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
137
138
139
|
% STK_DISTRIB_STUDENT_EI computes the Student expected improvement
%
% CALL: EI = stk_distrib_student_ei (Z, NU)
%
% computes the expected improvement of a Student random variable with NU
% degrees of freedom above the threshold Z.
%
% CALL: EI = stk_distrib_student_ei (Z, NU, MU, SIGMA)
%
% computes the expected improvement of a Student random variable with NU
% degrees of freedom, location parameter MU and scale parameter SIGMA,
% above the threshold Z.
%
% CALL: EI = stk_distrib_student_ei (Z, NU, MU, SIGMA, MINIMIZE)
%
% computes the expected improvement of a Student random variable with NU
% degrees of freedom, location parameter MU and scale parameter SIGMA,
% below the threshold Z if MINIMIZE is true, above the threshold Z
% otherwise.
%
% REFERENCES
%
% [1] R. Benassi, J. Bect and E. Vazquez. Robust Gaussian process-based
% global optimization using a fully Bayesian expected improvement
% criterion. In: Learning and Intelligent Optimization (LION 5),
% LNCS 6683, pp. 176-190, Springer, 2011
%
% [2] B. Williams, T. Santner and W. Notz. Sequential Design of Computer
% Experiments to Minimize Integrated Response Functions. Statistica
% Sinica, 10(4):1133-1152, 2000.
%
% See also stk_distrib_normal_ei
% Copyright Notice
%
% Copyright (C) 2018 CentraleSupelec
% Copyright (C) 2013, 2014 SUPELEC
%
% Authors: Julien Bect <julien.bect@centralesupelec.fr>
% Romain Benassi <romain.benassi@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 ei = stk_distrib_student_ei (z, nu, mu, sigma, minimize)
nu(nu < 0) = nan;
if nargin > 2,
delta = bsxfun (@minus, mu, z);
else
% Default: mu = 0;
delta = - z;
end
if nargin > 3,
sigma(sigma < 0) = nan;
else
% Default
sigma = 1;
end
% Default: compute the EI for a maximization problem
if nargin > 4,
minimize = logical (minimize);
else
minimize = false;
end
% Reduce to the maximization case
if minimize,
delta = - delta;
end
[delta, nu, sigma] = stk_commonsize (delta, nu, sigma);
ei = nan (size (delta));
b0 = ~ (isnan (delta) | isnan (nu) | isnan (sigma));
b1 = (nu > 1);
b2 = (sigma > 0);
% The EI is infinite for nu <= 1
ei(b0 & (~ b1)) = +inf;
b0 = b0 & b1;
% Compute the EI where nu > 1 and sigma > 0
b = b0 & b2;
if any (b)
u = delta(b) ./ sigma(b); nu = nu(b);
ei(b) = sigma(b) .* ((nu + u .^ 2) ./ (nu - 1) ...
.* stk_distrib_student_pdf (u, nu) ...
+ u .* stk_distrib_student_cdf (u, nu));
end
% Compute the EI where nu > 1 and sigma == 0
b = b0 & (~ b2);
ei(b) = max (0, delta(b));
% Correct numerical inaccuracies
ei(ei < 0) = 0;
end % function
%!assert (stk_isequal_tolrel (stk_distrib_student_ei (0, 2), 1 / sqrt (2), eps))
%!test % Decreasing as a function of z
%! ei = stk_distrib_student_ei (linspace (-10, 10, 200), 3.33);
%! assert (all (diff (ei) < 0))
%!shared M, mu, sigma, ei, nu
%! M = randn (1, 10);
%! mu = randn (5, 1);
%! sigma = 1 + rand (1, 1, 7);
%! nu = 2;
%! ei = stk_distrib_student_ei (M, nu, mu, sigma);
%!assert (isequal (size (ei), [5, 10, 7]))
%!assert (all (ei(:) >= 0))
%!assert (isequal (ei, stk_distrib_student_ei (M, nu, mu, sigma, false)));
%!assert (isequal (ei, stk_distrib_student_ei (-M, nu, -mu, sigma, true)));
|
