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% STK_EXPCOV_ANISO computes the anisotropic exponential covariance function
%
% CALL: K = stk_expcov_aniso (PARAM, X, Y)
%
% computes the covariance matrix K between the sets of locations X and Y,
% using the exponential 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 DIM + 1 elements,
% where DIM is the common number of columns of X and Y:
%
% * PARAM(1) = log (SIGMA ^ 2), where SIGMA is the standard deviation,
%
% * PARAM(1+i) = - log (RHO(i)), where RHO(i) is the range parameter
% for the ith dimension.
%
% CALL: dK = stk_expcov_aniso (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_aniso
% (PARAM, X, Y)).
%
% CALL: K = stk_expcov_aniso (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
% (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 k = stk_expcov_aniso (param, x, y, diff, pairwise)
persistent x0 y0 xs ys param0 pairwise0 D Kx_cache compute_Kx_cache
% process input arguments
x = double (x);
y = double (y);
if nargin < 4, diff = -1; end
if nargin < 5, pairwise = false; end
% check consistency for the number of factors
dim = size (x, 2);
if (size (y, 2) ~= dim)
stk_error ('xi and yi have incompatible sizes.', 'InvalidArgument');
end
% check param
nb_params = dim + 1;
if (numel (param) ~= nb_params)
stk_error ('xi and param have incompatible sizes.', 'InvalidArgument');
else
param = reshape (param, 1, nb_params); % row vector
end
% extract parameters from the "param" vector
Sigma2 = exp (param(1));
invRho = exp (param(2:end));
% check parameter values
if ~ (Sigma2 > 0) || ~ all (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
xs = bsxfun (@times, x, invRho);
ys = bsxfun (@times, y, invRho);
D = stk_dist (xs, ys, pairwise);
% save arguments for the next call
x0 = x; y0 = y; param0 = param; pairwise0 = pairwise;
% recomputation of Kx_cache is required
compute_Kx_cache = true;
end
if diff == -1
%%% compute the value (not a derivative)
k = Sigma2 * stk_rbf_exponential (D, -1);
elseif diff == 1
%%% diff wrt param(1) = log(Sigma2)
k = Sigma2 * stk_rbf_exponential (D, -1);
elseif (diff >= 2) && (diff <= nb_params)
%%% diff wrt param(diff) = - log(invRho(diff-1))
ind = diff - 1;
if compute_Kx_cache || isempty (Kx_cache)
Kx_cache = 1 ./ (D + eps) .* (Sigma2 * stk_rbf_exponential (D, 1));
compute_Kx_cache = false;
end
if pairwise
k = (xs(:, ind) - ys(:, ind)).^2 .* Kx_cache;
else
k = (bsxfun (@minus, xs(:, ind), ys(:, ind)')) .^ 2 .* Kx_cache;
end
else
stk_error ('Incorrect value for the ''diff'' parameter.', ...
'InvalidArgument');
end
end % function
%%
% 1D, 5x5
%!shared param, x, y, K1, K2, K3
%! dim = 1;
%! param = log ([1.0; 2.5]);
%! x = stk_sampling_randunif (5, dim);
%! y = stk_sampling_randunif (6, dim);
%!error K0 = stk_expcov_aniso ();
%!error K0 = stk_expcov_aniso (param);
%!error K0 = stk_expcov_aniso (param, x);
%!test K1 = stk_expcov_aniso (param, x, y);
%!test K2 = stk_expcov_aniso (param, x, y, -1);
%!test K3 = stk_expcov_aniso (param, x, y, -1, false);
%!assert (isequal (K1, K2));
%!assert (isequal (K1, K3));
%!test % df versus ordinary array
%! u = double (x); v = double (y);
%! K1 = stk_expcov_aniso (param, u, v, -1);
%! K2 = stk_expcov_aniso (param, stk_dataframe (u), stk_dataframe (v), -1);
%!error stk_expcov_aniso (param, x, y, -2);
%!test stk_expcov_aniso (param, x, y, -1);
%!error stk_expcov_aniso (param, x, y, 0);
%!test stk_expcov_aniso (param, x, y, 1);
%!test stk_expcov_aniso (param, x, y, 2);
%!error stk_expcov_aniso (param, x, y, 3);
%!error stk_expcov_aniso (param, x, y, nan);
%!error stk_expcov_aniso (param, x, y, inf);
%%
% 3D, 4x10
%!shared dim, param, x, y, nx, ny
%! dim = 3;
%! param = log ([1.0; 2.5; 2.4; 2.6]);
%! nx = 4; ny = 10;
%! x = stk_sampling_randunif (nx, dim);
%! y = stk_sampling_randunif (ny, dim);
%!test
%! K1 = stk_expcov_aniso (param, x, y);
%! K2 = stk_expcov_aniso (param, x, y, -1);
%! assert (isequal (size(K1), [nx ny]));
%! assert (stk_isequal_tolabs (K1, K2));
%!test
%! for i = 1:(dim + 1),
%! dK = stk_expcov_aniso (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_expcov_aniso (param, x, y);
%! K2 = stk_expcov_aniso (param, x, y, -1, true);
%! assert (isequal (size (K1), [n n]));
%! assert (stk_isequal_tolabs (K2, diag (K1)));
%!
%! for i = 1:(dim + 1),
%! dK1 = stk_expcov_aniso (param, x, y, i);
%! dK2 = stk_expcov_aniso (param, x, y, i, true);
%! assert (isequal (size (dK1), [n n]));
%! assert (stk_isequal_tolabs (dK2, diag (dK1)));
%! end
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