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% STK_SAMPCRIT_EHVI_EVAL computes the EHVI criterion
%
% CALL: EHVI = stk_sampcrit_ehvi_eval (ZP_MEAN, ZP_STD, ZI, ZR)
%
% computes the value EHVI of the Expected HyperVolume Improvement (EHVI) for
% a multi-objective minimization problem, with respect to the observed values
% ZI and the reference point ZR, assuming Gaussian predictive distributions
% with means ZP_MEAN and standard deviations ZP_STD. The input arguments
% must have the following sizes:
%
% * ZP_MEAN M x P,
% * ZP_STD M x P,
% * ZI N x P,
% * ZR 1 x P,
%
% where M is the number of points where the EHVI 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.
%
% 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.
%
% REFERENCES
%
% [1] Emmerich, M. T., Giannakoglou, K. C., & Naujoks, B. Single- and
% multiobjective evolutionary optimization assisted by gaussian random
% field metamodels. IEEE Transactions on Evolutionary Computation,
% 10(4), 421-439, 2006.
%
% See also: stk_sampcrit_emmi_eval, stk_sampcrit_ei_eval
% Copyright Notice
%
% Copyright (C) 2015, 2017, 2018 CentraleSupelec
%
% Author: Julien Bect <julien.bect@centralesupelec.fr>
% Copying Permission Statement
%
% This file is part of
%
% STK: a Small (Matlab/Octave) Toolbox for Kriging
% (http://sourceforge.net/projects/kriging)
%
% 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 EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr)
% EHVI with respect to the reference
EIr = stk_distrib_normal_ei (zr, zp_mean, zp_std, 1); % m x p
EHVI = prod (EIr, 2); % m x 1
% Compute signed decomposition wrt to the reference zr
% (note: stk_dominatedhv removes non-dominated points and duplicates from zi)
S = stk_dominatedhv (zi, zr, 1);
if ~ isempty (S.sign)
% Shift rectangle number to third dimension
Rs = shiftdim (S.sign, -2);
Ra = shiftdim (S.xmin', -1);
Rb = shiftdim (S.xmax', -1);
% Number of rectangles
R = size (Ra, 3);
% Deal with BLOCK_SIZE rectangles at a time to avoid OOM
BLOCK_SIZE = ceil (1e7 / (numel (EIr)));
nb_blocks = ceil (R / BLOCK_SIZE);
r2 = 0;
for b = 1:nb_blocks
r1 = r2 + 1;
r2 = min (r1 + BLOCK_SIZE - 1, R);
% Both EIa and EIb will have size m x p x BLOCK_SIZE
EIa = stk_distrib_normal_ei (Ra(:, :, r1:r2), zp_mean, zp_std, 1);
EIb = stk_distrib_normal_ei (Rb(:, :, r1:r2), zp_mean, zp_std, 1);
EHVI = EHVI - sum (bsxfun (@times, ...
Rs(:, :, r1:r2), prod (EIb - EIa, 2)), 3);
end % if
end % if
end % function
%!shared zr, zi
%! zr = [1 1];
%! zi = [0.25 0.75; 0.5 0.5; 0.75 0.25];
%!test % no improvement (1 computation)
%! zp_mean = [0.6 0.6]; zp_std = [0 0];
%! EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr);
%! assert (stk_isequal_tolabs (EHVI, 0, 1e-12));
%!test % guaranteed improvement (1 computation)
%! zp_mean = [0 0]; zp_std = [0 0];
%! EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr);
%! assert (stk_isequal_tolabs (EHVI, 10 * 0.25 ^ 2));
%!test % no improvement again (2 computations)
%! zp_mean = [0.5 0.5; 0.6 0.6]; zp_std = [0 0; 0 0];
%! EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr);
%! assert (stk_isequal_tolabs (EHVI, [0; 0], 1e-12));
%!test % no observation -> EHVI wrt zr
%! zp_mean = [0.6 0.6]; zp_std = 0.01 * [1 1]; zi = [];
%! EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr);
%! assert (stk_isequal_tolabs (EHVI, (1 - 0.6)^2, 1e-12));
%!test % no observation below zr -> EHVI wrt zr
%! zp_mean = [0.6 0.6]; zp_std = 0.01 * [1 1]; zi = [2 2];
%! EHVI = stk_sampcrit_ehvi_eval (zp_mean, zp_std, zi, zr);
%! assert (stk_isequal_tolabs (EHVI, (1 - 0.6)^2, 1e-12));
% FIXME: add MORE unit tests
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