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% STK_GENERATE_SAMPLEPATHS generates sample paths of a Gaussian process
%
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XT)
%
% generates one sample path ZSIM of the Gaussian process MODEL discretized on
% the evaluation points XT. The input argument XT can be either a numerical
% matrix or a dataframe. The output argument ZSIM has the same number of
% rows as XT. More precisely, on a factor space of dimension DIM,
%
% * XT must have size NS x DIM,
% * ZSIM will have size NS x 1,
%
% where NS is the number of simulation points.
%
% Note that, in the case where MODEL is a model for noisy observations, this
% function simulates sample paths of the underlying (latent) Gaussian
% process, i.e., noiseless observations.
%
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XT, NB_PATHS)
%
% generates NB_PATHS sample paths at once. In this case, the output argument
% ZSIM has size NS x NB_PATHS.
%
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XI, ZI, XT)
%
% generates one sample path ZSIM, using the kriging model MODEL and the
% evaluation points XT, conditional on the evaluations (XI, ZI).
%
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XI, ZI, XT, NB_PATHS)
%
% generates NB_PATHS conditional sample paths at once.
%
% NOTE: Sample size limitation
%
% This function generates (discretized) sample paths using a Cholesky
% factorization of the covariance matrix, and is therefore restricted to
% moderate values of the number of evaluation points.
%
% NOTE: Output type
%
% The output argument ZSIM is a plain (double precision) numerical array,
% even if XT is a data frame. Row names can be added afterwards as follows:
%
% ZSIM = stk_generate_samplepaths (MODEL, XT);
% ZSIM = stk_dataframe (ZSIM, {}, XT.rownames);
%
% EXAMPLES: see stk_example_kb05, stk_example_kb07
%
% See also stk_conditioning, stk_cholcov
% Copyright Notice
%
% Copyright (C) 2015-2018, 2021 CentraleSupelec
% Copyright (C) 2011-2014 SUPELEC
%
% Authors: Julien Bect <julien.bect@centralesupelec.fr>
% Emmanuel Vazquez <emmanuel.vazquez@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 zsim = stk_generate_samplepaths (model, varargin)
switch nargin
case {0, 1}
stk_error ('Not enough input arguments.', 'NotEnoughInputArgs');
case 2
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XT)
xt = varargin{1};
nb_paths = 1;
conditional = false;
case 3
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XT, NB_PATHS)
xt = varargin{1};
nb_paths = varargin{2};
conditional = false;
case 4
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XI, ZI, XT)
xi = varargin{1};
zi = varargin{2};
xt = varargin{3};
nb_paths = 1;
conditional = true;
otherwise
% CALL: ZSIM = stk_generate_samplepaths (MODEL, XI, ZI, XT, NB_PATHS)
xi = varargin{1};
zi = varargin{2};
xt = varargin{3};
nb_paths = varargin{4};
conditional = true;
end
%--- Process input arguments ---------------------------------------------------
xt = double (xt);
% Check nb_paths argument
nb_paths = double (nb_paths);
if ~ isscalar (nb_paths) || ~ (nb_paths > 0)
stk_error ('nb_paths must be a positive scalar', 'Invalid argument');
end
%--- Extend xt with the observation points, if needed --------------------------
if conditional
% Keep only numerical data for xi, zi
xi = double (xi);
zi = double (zi);
% Conditioning by kriging => we must simulate on the observation points too
xt = [xi; xt];
xi_ind = 1:(size (xi, 1));
end
% FIXME: Avoid duplicating observations points if xi is a subset of xt
%--- Generate unconditional sample paths --------------------------------------
% Pick unique simulation points
[xt_unique, ~, j] = unique (xt, 'rows');
% Did we actually find duplicates in xt ?
duplicates_detected = (size (xt_unique, 1) < size (xt, 1));
% Compute the covariance matrix
% (even if there no duplicates, it is not guaranteed
% that xt_unique and xt are equal)
if duplicates_detected
K = stk_make_matcov (model, xt_unique, xt_unique);
else
K = stk_make_matcov (model, xt, xt);
end
% Cholesky factorization of the covariance matrix
V = stk_cholcov (K);
% Generates samplepaths
zsim = V' * randn (size (K, 1), nb_paths);
% Duplicate simulated values, if necessary
if duplicates_detected, zsim = zsim(j, :); end
%--- Generate conditional sample paths ----------------------------------------
if conditional
% Carry out the kriging prediction at points xt
[~, lambda] = stk_predict (model, xi, zi, xt);
if ~ stk_isnoisy (model)
% Simulate sample paths conditioned on noiseless observations
zsim = stk_conditioning (lambda, zi, zsim, xi_ind);
else % Noisy case
% Simulate noise values
noise_sim = stk_simulate_noise (model, xi, nb_paths);
% Simulate sample paths conditioned on noisy observations
zsim = stk_conditioning (lambda, zi, zsim, xi_ind, noise_sim);
end
% TEMPORARY FIX (until stk_conditioning is fixed as well)
zsim = double (zsim);
zsim(xi_ind, :) = [];
end
end % function
%!shared model, xi, zi, xt, n, nb_paths
%! dim = 1; n = 50; nb_paths = 5;
%! model = stk_model (@stk_materncov32_iso, dim);
%! model.param = log ([1.0; 2.9]);
%! xt = stk_sampling_regulargrid (n, dim, [-1.0; 1.0]);
%! xi = [xt(1, :); xt(end, :)]; zi = [0; 0];
%!error zsim = stk_generate_samplepaths ();
%!error zsim = stk_generate_samplepaths (model);
%!test zsim = stk_generate_samplepaths (model, xt);
%!test zsim = stk_generate_samplepaths (model, xt, nb_paths);
%!test zsim = stk_generate_samplepaths (model, xi, zi, xt);
%!test zsim = stk_generate_samplepaths (model, xi, zi, xt, nb_paths);
%!test
%! zsim = stk_generate_samplepaths (model, xt);
%! assert (isequal (size (zsim), [n, 1]));
%!test
%! zsim = stk_generate_samplepaths (model, xt, nb_paths);
%! assert (isequal (size (zsim), [n, nb_paths]));
%!test % duplicate simulation points
%! zsim = stk_generate_samplepaths (model, [xt; xt], nb_paths);
%! assert (isequal (size (zsim), [2 * n, nb_paths]));
%! assert (isequal (zsim(1:n, :), zsim((n + 1):end, :)));
%!test % simulation points equal to observation points (noiseless model)
%! % https://sourceforge.net/p/kriging/tickets/14/
%! zsim = stk_generate_samplepaths (model, xt, zeros (n, 1), xt);
%! assert (isequal (zsim, zeros (n, 1)));
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