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% STK_MATERNCOV_ISO computes the isotropic Matern covariance
%
% CALL: K = stk_materncov_iso (PARAM, X, Y)
%
% computes the covariance matrix K between the sets of locations X and Y,
% using the isotropic Matern covariance function with parameters PARAM. The
% output matrix K has size NX x NY, where NX is the number of rows in X
% and NY the number of rows in Y. The vector of parameters must have
% three elements:
%
% * PARAM(1) = log (SIGMA ^ 2), where SIGMA is the standard deviation,
%
% * PARAM(2) = log(NU), where NU is the regularity parameter,
%
% * PARAM(3) = - log (RHO), where RHO is the range parameter.
%
% CALL: dK = stk_materncov_iso (PARAM, X, Y, DIFF)
%
% computes the derivative of the covariance matrix with respect to
% PARAM(DIFF) if DIFF~= -1, or the covariance matrix itself if DIFF is
% equal to -1 (in which case this is equivalent to stk_materncov_iso
% (PARAM, X, Y)).
%
% CALL: K = stk_materncov_iso (PARAM, X, Y, DIFF, PAIRWISE)
%
% computes the covariance vector (or a derivative of it if DIFF > 0)
% between the sets of locations X and Y. The output K is a vector of
% length N, where N is the common number of rows of X and Y.
% Copyright Notice
%
% Copyright (C) 2015, 2016, 2018 CentraleSupelec
% Copyright (C) 2011-2014 SUPELEC
%
% Authors: Julien Bect <julien.bect@centralesupelec.fr>
% Emmanuel Vazquez <emmanuel.vazquez@centralesupelec.fr>
% Paul Feliot <paul.feliot@irt-systemx.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_materncov_iso (param, x, y, diff, pairwise)
persistent x0 y0 param0 pairwise0 D
% process input arguments
x = double (x);
y = double (y);
if nargin < 4, diff = -1; end
if nargin < 5, pairwise = false; end
% extract parameters from the "param" vector
Sigma2 = exp (param(1));
Nu = exp (param(2));
invRho = exp (param(3));
% check parameter values
if ~ (Sigma2 > 0) || ~ (Nu > 0) || ~ (invRho >= 0),
error ('Incorrect parameter value.');
end
% check if all input arguments are the same as before
% (or if this is the first call to the function)
if isempty (x0) || isempty (y0) || isempty (param0) ...
|| ~ isequal ({x, y, param}, {x0, y0, param0}) ...
|| ~ isequal (pairwise, pairwise0)
% compute the distance matrix
D = invRho * stk_dist (x, y, pairwise);
% save arguments for the nex call
x0 = x; y0 = y; param0 = param; pairwise0 = pairwise;
end
if diff == -1,
%%% compute the value (not a derivative)
k = Sigma2 * stk_rbf_matern (Nu, D, -1);
elseif diff == 1,
%%% diff wrt param(1) = log(Sigma2)
k = Sigma2 * stk_rbf_matern (Nu, D, -1);
elseif diff == 2,
%%% diff wrt param(2) = log(Nu)
k = Nu * Sigma2 * stk_rbf_matern (Nu, D, 1);
elseif diff == 3,
%%% diff wrt param(3) = - log(invRho)
k = D .* (Sigma2 * stk_rbf_matern (Nu, D, 2));
else
error ('there must be something wrong here !');
end
end % function
%%
% 1D, 5x5
%!shared param, x, y
%! dim = 1;
%! param = log ([1.0; 1.5; 2.9]);
%! x = stk_sampling_randunif(5, dim);
%! y = stk_sampling_randunif(5, dim);
%!error stk_materncov_iso();
%!error stk_materncov_iso(param);
%!error stk_materncov_iso(param, x);
%!test stk_materncov_iso(param, x, y);
%!test stk_materncov_iso(param, x, y, -1);
%!test stk_materncov_iso(param, x, y, -1, false);
%!error stk_materncov_iso(param, x, y, -2);
%!test stk_materncov_iso(param, x, y, -1);
%!error stk_materncov_iso(param, x, y, 0);
%!test stk_materncov_iso(param, x, y, 1);
%!test stk_materncov_iso(param, x, y, 2);
%!test stk_materncov_iso(param, x, y, 3);
%!error stk_materncov_iso(param, x, y, 4);
%!error stk_materncov_iso(param, x, y, nan);
%!error stk_materncov_iso(param, x, y, inf);
%%
% 3D, 4x10
%!shared dim, param, x, y, nx, ny
%! dim = 3;
%! param = log ([1.0; 1.5; 2.9]);
%! nx = 4; ny = 10;
%! x = stk_sampling_randunif (nx, dim);
%! y = stk_sampling_randunif (ny, dim);
%!test
%! K1 = stk_materncov_iso (param, x, y);
%! K2 = stk_materncov_iso (param, x, y, -1);
%! assert (isequal (size (K1), [nx ny]));
%! assert (stk_isequal_tolabs (K1, K2));
%!test
%! for i = 1:3,
%! dK = stk_materncov_iso (param, x, y, i);
%! assert (isequal (size (dK), [nx ny]));
%! end
%!test
%! n = 7;
%! x = stk_sampling_randunif(n, dim);
%! y = stk_sampling_randunif(n, dim);
%!
%! K1 = stk_materncov_iso(param, x, y);
%! K2 = stk_materncov_iso(param, x, y, -1, true);
%! assert(isequal(size(K1), [n n]));
%! assert(stk_isequal_tolabs(K2, diag(K1)));
%!
%! for i = 1:3,
%! dK1 = stk_materncov_iso(param, x, y, i);
%! dK2 = stk_materncov_iso(param, x, y, i, true);
%! assert(isequal(size(dK1), [n n]));
%! assert(stk_isequal_tolabs(dK2, diag(dK1)));
%! end
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