrd , id_z4_xrd , jp3) = dhx_jp3;
dA(id_z4_xrd , id_z5_xf_xs , jp3) = dhxx_jp3;
dA(id_z4_xrd , id_z6_xf_xf_xf , jp3) = 1/6*dhxxx_jp3;
dA(id_z5_xf_xs , id_z1_xf , jp3) = dhx_hss2_jp3;
dA(id_z5_xf_xs , id_z5_xf_xs , jp3) = dhx_hx_jp3;
dA(id_z5_xf_xs , id_z6_xf_xf_xf , jp3) = dhx_hxx2_jp3;
dA(id_z6_xf_xf_xf , id_z6_xf_xf_xf , jp3) = dhx_hx_hx_jp3;
dB(id_z1_xf , id_inov1_u , jp3) = dhu_jp3;
dB(id_z2_xs , id_inov2_u_u , jp3) = 1/2*dhuu_jp3;
dB(id_z2_xs , id_inov3_xf_u , jp3) = dhxu_jp3;
dB(id_z3_xf_xf , id_inov2_u_u , jp3) = dhu_hu_jp3;
dB(id_z3_xf_xf , id_inov3_xf_u , jp3) = (I_xx+K_x_x)*dhx_hu_jp3;
dB(id_z4_xrd , id_inov1_u , jp3) = 3/6*dhuss_jp3;
dB(id_z4_xrd , id_inov4_xs_u , jp3) = dhxu_jp3;
dB(id_z4_xrd , id_inov5_xf_xf_u , jp3) = 3/6*dhxxu_jp3;
dB(id_z4_xrd , id_inov6_xf_u_u , jp3) = 3/6*dhxuu_jp3;
dB(id_z4_xrd , id_inov7_u_u_u , jp3) = 1/6*dhuuu_jp3;
dB(id_z5_xf_xs , id_inov1_u , jp3) = dhu_hss2_jp3;
dB(id_z5_xf_xs , id_inov4_xs_u , jp3) = K_x_x*dhx_hu_jp3;
dB(id_z5_xf_xs , id_inov5_xf_xf_u , jp3) = dhx_hxu_jp3 + K_x_x*dhxx2_hu_jp3;
dB(id_z5_xf_xs , id_inov6_xf_u_u , jp3) = dhx_huu2_jp3 + K_x_x*dhxu_hu_jp3;
dB(id_z5_xf_xs , id_inov7_u_u_u , jp3) = dhu_huu2_jp3;
dB(id_z6_xf_xf_xf , id_inov5_xf_xf_u , jp3) = dhx_hx_hu_jp3 + kron(dhx_jp3,K_x_x*hx_hu) + kron(hx,K_x_x*dhx_hu_jp3) + dhu_hx_hx_jp3*K_u_xx;
dB(id_z6_xf_xf_xf , id_inov6_xf_u_u , jp3) = dhx_hu_hu_jp3 + dhu_hx_hu_jp3*K_u_ux + kron(dhu_jp3,K_x_x*hx_hu)*K_u_ux + kron(hu,K_x_x*dhx_hu_jp3)*K_u_ux;
dB(id_z6_xf_xf_xf , id_inov7_u_u_u , jp3) = dhu_hu_hu_jp3;
dc(id_z2_xs , jp3) = 1/2*dhss_jp3 + 1/2*dhuu_jp3*E_uu(:) + 1/2*huu*dE_uu_jp3(:);
dc(id_z3_xf_xf , jp3) = dhu_hu_jp3*E_uu(:) + hu_hu*dE_uu_jp3(:);
dC(: , id_z1_xf , jp3) = dgx_jp3 + 3/6*dgxss_jp3;
dC(: , id_z2_xs , jp3) = dgx_jp3;
dC(: , id_z3_xf_xf , jp3) = 1/2*dgxx_jp3;
dC(: , id_z4_xrd , jp3) = dgx_jp3;
dC(: , id_z5_xf_xs , jp3) = dgxx_jp3;
dC(: , id_z6_xf_xf_xf , jp3) = 1/6*dgxxx_jp3;
