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% STK_SAMPCRIT_EMMI_EVAL computes the EMMI criterion
%
% CALL: EMMI = stk_sampcrit_emmi_eval (ZP_MEAN, ZP_STD, ZI)
%
% computes the value EMMI of the Expected MaxiMin Improvement (EMMI) for a
% multi-objective minimization problem, with respect to the observed values
% ZI, assuming Gaussian predictive distributions with means ZP_MEAN and
% standard deviations ZP_STD. The value of the criterion is computed
% approximately, using Monte Carlo simulations. The input arguments must
% have the following sizes:
%
% * ZP_MEAN M x P,
% * ZP_STD M x P,
% * ZI N x P,
%
% where M is the number of points where the EMMI must be computed, P the
% number of objective functions to be minimized, and N the current number of
% Pareto optimal solutions. The output has size M x 1.
%
% CALL: EMMI = stk_sampcrit_emmi_eval (ZP_MEAN, ZP_STD, ZI, NSIMU)
%
% allows to change the number of simulations NSIMU used in the calculation of
% the criterion.
%
% NOTE
%
% 1) The result depends only on the non-dominated rows of ZI.
%
% 2) Multi-objective maximization problems, or mixed minimization/maximization
% problems, can be handled by changing the sign of the corresponding
% components of ZP_MEAN and ZI.
%
% 3) Objective functions should be normalized for better performances.
%
% REFERENCES
%
% [1] Svenson, J., & Santner, T. (2016). Multiobjective optimization of
% expensive-to-evaluate deterministic computer simulator models.
% Computational Statistics & Data Analysis, 94, 250-264,
% DOI: 10.1016/j.csda.2015.08.011.
%
% See also: stk_sampcrit_ehvi_eval
% Copyright Notice
%
% Copyright (C) 2017, 2018, 2020 CentraleSupelec
% Copyright (C) 2016 IRT SystemX
%
% Author: Paul Feliot <paul.feliot@irt-systemx.fr>
% Copying Permission Statement
%
% This file is part of
%
% STK: a Small (Matlab/Octave) Toolbox for Kriging
% (https://github.com/stk-kriging/stk/)
%
% STK 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.
%
% STK 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 STK. If not, see <http://www.gnu.org/licenses/>.
function EMMI = stk_sampcrit_emmi_eval (zp_mean, zp_std, zi, nsimu)
% Handle empty zp
if isempty (zp_mean) || isempty (zp_std)
EMMI = [];
return
end
% Handle empty zi
if isempty (zi)
stk_error ('Empty zi.', 'EmptyZi')
else
zi = double (zi);
% Keep only non-dominated points, and remove duplicates
zi = unique (zi(stk_paretofind (zi), :), 'rows');
end
% Size parameters
n = size (zp_mean, 1);
[np, p] = size(zi);
% Number of simulation not provided: Defaults to nsimu = 200 x p
if nargin == 3
nsimu = 200 * p;
end
% The criterion is calculated using Monte Carlo simulation
EMMI = zeros (n, 1);
for i = 1:n
% Simulate normal distribution
simu = bsxfun (@plus, zp_mean(i,1:p), ...
bsxfun (@times, zp_std(i,1:p), randn (nsimu, p)));
% Compute maximin improvement
maximin = min (bsxfun(@minus, simu, zi(1,:)), [], 2);
for k = 2:np
maximin = max (maximin, min (bsxfun(@minus, simu, zi(k,:)), [], 2));
end
isdom = stk_isdominated (simu, zi);
improvement = - maximin .* ~isdom;
% Approximate expectation
EMMI(i) = 1/nsimu * sum (improvement);
end
end % function
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