1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
|
% STK_EXAMPLE_MISC02 How to use priors on the covariance parameters
%
% A Matern covariance in dimension one is considered as an example. A Gaussian
% prior is used for all three parameters: log-variance, log-regularity and log-
% inverse-range. The corresponding parameter estimates are Maximum A Posteriori
% (MAP) estimates or, more precisely, Restricted MAP (ReMAP) estimates.
%
% Several values for the variance of the prior are successively considered, to
% illustrate the effect of this prior variance on the parameter estimates. When
% the variance is small, the MAP estimate is close to the mode of the prior. On
% the other hand, when the variance is large, the prior becomes "flat" and the
% MAP estimate is close to the ReML estimate (see figure b).
% Copyright Notice
%
% Copyright (C) 2012-2014 SUPELEC
%
% Author: Julien Bect <julien.bect@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
%% DEFINE AN ARTIFICIAL ONE-DIMENSIONAL DATASET
DIM = 1; % dimension of the factor space
BOX = [0.0; 1.0]; % factor space
xi = [0.25; 0.26; 0.50; 0.60];
zi = [1.00; 1.10; 0.20; 0.35];
%% SPECIFICATION OF THE MODEL & REML ESTIMATION
model = stk_model ('stk_materncov_iso');
% small "regularization noise".
model.lognoisevariance = 2 * log (1e-6);
% Mode of the prior on the parameters of the Matern covariance
% (also serves as an initial guess for the optimization)
SIGMA2 = 1.0; % variance parameter
NU = 2.0; % regularity parameter
RHO1 = 0.4; % scale (range) parameter
param0 = log ([SIGMA2; NU; 1/RHO1]);
% Estimate covariance parameters (without prior)
model.param = stk_param_estim (model, xi, zi, param0);
param_opt_reml = model.param;
%% EXPERIMENT WITH SEVERAL VALUES FOR PRIOR VARIANCES
std_list = [10 1 0.2 0.01];
NT = 400; % nb of points in the grid
xt = stk_sampling_regulargrid (NT, DIM, BOX);
stk_figure ('stk_example_misc02 (a)');
param_opt = zeros (length (param0), length (std_list));
for k = 1:length (std_list),
% Prior on the parameters of the Matern covariance
model.prior.mean = param0;
model.prior.invcov = eye (length (param0)) ./ (std_list(k)^2);
% Estimate covariance parameters (with a prior)
model.param = stk_param_estim (model, xi, zi, param0);
param_opt(:, k) = model.param;
% Carry out kriging prediction
zp = stk_predict (model, xi, zi, xt);
% Plot predicted values and pointwise confidences intervals
stk_subplot (2, 2, k); stk_plot1d (xi, zi, xt, [], zp);
stk_labels ('input x', 'predicted output z');
stk_title (sprintf ('prior std = %.2f', std_list(k)));
end
%% FIGURE: ESTIMATED PARAMETER VERSUS PRIOR STD
stk_figure ('stk_example_misc02 (b)');
param_name = {'SIGMA2', 'NU', '1/RHO'};
for j = 1:3,
stk_subplot (2, 2, j);
% estimated parameter versus prior std
h = semilogx (std_list, exp (param_opt(j, :)), 'ko-');
set (h, 'LineWidth', 2, 'MarkerFaceColor', 'y');
stk_labels ('prior std', param_name{j});
% add an horizontal line showing the value of REML estimation
hold on; semilogx (xlim, exp (param_opt_reml(j)) * [1 1], 'r--');
% add a second horizontal line showing the mode of the prior
hold on; semilogx (xlim, exp (param0(j)) * [1 1], 'b--');
% adjust ylim
yy = exp ([param_opt(j, :) param_opt_reml(j) param0(j)]);
ylim_min = min (yy); ylim_max = max (yy); delta = ylim_max - ylim_min;
ylim ([ylim_min - 0.05*delta ylim_max + 0.05*delta]);
end
if ~ strcmp (graphics_toolkit (), 'gnuplot')
h1 = legend ('MAP estimates', 'REML estimate', 'mode of the prior');
h2 = stk_subplot (2, 2, 4); axis off;
set (h1, 'Position', get (h2, 'Position'));
end
%!test stk_example_misc02; close all;
|