dD(: , id_inov1_u , jp3) = dgu_jp3 + 3/6*dguss_jp3;
dD(: , id_inov2_u_u , jp3) = 1/2*dguu_jp3;
dD(: , id_inov3_xf_u , jp3) = dgxu_jp3;
dD(: , id_inov4_xs_u , jp3) = dgxu_jp3;
dD(: , id_inov5_xf_xf_u , jp3) = 3/6*dgxxu_jp3;
dD(: , id_inov6_xf_u_u , jp3) = 3/6*dgxuu_jp3;
dD(: , id_inov7_u_u_u , jp3) = 1/6*dguuu_jp3;
dd(:,jp3) = 1/2*dgss_jp3 + 1/2*dguu_jp3*E_uu(:) + 1/2*guu*dE_uu_jp3(:);
dVarinov(id_inov1_u , id_inov1_u , jp3) = dE_uu_jp3;
dVarinov(id_inov1_u , id_inov4_xs_u , jp3) = kron(dE_xs_jp3',E_uu) + kron(E_xs',dE_uu_jp3);
dVarinov(id_inov1_u , id_inov5_xf_xf_u , jp3) = kron(dE_xfxf_jp3(:)',E_uu) + kron(E_xfxf(:)',dE_uu_jp3);
dVarinov(id_inov1_u , id_inov7_u_u_u , jp3) = reshape(dE_u_u_u_u_jp3,u_nbr,u_nbr^3);
dVarinov(id_inov2_u_u , id_inov2_u_u , jp3) = reshape(dE_u_u_u_u_jp3,u_nbr^2,u_nbr^2) - dE_uu_jp3(:)*E_uu(:)' - E_uu(:)*dE_uu_jp3(:)';
dVarinov(id_inov3_xf_u , id_inov3_xf_u , jp3) = dE_xfxf_uu_jp3;
dVarinov(id_inov4_xs_u , id_inov1_u , jp3) = kron(dE_xs_jp3,E_uu) + kron(E_xs,dE_uu_jp3);
dVarinov(id_inov4_xs_u , id_inov4_xs_u , jp3) = kron(dE_xsxs_jp3,E_uu) + kron(E_xsxs,dE_uu_jp3);
dVarinov(id_inov4_xs_u , id_inov5_xf_xf_u , jp3) = kron(dE_xsxf_xf_jp3, E_uu) + kron(E_xsxf_xf, dE_uu_jp3);
dVarinov(id_inov4_xs_u , id_inov7_u_u_u , jp3) = kron(dE_xs_jp3,reshape(QPu*E_u_u_u_u,u_nbr,u_nbr^3)) + kron(E_xs,reshape(dE_u_u_u_u_jp3,u_nbr,u_nbr^3));
dVarinov(id_inov5_xf_xf_u , id_inov1_u , jp3) = kron(dE_xfxf_jp3(:),E_uu) + kron(E_xfxf(:),dE_uu_jp3);
dVarinov(id_inov5_xf_xf_u , id_inov4_xs_u , jp3) = kron(dE_xf_xfxs_jp3,E_uu) + kron(E_xf_xfxs,dE_uu_jp3);
dVarinov(id_inov5_xf_xf_u , id_inov5_xf_xf_u , jp3) = kron(dE_xf_xfxf_xf_jp3,E_uu) + kron(E_xf_xfxf_xf,dE_uu_jp3);
dVarinov(id_inov5_xf_xf_u , id_inov7_u_u_u , jp3) = kron(dE_xfxf_jp3(:),reshape(QPu*E_u_u_u_u,u_nbr,u_nbr^3)) + kron(E_xfxf(:),reshape(dE_u_u_u_u_jp3,u_nbr,u_nbr^3));
