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% STK_PHIPCRIT computes the "phi_p" criterion of Morris & Mitchell
%
% CALL: D = stk_phipcrit (X, P)
%
% computes the phi_P criterion on the set of points X, which is defined for
% an n x d array X as
%
% D = (sum_{1 <= i < j <= n} d_ij ^ (-p)) ^ (1/p)
%
% where d_ij is the Euclidean distance in R^d between X(i,:) and X(j,:).
%
% CALL: D = stk_phipcrit (X)
%
% computes the phi_P criterion with P = 50.
%
% NOTES:
%
% * In the special case P = 2, this criterion has first been introduced by
% Audze & Eglais (1977).
%
% * When p -> +Inf, the value of the phi_p criterion tends to the inverse of
% the mindist criterion. The phi_p criterion with a high value of p is
% often used in place of the mindist criterion for its being easier to
% optimize. Morris & Mitchell recommend using p in the range 20-50 for this
% purpose.
%
% REFERENCES
%
% [1] Max D. Morris and Toby J. Mitchell, "Exploratory Designs for Computer
% Experiments", Journal of Statistical Planning and Inference,
% 43(3):381-402, 1995.
%
% [2] P. Audze and V. Eglais, "New approach for planning out experiments",
% Problems of Dynamics and Strengths, 35:104-107, 1977.
%
% [3] Luc Pronzato and Werner G. Muller, "Design of computer
% experiments: space filling and beyond", Statistics and Computing,
% 22(3):681-701, 2012.
%
% [4] G. Damblin, M. Couplet and B. Iooss, "Numerical studies of space filling
% designs: optimization of Latin hypercube samples and subprojection
% properties", Journal of Simulation, in press.
%
% See also: stk_mindist, stk_filldist
% Copyright Notice
%
% Copyright (C) 2017, 2018 CentraleSupelec
% Copyright (C) 2013, 2014 SUPELEC
%
% Author: Julien Bect <julien.bect@centralesupelec.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 phi = stk_phipcrit (x, p)
if nargin < 2,
p = 50;
end
% compute the distance matrix
D = stk_dist (x);
% compute mindist
D = D + diag (inf (1, size (x, 1)));
z = min (D(:));
% compute the value of the criterion
if z > 0
tmp = triu ((D / z) .^ (-p), 1);
phi = 1 / z * sum(tmp(:)) .^ (1/p);
else
phi = Inf;
end
end % function
%!shared x
%! x = [0, 0.2, 0.4, 0.6, 0.8, 1.0;
%! 0, 0.6, 0.8, 1.0, 0.2, 0.4]';
%!assert (stk_isequal_tolabs ...
%! (stk_phipcrit (x, 10), 3.946317664423303, 1e-15))
%!assert (stk_isequal_tolabs ...
%! (stk_phipcrit (x, 50), 3.614077252813102, 1e-15));
%!assert (stk_isequal_tolabs ...
%! (stk_phipcrit (x, 100), 3.574589859827413, 1e-15));
%!assert (stk_isequal_tolabs ...
%! (stk_phipcrit (x, 1e9), 1 / stk_mindist (x), 1e-8));
%!assert (isequal (stk_phipcrit (ones (2)), Inf));
% library (DiceDesign) # load DiceDesign 1.2
% options (digits = 16) # display 16 significat digits
%
% x <- data.frame (x1 = c(0, 0.2, 0.4, 0.6, 0.8, 1.0),
% x2 = c(0, 0.6, 0.8, 1.0, 0.2, 0.4))
%
% phiP (x, 10) # 3.946317664423303
% phiP (x, 50) # 3.614077252813102
% phiP (x, 100) # 3.574589859827413
% phiP (x, 1000) # Inf, but we can do better
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