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
|
% STK_PARAM_GLS computes a generalised least squares estimate
%
% CALL: BETA = stk_param_gls (MODEL, XI, ZI)
%
% computes the generalised least squares estimate BETA of the vector of
% coefficients for the linear part of MODEL, where XI and ZI stand for
% the evaluation points and observed responses, respectively.
%
% CALL: [BETA, SIGMA2] = stk_param_gls (MODEL, XI, ZI)
%
% also returns the associated unbiased estimate SIGMA2 of sigma^2, assu-
% ming that the actual covariance matrix of the Gaussian process part of
% the model is sigma^2 K, with K the covariance matrix built from MODEL.
%
% SIGMA2 is actually the "best" unbiased estimate of sigma^2 :
%
% 1
% SIGMA2 = ----- * || ZI - P BETA ||^2_{K^{-1}}
% n - r
%
% where n is the number of observations, r the length of BETA, P the
% design matrix for the linear part of the model, and || . ||_{K^{-1}}
% the norm associated to the positive definite matrix K^{-1}. It is the
% best estimate with respect to the quadratic risk, among all unbiased
% estimates which are quadratic in the residuals.
% Copyright Notice
%
% Copyright (C) 2015, 2016 CentraleSupelec
% Copyright (C) 2014 SUPELEC & A. Ravisankar
% Copyright (C) 2011-2013 SUPELEC
%
% Authors: Julien Bect <julien.bect@centralesupelec.fr>
% Emmanuel Vazquez <emmanuel.vazquez@centralesupelec.fr>
% Ashwin Ravisankar <ashwinr1993@gmail.com>
% 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/>.
function [beta, sigma2, L] = stk_param_gls (model, xi, zi)
n = size (xi, 1);
% Build the covariance matrix and the design matrix
[K, P] = stk_make_matcov (model, xi);
% Cast zi into a double-precision array
zi = double (zi);
% Compute the Generalized Least Squares (GLS) estimate
L = stk_cholcov (K, 'lower');
W = L \ P;
u = L \ zi;
beta = (W' * W) \ (W' * u);
if nargin > 1
% Assuming that the actual covariance matrice is sigma^2 K, compute the
% "best" unbiased estimate of sigma2 (best wrt the quadratic risk, among
% all unbiased estimates which are quadratic in the residuals)
r = length (beta);
sigma2 = 1 / (n - r) * sum ((u - W * beta) .^ 2);
end
end % end function stk_param_gls
%!shared xi, zi, model, beta, sigma2
%! xi = (1:10)'; zi = sin (xi);
%! model = stk_model ('stk_materncov52_iso');
%! model.param = [0.0 0.0];
%!test
%! model.lm = stk_lm_constant ();
%! [beta, sigma2] = stk_param_gls (model, xi, zi);
%!assert (stk_isequal_tolabs (beta, 0.1346064, 1e-6))
%!assert (stk_isequal_tolabs (sigma2, 0.4295288, 1e-6))
%!test
%! model.lm = stk_lm_affine ();
%! [beta, sigma2] = stk_param_gls (model, xi, zi);
%!assert (stk_isequal_tolabs (beta, [0.4728342; -0.0614960], 1e-6))
%!assert (stk_isequal_tolabs (sigma2, 0.4559431, 1e-6))
%!test
%! model.lm = stk_lm_null ();
%! [beta, sigma2] = stk_param_gls (model, xi, zi);
%!assert (isequal (beta, zeros (0, 1)))
%!assert (stk_isequal_tolabs (sigma2, 0.3977993, 1e-6))
|