dVarinov(id_inov6_xf_u_u , id_inov6_xf_u_u , jp3) = kron(dE_xfxf_jp3,reshape(QPu*E_u_u_u_u,u_nbr^2,u_nbr^2)) + kron(E_xfxf,reshape(dE_u_u_u_u_jp3,u_nbr^2,u_nbr^2));
dVarinov(id_inov7_u_u_u , id_inov1_u , jp3) = reshape(dE_u_u_u_u_jp3,u_nbr^3,u_nbr);
dVarinov(id_inov7_u_u_u , id_inov4_xs_u , jp3) = kron(dE_xs_jp3',reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr)) + kron(E_xs',reshape(dE_u_u_u_u_jp3,u_nbr^3,u_nbr));
dVarinov(id_inov7_u_u_u , id_inov5_xf_xf_u , jp3) = kron(transpose(dE_xfxf_jp3(:)),reshape(QPu*E_u_u_u_u,u_nbr^3,u_nbr)) + kron(transpose(E_xfxf(:)),reshape(dE_u_u_u_u_jp3,u_nbr^3,u_nbr));
dVarinov(id_inov7_u_u_u , id_inov7_u_u_u , jp3) = reshape(dE_u_u_u_u_u_u_jp3,u_nbr^3,u_nbr^3);
dE_inovzlag1(id_inov6_xf_u_u , id_z1_xf , jp3) = kron(dE_xfxf_jp3,E_uu(:)) + kron(E_xfxf,dE_uu_jp3(:));
dE_inovzlag1(id_inov6_xf_u_u , id_z4_xrd , jp3) = kron(dE_xrdxf_jp3',E_uu(:)) + kron(E_xrdxf',dE_uu_jp3(:));
dE_inovzlag1(id_inov6_xf_u_u , id_z5_xf_xs , jp3) = kron(reshape(commutation(x_nbr^2,x_nbr)*vec(dE_xsxf_xf_jp3),x_nbr,x_nbr^2),vec(E_uu)) + kron(reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu_jp3)) ;
dE_inovzlag1(id_inov6_xf_u_u , id_z6_xf_xf_xf , jp3) = kron(reshape(dE_xf_xfxf_xf_jp3,x_nbr,x_nbr^3),E_uu(:)) + kron(reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),dE_uu_jp3(:));
dBinovzlag1A_jp3 = dB(:,:,jp3)*E_inovzlag1*transpose(A) + B*dE_inovzlag1(:,:,jp3)*transpose(A) + B*E_inovzlag1*transpose(dA(:,:,jp3));
dOm_z_jp3 = dB(:,:,jp3)*Varinov*transpose(B) + B*dVarinov(:,:,jp3)*transpose(B) + B*Varinov*transpose(dB(:,:,jp3)) + dBinovzlag1A_jp3 + transpose(dBinovzlag1A_jp3);
[dVar_z(:,:,jp3), errorflag] = disclyap_fast(A, dA(:,:,jp3)*Var_z*A' + A*Var_z*dA(:,:,jp3)' + dOm_z_jp3, options.lyapunov_doubling_tol);
if errorflag
dVar_z(:,:,jp3) = lyapunov_symm(A, dA(:,:,jp3)*Var_z*A' + A*Var_z*dA(:,:,jp3)' + dOm_z_jp3,...
options.lyapunov_fixed_point_tol,options.qz_criterium,options.lyapunov_complex_threshold,...
lyapunov_symm_method,...
