142 lines
4.4 KiB
Matlab
142 lines
4.4 KiB
Matlab
function [f, df, d2f, R2] = likelihood_quadratic_approximation(particles, likelihoodvalues)
|
||
|
||
% Approximate the shape of the likelihood function with a multivariate second order polynomial.
|
||
%
|
||
%
|
||
% INPUTS
|
||
% - particles [double] n×p matrix of (p) particles around the estimated posterior mode.
|
||
% - likelihoodvalues [double] p×1 vector of corresponding values for the likelihood (or posterior kernel).
|
||
%
|
||
% OUTPUTS
|
||
% - f [handle] function handle for the approximated likelihood.
|
||
% - df [handle] function handle for the gradient of the approximated likelihood.
|
||
% - d2f [handle] Hessian matrix of the approximated likelihood (constant since we consider a second order multivariate polynomial)
|
||
% - R2 [double] scalar, goodness of fit measure.
|
||
%
|
||
% REMARKS
|
||
% [1] Function f takes a n×m matrix as input argument (the function is evaluated in m points) and returns a m×1 vector.
|
||
% [2] Funtion df takes a n×1 vector as input argument (the point where the gradient is computed) and returns a n×1 vector.
|
||
|
||
% Copyright © 2024 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/>.
|
||
|
||
n = rows(particles); % Number of parmaeters
|
||
p = columns(particles); % Number of particles
|
||
q = 1 + n + n*(n+1)/2; % Number of regressors (with a constant)
|
||
|
||
if p<=q
|
||
error('Quadratic approximation requires more than %u particles.', q)
|
||
end
|
||
|
||
%
|
||
% Build the set of regressors.
|
||
%
|
||
|
||
X = NaN(p, q);
|
||
X(:,1) = 1; % zero order term
|
||
X(:,2:n+1) = transpose(particles); % first order terms
|
||
X(:,n+2:end) = crossproducts(particles); % second order terms
|
||
|
||
%
|
||
% Perform the regression
|
||
%
|
||
|
||
parameters = X\likelihoodvalues(:);
|
||
|
||
%
|
||
% Return a function to evaluate the approximation at x (a n×1 vector).
|
||
%
|
||
|
||
f = @(X) parameters(1) + transpose(X)*parameters(2:n+1) + crossproducts(X)*parameters(n+2:end);
|
||
|
||
if nargout>1
|
||
%
|
||
% Return a function to evaluate the gradient of the approximation at x (a n×1 vector)
|
||
%
|
||
|
||
df = @(X) parameters(2:n+1) + dcrossproducts(X)*parameters(n+2:end);
|
||
|
||
if nargout>2
|
||
%
|
||
% Return the hessian matrix of the approximation.
|
||
%
|
||
|
||
d2f = NaN(n,n);
|
||
|
||
h = 1;
|
||
for i=1:n
|
||
for j=i:n
|
||
d2f(i,j) = parameters(n+1+h);
|
||
if ~isequal(j, i)
|
||
d2f(j,i) = d2f(i,j);
|
||
end
|
||
h = h+1;
|
||
end
|
||
end
|
||
|
||
if nargout>3
|
||
%
|
||
% Return a measure of fit goodness
|
||
%
|
||
|
||
R2 = 1-sum((likelihoodvalues(:)-X*parameters).^2)/sum(demean(likelihoodvalues(:)).^2);
|
||
|
||
end
|
||
|
||
end
|
||
end
|
||
|
||
|
||
function XX = crossproducts(X)
|
||
% n n
|
||
% XX*ones(1,(n+1)*n/2) = ∑ xᵢ² + 2 ∑ xᵢxⱼ
|
||
% i=1 i=1
|
||
% j>i
|
||
XX = NaN(columns(X), n*(n+1)/2);
|
||
column = 1;
|
||
for i=1:n
|
||
XX(:,column) = transpose(X(i,:).*X(i,:));
|
||
column = column+1;
|
||
for j=i+1:n
|
||
XX(:,column) = 2*transpose(X(i,:).*X(j,:));
|
||
column = column+1;
|
||
end
|
||
end
|
||
end
|
||
|
||
|
||
function xx = dcrossproducts(x)
|
||
xx = zeros(n, n*(n+1)/2);
|
||
size(xx)
|
||
for i = 1:n
|
||
base = (i-1)*n-sum(0:i-2);
|
||
incol = 1;
|
||
xx(i,base+incol) = 2*x(i);
|
||
for j = i+1:n
|
||
incol = incol+1;
|
||
xx(i,incol) = 2*x(j);
|
||
end
|
||
for j=1:i-1
|
||
base = (j-1)*n-sum(0:j-2)+1;
|
||
colid = base+i-j;
|
||
xx(i,colid) = 2*x(j);
|
||
end
|
||
end
|
||
end
|
||
|
||
end
|