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% STK_EXAMPLE_KB01N Ordinary kriging in 1D, with noisy data
%
% This example shows how to compute ordinary kriging predictions on a
% one-dimensional noisy dataset.
%
% The Gaussian Process (GP) prior is the same as in stk_example_kb01.
%
% The observation noise is Gaussian and homoscedastic (constant variance).
% Its variance is assumed to be known.
%
% Note that the kriging predictor, which is the posterior mean of the GP,
% does NOT interpolate the data in this noisy example.
%
% See also: stk_example_kb01, stk_example_kb02n
% 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>
% 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
%% Dataset
% Load a 1D noisy dataset (homoscedastic Gaussian noise)
[xi, zi, ref] = stk_dataset_twobumps ('noisy1');
% The grid where predictions must be made
xt = ref.xt;
% Reference values on the grid
zt = ref.zt;
stk_figure ('stk_example_kb01n (a)');
stk_plot1d (xi, zi, xt, zt); stk_legend;
stk_title ('True function and observed data');
stk_labels ('input variable x', 'response z');
%% 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');
% NOTE: the suffix '_iso' indicates an ISOTROPIC covariance function, but the
% distinction isotropic / anisotropic is irrelevant here since DIM = 1.
% Parameters for the Matern covariance function
% ("help stk_materncov_iso" for more information)
SIGMA2 = 0.5; % variance parameter
NU = 4.0; % regularity parameter
RHO1 = 0.4; % scale (range) parameter
model.param = log ([SIGMA2; NU; 1/RHO1]);
% It is assumed in this example that the variance of the noise is known
model.lognoisevariance = 2 * log (ref.noise_std);
model
%% Carry out the kriging prediction and display the result
%
% The result of a kriging predicition is provided by stk_predict() in an object
% zp of type stk_dataframe, with two columns: "zp.mean" (the kriging mean) and
% "zp.var" (the kriging variance).
%
% Carry out the kriging prediction at points xt
zp = stk_predict (model, xi, zi, xt);
% Display the result
stk_figure ('stk_example_kb01n (b)');
stk_plot1d (xi, zi, xt, zt, zp); stk_legend;
stk_title ('Kriging prediction based on noisy observations');
stk_labels ('input variable x', 'response z');
%#ok<*NOPTS>
%!test stk_example_kb01n; close all;
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