options.debug);
if lyapunov_symm_method == 1
lyapunov_symm_method = 2; %now we can reuse persistent schur
end
end
%make sure some stuff is zero due to Gaussianity
dVar_z(id_z1_xf , id_z2_xs , jp3) = zeros(x_nbr,x_nbr);
dVar_z(id_z1_xf , id_z3_xf_xf , jp3) = zeros(x_nbr,x_nbr^2);
dVar_z(id_z2_xs , id_z1_xf , jp3) = zeros(x_nbr,x_nbr);
dVar_z(id_z2_xs , id_z4_xrd , jp3) = zeros(x_nbr,x_nbr);
dVar_z(id_z2_xs , id_z5_xf_xs , jp3) = zeros(x_nbr,x_nbr^2);
dVar_z(id_z2_xs , id_z6_xf_xf_xf , jp3) = zeros(x_nbr,x_nbr^3);
dVar_z(id_z3_xf_xf , id_z1_xf , jp3) = zeros(x_nbr^2,x_nbr);
dVar_z(id_z3_xf_xf , id_z4_xrd , jp3) = zeros(x_nbr^2,x_nbr);
dVar_z(id_z3_xf_xf , id_z5_xf_xs , jp3) = zeros(x_nbr^2,x_nbr^2);
dVar_z(id_z3_xf_xf , id_z6_xf_xf_xf , jp3) = zeros(x_nbr^2,x_nbr^3);
dVar_z(id_z4_xrd , id_z2_xs , jp3) = zeros(x_nbr,x_nbr);
dVar_z(id_z4_xrd , id_z3_xf_xf , jp3) = zeros(x_nbr,x_nbr^2);
dVar_z(id_z5_xf_xs , id_z2_xs , jp3) = zeros(x_nbr^2,x_nbr);
dVar_z(id_z5_xf_xs , id_z3_xf_xf , jp3) = zeros(x_nbr^2,x_nbr^2);
dVar_z(id_z6_xf_xf_xf , id_z2_xs , jp3) = zeros(x_nbr^3,x_nbr);
dVar_z(id_z6_xf_xf_xf , id_z3_xf_xf , jp3) = zeros(x_nbr^3,x_nbr^2);
end
end
end
end
%% Covariance/Correlation of control variables
Var_y = NaN*ones(y_nbr,y_nbr);
if order < 3
Var_y(stationary_vars,stationary_vars) = C(stationary_vars,:)*Var_z*C(stationary_vars,:)'...
+ D(stationary_vars,:)*Varinov*D(stationary_vars,:)';
else
Var_y(stationary_vars,stationary_vars) = C(stationary_vars,:)*Var_z*C(stationary_vars,:)'...
+ D(stationary_vars,:)*E_inovzlag1*C(stationary_vars,:)'...
+ C(stationary_vars,:)*transpose(E_inovzlag1)*D(stationary_vars,:)'...
+ D(stationary_vars,:)*Varinov*D(stationary_vars,:)';
end
indzeros = find(abs(Var_y) < 1e-12); %find values that are numerical zero
Var_y(indzeros) = 0;
if useautocorr
sdy = sqrt(diag(Var_y)); %theoretical standard deviation
sdy = sdy(stationary_vars);
sy = sdy*sdy'; %cross products of standard deviations
Corr_y = NaN*ones(y_nbr,y_nbr);
Corr_y(stationary_vars,stationary_vars) = Var_y(stationary_vars,stationary_vars)./sy;
Corr_yi = NaN*ones(y_nbr,y_nbr,nlags);
end
if compute_derivs
dVar_y = NaN*ones(y_nbr,y_nbr,totparam_nbr);
if useautocorr
dCorr_y = NaN*ones(y_nbr,y_nbr,totparam_nbr);
dCorr_yi = NaN*ones(y_nbr,y_nbr,nlags,totparam_nbr);
end
for jpV=1:totparam_nbr
if order < 3
dVar_y_tmp = dC(stationary_vars,:,jpV)*Var_z*C(stationary_vars,:)' + C(stationary_vars,:)*dVar_z(:,:,jpV)*C(stationary_vars,:)' + C(stationary_vars,:)*Var_z*dC(stationary_vars,:,jpV)'...
+ dD(stationary_vars,:,jpV)*Varinov*D(stationary_vars,:)' + D(stationary_vars,:)*dVarinov(:,:,jpV)*D(stationary_vars,:)' + D(stationary_vars,:)*Varinov*dD(stationary_vars,:,jpV)';
else
dVar_y_tmp = dC(stationary_vars,:,jpV)*Var_z*C(stationary_vars,:)' + C(stationary_vars,:)*dVar_z(:,:,jpV)*C(stationary_vars,:)' + C(stationary_vars,:)*Var_z*dC(stationary_vars,:,jpV)'...
