Approximation of the likelihood with a second order multivariate polynomial.
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function [f, df, d2f, R2] = likelihood_quadratic_approximation(particles, likelihoodvalues)
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% Approximate the shape of the likelihood function with a multivariate second order polynomial.
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%
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%
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% INPUTS
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% - particles [double] n×p matrix of (p) particles around the estimated posterior mode.
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% - likelihoodvalues [double] p×1 vector of corresponding values for the likelihood (or posterior kernel).
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%
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% OUTPUTS
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% - f [handle] function handle for the approximated likelihood.
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% - df [handle] function handle for the gradient of the approximated likelihood.
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% - d2f [handle] Hessian matrix of the approximated likelihood (constant since we consider a second order multivariate polynomial)
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% - R2 [double] scalar, goodness of fit measure.
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%
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% REMARKS
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% [1] Function f takes a n×m matrix as input argument (the approximated likelihood is evaluated in m points) and returns a m×1 vector.
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% [2] Funtion df takes a n×1 vector as input argument (the point where the gradient is computed) and returns a n×1 vector.
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% Copyright © 2024 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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n = rows(particles); % Number of parmaeters
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p = columns(particles); % Number of particles
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q = 1 + n + n*(n+1)/2; % Number of regressors (with a constant)
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if p<=q
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error('Quadratic approximation requires more than %u particles.', q)
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end
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%
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% Build the set of regressors.
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%
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X = NaN(p, q);
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X(:,1) = 1; % zero order term
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X(:,2:n+1) = transpose(particles); % first order terms
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X(:,n+2:end) = crossproducts(particles); % second order terms
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%
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% Perform the regression
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%
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parameters = X\likelihoodvalues(:);
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%
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% Return a function to evaluate the approximation at x (a n×1 vector).
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%
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f = @(X) parameters(1) + transpose(X)*parameters(2:n+1) + crossproducts(X)*parameters(n+2:end);
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if nargout>1
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%
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% Return a function to evaluate the gradient of the approximation at x (a n×1 vector)
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%
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df = @(X) parameters(2:n+1) + dcrossproducts(X)*parameters(n+2:end);
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if nargout>2
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%
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% Return the hessian matrix of the approximation.
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%
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d2f = NaN(n,n);
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h = 1;
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for i=1:n
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for j=i:n
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d2f(i,j) = parameters(n+1+h);
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if ~isequal(j, i)
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d2f(j,i) = d2f(i,j);
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end
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h = h+1;
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end
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end
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if nargout>3
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%
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% Return a measure of fit goodness
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%
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R2 = 1-sum((likelihoodvalues(:)-X*parameters).^2)/sum(demean(likelihoodvalues(:)).^2);
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end
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end
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end
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function XX = crossproducts(X)
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% n n
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% XX*ones(1,(n+1)*n/2) = ∑ xᵢ² + 2 ∑ xᵢxⱼ
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% i=1 i=1
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% j>i
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XX = NaN(columns(X), n*(n+1)/2);
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column = 1;
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for i=1:n
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XX(:,column) = transpose(X(i,:).*X(i,:));
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column = column+1;
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for j=i+1:n
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XX(:,column) = 2*transpose(X(i,:).*X(j,:));
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column = column+1;
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end
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end
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end
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function xx = dcrossproducts(x)
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xx = zeros(n, n*(n+1)/2);
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for i = 1:n
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base = (i-1)*n-sum(0:i-2);
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incol = 1;
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xx(i,base+incol) = 2*x(i);
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for j = i+1:n
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incol = incol+1;
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xx(i,incol) = 2*x(j);
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end
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for j=1:i-1
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base = (j-1)*n-sum(0:j-2)+1;
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colid = base+i-j;
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xx(i,colid) = 2*x(j);
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end
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end
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end
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return % --*-- Unit tests --*--
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%@test:1
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% Create data
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X = randn(10,1000);
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Y = 1 + rand(1,10)*X.^2+0.01*randn(1,1000);
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% Perform approximation
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try
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[f,df, d2f,R2] = likelihood_quadratic_approximation(X,Y);
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t(1) = true;
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catch
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t(1) = false;
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end
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% Test returned arguments
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if t(1)
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try
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y = f(randn(10,100));
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t(2) = true;
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if ~(rows(y)==100 && columns(y)==1)
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t(2) = false;
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end
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catch
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t(2) = false;
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end
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try
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dy = df(zeros(10,1));
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t(3) = true;
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if ~(rows(dy)==10 && columns(dy)==1)
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t(3) = false;
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end
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catch
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t(3) = false;
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end
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t(4) = true;
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if ~(rows(d2f)==10 && columns(d2f)==10)
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t(4) = false
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end
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if ~(rows(d2f)==10 && columns(d2f)==10)
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t(4) = false
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end
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t(4) = issymmetric(d2f);
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t(4) = ispd(d2f);
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t(5) = isscalar(R2);
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t(5) = (R2>0) & (R2<1); % Note that in a nonlinear model nothing ensures that these inequalities are satisfied.
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end
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T = all(t);
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%@eof:1
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end
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