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% STK_EXAMPLE_KB06 Ordinary kriging VS kriging with a linear trend
%
% The same dataset is analyzed using two variants of kriging.
%
% The left panel shows the result of ordinary kriging, in other words, Gaussian
% process interpolation assuming a constant (but unknown) mean. The right panel
% shows the result of adding a linear trend in the mean of the Gaussian process.
%
% The difference with the left plot is clear in extrapolation: the first predic-
% tor exhibits a "mean reverting" behaviour, while the second one captures an
% increasing trend in the data.
% Copyright Notice
%
% Copyright (C) 2016 CantraleSupelec
% 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_kb06');
%% Preliminaries
DIM = 1; % Dimension of the factor space
BOX = [-2.0; 2.0]; % Factor space
NT = 1e3; % Number of points in the grid
xt = stk_sampling_regulargrid (NT, DIM, BOX); % Construct a regular grid
%% Data
xi = stk_dataframe ([0.00; 0.10; 0.20], {'x'}); % Evaluation points
zi = stk_dataframe ([0.00; 0.09; 0.21], {'z'}); % Evaluation results
%% Default parameters for the Matern covariance
% Parameters used as initial values for stk_param_estim()
SIGMA2 = 1.0; % variance parameter
NU = 2.0; % regularity parameter
RHO1 = 0.4; % scale (range) parameter
param0 = log ([SIGMA2; NU; 1/RHO1]);
%% Ordinary kriging (constant mean)
model = stk_model ('stk_materncov_iso', DIM);
model.lognoisevariance = 2 * log (1e-10);
% model.lm = stk_lm_constant is the default
% Estimate the parameters of the covariance
model.param = stk_param_estim (model, xi, zi, param0);
% Carry out kriging prediction
zp = stk_predict (model, xi, zi, xt);
% Plot the result
stk_subplot (1, 2, 1); stk_plot1d (xi, zi, xt, [], zp);
stk_title ('Ordinary kriging'); ylim ([-5 5]);
%% Linear trend (aka "universal kriging")
% We just need to change the value of 'order' in the model
model.lm = stk_lm_affine;
% Re-estimate the parameters of the covariance
model.param = stk_param_estim (model, xi, zi, param0);
% Carry out kriging prediction
zp = stk_predict (model, xi, zi, xt);
% Plot the result
stk_subplot (1, 2, 2); stk_plot1d (xi, zi, xt, [], zp);
stk_title ('Kriging with linear trend'); ylim ([-5 5]);
%!test stk_example_kb06; close all;
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