+ dD(stationary_vars,:,jpV)*E_inovzlag1*C(stationary_vars,:)' + D(stationary_vars,:)*dE_inovzlag1(:,:,jpV)*C(stationary_vars,:)' + D(stationary_vars,:)*E_inovzlag1*dC(stationary_vars,:,jpV)'...
+ dC(stationary_vars,:,jpV)*transpose(E_inovzlag1)*D(stationary_vars,:)' + C(stationary_vars,:)*transpose(dE_inovzlag1(:,:,jpV))*D(stationary_vars,:)' + C(stationary_vars,:)*transpose(E_inovzlag1)*dD(stationary_vars,:,jpV)'...
+ dD(stationary_vars,:,jpV)*Varinov*D(stationary_vars,:)' + D(stationary_vars,:)*dVarinov(:,:,jpV)*D(stationary_vars,:)' + D(stationary_vars,:)*Varinov*dD(stationary_vars,:,jpV)';
end
indzeros = find(abs(dVar_y_tmp) < 1e-12); %find values that are numerical zero
dVar_y_tmp(indzeros) = 0;
dVar_y(stationary_vars,stationary_vars,jpV) = dVar_y_tmp;
if useautocorr
dsy = 1/2./sdy.*diag(dVar_y(:,:,jpV));
dsy = dsy(stationary_vars);
dsy = dsy*sdy'+sdy*dsy';
dCorr_y(stationary_vars,stationary_vars,jpV) = (dVar_y(stationary_vars,stationary_vars,jpV).*sy-dsy.*Var_y(stationary_vars,stationary_vars))./(sy.*sy);
dCorr_y(stationary_vars,stationary_vars,jpV) = dCorr_y(stationary_vars,stationary_vars,jpV)-diag(diag(dCorr_y(stationary_vars,stationary_vars,jpV)))+diag(diag(dVar_y(stationary_vars,stationary_vars,jpV)));
end
end
end
%% Autocovariances/autocorrelations of lagged control variables
Var_yi = NaN*ones(y_nbr,y_nbr,nlags);
Ai = eye(z_nbr); %this is A^0
hxi = eye(x_nbr);
E_inovzlagi = E_inovzlag1;
Var_zi = Var_z;
if order <= 2
tmp = A*Var_z*C(stationary_vars,:)' + B*Varinov*D(stationary_vars,:)';
else
tmp = A*E_inovzlag1'*D(stationary_vars,:)' + B*Varinov*D(stationary_vars,:)';
end
for i = 1:nlags
if order <= 2
Var_yi(stationary_vars,stationary_vars,i) = C(stationary_vars,:)*Ai*tmp;
else
Var_zi = A*Var_zi + B*E_inovzlagi;
hxi = hx*hxi;
E_inovzlagi = zeros(inov_nbr,z_nbr);
E_inovzlagi(id_inov6_xf_u_u , id_z1_xf ) = kron(hxi*E_xfxf,E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z4_xrd ) = kron(hxi*E_xrdxf',E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
E_inovzlagi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
Var_yi(stationary_vars,stationary_vars,i) = C(stationary_vars,:)*Var_zi*C(stationary_vars,:)' + C(stationary_vars,:)*Ai*tmp + D(stationary_vars,:)*E_inovzlagi*C(stationary_vars,:)';
end
if useautocorr
Corr_yi(stationary_vars,stationary_vars,i) = Var_yi(stationary_vars,stationary_vars,i)./sy;
end
Ai = Ai*A; %note that this is A^(i-1)
end
if compute_derivs
dVar_yi = NaN*ones(y_nbr,y_nbr,nlags,totparam_nbr);
for jpVi=1:totparam_nbr
Ai = eye(z_nbr); dAi_jpVi = zeros(z_nbr,z_nbr);
hxi = eye(x_nbr); dhxi_jpVi = zeros(x_nbr,x_nbr);
E_inovzlagi = E_inovzlag1; dE_inovzlagi_jpVi = dE_inovzlag1(:,:,jpVi);
Var_zi = Var_z; dVar_zi_jpVi = dVar_z(:,:,jpVi);
if order <= 2
dtmp_jpVi = dA(:,:,jpVi)*Var_z*C(stationary_vars,:)' + A*dVar_z(:,:,jpVi)*C(stationary_vars,:)' + A*Var_z*dC(stationary_vars,:,jpVi)'...
