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% STK_RBF_EXPONENTIAL computes the exponential correlation function
%
% CALL: K = stk_rbf_exponential (H)
%
% computes the value of the exponential correlation function at distance H:
%
% K = exp (- sqrt(2) |H|).
%
% Note that this correlation function is a special of the Matern correlation
% function (NU = 1/2).
%
% CALL: K = stk_rbf_exponential (H, DIFF)
%
% computes the derivative of the exponential correlation function with
% respect the H if DIFF is equal to 1, and simply returns the value of the
% exponential correlation function if DIFF <= 0 (in which case it is
% equivalent to K = stk_rbf_exponential (H)).
%
% ADMISSIBILITY
%
% The exponential correlation is a valid correlation function for all
% dimensions.
%
% REMARK
%
% The constant sqrt (2) is consistent with the definition of the Matern
% correlation function in STK. Other references may use different constants.
%
% See also: stk_rbf_matern, stk_rbf_matern32, stk_rbf_matern52
% Copyright Notice
%
% Copyright (C) 2016, 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 k = stk_rbf_exponential (h, diff)
C = 1.4142135623730951; % 2 * sqrt (Nu) with Nu = 1/2
% value of the exponential correlation at h
k = exp (- C * abs (h));
% for diff <= 0, we return the value of the correlation function; otherwise...
if (nargin > 1) && (diff > 0)
% for diff == 1, we return the derivative
if diff == 1
b = (k > 0);
% convention: k'(0) = 0, even though k is not derivable at h=0
k(b) = - C * (sign (h(b))) .* k(b);
else
error ('incorrect value for diff.');
end
end
if any (isnan (k))
keyboard
end
end % function
%!shared h, diff
%! h = 1.0; diff = -1;
%!error stk_rbf_exponential ();
%!test stk_rbf_exponential (h);
%!test stk_rbf_exponential (h, diff);
%!test %% h = 0.0 => correlation = 1.0
%! x = stk_rbf_exponential (0.0);
%! assert (stk_isequal_tolrel (x, 1.0, 1e-8));
%!test %% check derivative numerically
%! h = [-1 -0.5 -0.1 0.1 0.5 1]; delta = 1e-9;
%! d1 = (stk_rbf_exponential (h + delta) - stk_rbf_exponential (h)) / delta;
%! d2 = stk_rbf_exponential (h, 1);
%! assert (stk_isequal_tolabs (d1, d2, 1e-4));
%!test %% consistency with stk_rbf_matern: function values
%! for h = 0.1:0.1:2.0,
%! x = stk_rbf_matern (1/2, h);
%! y = stk_rbf_exponential (h);
%! assert (stk_isequal_tolrel (x, y, 1e-8));
%! end
%!test %% consistency with stk_rbf_matern: derivatives
%! for h = 0.1:0.1:2.0,
%! x = stk_rbf_matern (1/2, h, 2);
%! y = stk_rbf_exponential (h, 1);
%! assert (stk_isequal_tolrel (x, y, 1e-8));
%! end
%!assert (stk_rbf_exponential (inf) == 0)
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