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% STK_EXAMPLE_KB05 Generation of conditioned sample paths
%
% A Matern Gaussian process model is used, with constant but unknown mean
% (ordinary kriging) and known covariance parameters.
%
% Given noiseless observations from the unknown function, a batch of conditioned
% sample paths is drawn using the "conditioning by kriging" technique. In short,
% this means that unconditioned sample path are simulated first (using
% stk_generate_samplepaths), and then conditioned on the observations by kriging
% (using stk_conditioning).
%
% Note: in this example, for pedagogical purposes, conditioned samplepaths are
% simulated in two steps: first, unconditioned samplepaths are simulated;
% second, conditioned samplepaths are obtained using conditioning by kriging.
% In practice, these two steps can be carried out all at once using
% stk_generate_samplepath (see, e.g., stk_example_kb09).
%
% See also: stk_generate_samplepaths, stk_conditioning, stk_example_kb09
% Copyright Notice
%
% Copyright (C) 2016, 2018 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
% (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/>.
stk_disp_examplewelcome; stk_figure ('stk_example_kb05');
%% Dataset
% Load a 1D noiseless dataset
[xi, zi, ref] = stk_dataset_twobumps ('noiseless');
% The grid where predictions must be made
xt = ref.xt;
% Reference values on the grid
zt = ref.zt;
% Indices of the evaluation points xi in the grid
xi_ind = ref.xi_ind;
%% Specification of the model
%
% We choose a Matern covariance with "fixed parameters" (in other
% words, the parameters of the covariance function are provided by the user
% rather than estimated from data).
%
% The following line defines a model with a constant but unknown mean
% (ordinary kriging) and a Matern covariance function. (Some default
% parameters are also set, but we override them below.)
model = stk_model ('stk_materncov_iso');
% Parameters for the Matern covariance
% ("help stk_materncov_iso" for more information)
SIGMA2 = 1.0; % variance parameter
NU = 4.0; % regularity parameter
RHO1 = 0.4; % scale (range) parameter
model.param = log ([SIGMA2; NU; 1/RHO1]);
%% Generate (unconditional) sample paths
NB_PATHS = 10;
zsim = stk_generate_samplepaths (model, xt, NB_PATHS);
% Display the result
stk_subplot (2, 2, 1); plot (xt, zsim, 'LineWidth', 2);
stk_title ('Unconditional sample paths');
stk_labels ('input variable x', 'response z');
%% Carry out the kriging prediction and generate conditional sample paths
% Carry out the kriging prediction at points xt
[zp, lambda] = stk_predict (model, xi, zi, xt);
% Condition sample paths on the observations
zsimc = stk_conditioning (lambda, zi, zsim, xi_ind);
% Display the observations only
stk_subplot (2, 2, 2); stk_plot1d (xi, zi);
stk_title ('Observations');
stk_labels ('input variable x', 'response z');
% Display the conditional sample paths
stk_subplot (2, 2, 3); plot (xt, zsimc, 'LineWidth', 2); hold on;
plot (xi, zi, 'ko', 'MarkerSize', 6, 'MarkerFaceColor', 'k');
stk_title ('Conditional sample paths');
stk_labels ('input variable x', 'response z');
% Display the kriging and credible intervals
stk_subplot (2, 2, 4); stk_plot1d (xi, zi, xt, zt, zp, zsimc);
stk_title ('Prediction and credible intervals');
stk_labels ('input variable x', 'response z');
%!test stk_example_kb05; close all;
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