+ dB(:,:,jpVi)*Varinov*D(stationary_vars,:)' + B*dVarinov(:,:,jpVi)*D(stationary_vars,:)' + B*Varinov*dD(stationary_vars,:,jpVi)';
else
dtmp_jpVi = dA(:,:,jpVi)*E_inovzlag1'*D(stationary_vars,:)' + A*dE_inovzlag1(:,:,jpVi)'*D(stationary_vars,:)' + A*E_inovzlag1'*dD(stationary_vars,:,jpVi)'...
+ dB(:,:,jpVi)*Varinov*D(stationary_vars,:)' + B*dVarinov(:,:,jpVi)*D(stationary_vars,:)' + B*Varinov*dD(stationary_vars,:,jpVi)';
end
for i = 1:nlags
if order <= 2
dVar_yi(stationary_vars,stationary_vars,i,jpVi) = dC(stationary_vars,:,jpVi)*Ai*tmp + C(stationary_vars,:)*dAi_jpVi*tmp + C(stationary_vars,:)*Ai*dtmp_jpVi;
else
Var_zi = A*Var_zi + B*E_inovzlagi;
dVar_zi_jpVi = dA(:,:,jpVi)*Var_zi + A*dVar_zi_jpVi + dB(:,:,jpVi)*E_inovzlagi + + B*dE_inovzlagi_jpVi;
dhxi_jpVi = dhx(:,:,jpVi)*hxi + hx*dhxi_jpVi;
hxi = hx*hxi;
E_inovzlagi = zeros(inov_nbr,z_nbr);
E_inovzlagi(id_inov6_xf_u_u , id_z1_xf ) = kron(hxi*E_xfxf,E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z4_xrd ) = kron(hxi*E_xrdxf',E_uu(:));
E_inovzlagi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu));
E_inovzlagi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:));
dE_inovzlagi_jpVi = zeros(inov_nbr,z_nbr);
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z1_xf ) = kron(dhxi_jpVi*E_xfxf,E_uu(:)) + kron(hxi*dE_xfxf(:,:,jpVi),E_uu(:)) + kron(hxi*E_xfxf,vec(dE_uu(:,:,jpVi)));
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z4_xrd ) = kron(dhxi_jpVi*E_xrdxf',E_uu(:)) + kron(hxi*dE_xrdxf(:,:,jpVi)',E_uu(:)) + kron(hxi*E_xrdxf',vec(dE_uu(:,:,jpVi)));
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z5_xf_xs ) = kron(dhxi_jpVi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(dE_xsxf_xf(:,:,jpVi)),x_nbr,x_nbr^2),vec(E_uu)) + kron(hxi*reshape(commutation(x_nbr^2,x_nbr)*vec(E_xsxf_xf),x_nbr,x_nbr^2),vec(dE_uu(:,:,jpVi)));
dE_inovzlagi_jpVi(id_inov6_xf_u_u , id_z6_xf_xf_xf ) = kron(dhxi_jpVi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),E_uu(:)) + kron(hxi*reshape(dE_xf_xfxf_xf(:,:,jpVi),x_nbr,x_nbr^3),E_uu(:)) + kron(hxi*reshape(E_xf_xfxf_xf,x_nbr,x_nbr^3),vec(dE_uu(:,:,jpVi)));
dVar_yi(stationary_vars,stationary_vars,i,jpVi) = dC(stationary_vars,:,jpVi)*Var_zi*C(stationary_vars,:)' + C(stationary_vars,:)*dVar_zi_jpVi*C(stationary_vars,:)' + C(stationary_vars,:)*Var_zi*dC(stationary_vars,:,jpVi)'...
+ dC(stationary_vars,:,jpVi)*Ai*tmp + C(stationary_vars,:)*dAi_jpVi*tmp + C(stationary_vars,:)*Ai*dtmp_jpVi...
+ dD(stationary_vars,:,jpVi)*E_inovzlagi*C(stationary_vars,:)' + D(stationary_vars,:)*dE_inovzlagi_jpVi*C(stationary_vars,:)' + D(stationary_vars,:)*E_inovzlagi*dC(stationary_vars,:,jpVi)';
end
if useautocorr
dsy = 1/2./sdy.*diag(dVar_y(:,:,jpVi));
dsy = dsy(stationary_vars);
dsy = dsy*sdy'+sdy*dsy';
dCorr_yi(stationary_vars,stationary_vars,i,jpVi) = (dVar_yi(stationary_vars,stationary_vars,i,jpVi).*sy-dsy.*Var_yi(stationary_vars,stationary_vars,i))./(sy.*sy);
end
dAi_jpVi = dAi_jpVi*A + Ai*dA(:,:,jpVi);
Ai = Ai*A;
end
end
end
%% Mean of control variables
E_z = E_xf;
if order > 1
E_z = [E_xf;E_xs;E_xfxf(:)];
end
if order > 2
E_xf_xs = zeros(x_nbr^2,1);
E_xf_xf_xf = zeros(x_nbr^3,1);
E_z = [E_xf;E_xs;E_xfxf(:);E_xrd;E_xf_xs;E_xf_xf_xf];
end
E_y = Yss(indy,:) + C*E_z + d;
if compute_derivs
dE_y = zeros(y_nbr,totparam_nbr);
for jpE = 1:totparam_nbr
if order == 1
dE_z_jpE = dE_xf(:,jpE);
elseif order == 2
dE_z_jpE = [dE_xf(:,jpE);dE_xs(:,jpE);vec(dE_xfxf(:,:,jpE))];
elseif order == 3
dE_xf_xs_jpE = zeros(x_nbr^2,1);
dE_xf_xf_xf_jpE = zeros(x_nbr^3,1);
dE_z_jpE = [dE_xf(:,jpE);dE_xs(:,jpE);vec(dE_xfxf(:,:,jpE)); dE_xrd(:,jpE); dE_xf_xs_jpE; dE_xf_xf_xf_jpE];
end
dE_y(:,jpE) = dC(:,:,jpE)*E_z + C*dE_z_jpE + dd(:,jpE);
if jpE > (stderrparam_nbr+corrparam_nbr)
dE_y(:,jpE) = dE_y(:,jpE) + dYss(indy,jpE-stderrparam_nbr-corrparam_nbr); %add steady state
end
end
end
non_stationary_vars = setdiff(1:y_nbr,stationary_vars);
E_y(non_stationary_vars) = NaN;
if compute_derivs
dE_y(non_stationary_vars,:) = NaN;
end
%% Store into output structure
pruned_state_space.indx = indx;
pruned_state_space.indy = indy;
pruned_state_space.A = A;
pruned_state_space.B = B;
pruned_state_space.C = C;
pruned_state_space.D = D;
pruned_state_space.c = c;
pruned_state_space.d = d;
pruned_state_space.Varinov = Varinov;
pruned_state_space.Var_z = Var_z; %remove in future [@wmutschl]
pruned_state_space.Var_y = Var_y;
pruned_state_space.Var_yi = Var_yi;
if useautocorr
pruned_state_space.Corr_y = Corr_y;
pruned_state_space.Corr_yi = Corr_yi;
end
pruned_state_space.E_y = E_y;
if compute_derivs == 1
pruned_state_space.dA = dA;
pruned_state_space.dB = dB;
pruned_state_space.dC = dC;
pruned_state_space.dD = dD;
pruned_state_space.dc = dc;
pruned_state_space.dd = dd;
pruned_state_space.dVarinov = dVarinov;
pruned_state_space.dVar_y = dVar_y;
pruned_state_space.dVar_yi = dVar_yi;
if useautocorr
pruned_state_space.dCorr_y = dCorr_y;
pruned_state_space.dCorr_yi = dCorr_yi;
end
pruned_state_space.dE_y = dE_y;
end
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