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Kriging_fit_single.m
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Kriging_fit_single.m
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function model = Kriging_fit_single(inputpar,hyperpar)
% Training a Kriging model with Gaussian kernel
%% Preparation
theta = hyperpar.theta ;
lb = hyperpar.lb;
ub = hyperpar.ub;
num = hyperpar.multistarts;
corr_fun = hyperpar.corr_fun;
opt_algorithm = hyperpar.opt_algorithm;
x = inputpar.x;
y = inputpar.y;
model.orig_output = y;
[m n] = size(x); % number of design sites and their dimension
dim = n;
u = x;
f = ones(m,1);
model.tran_input = u;
model.output = y;
model.orig_input = x;
model.input_dim = n;
model.sample_size = m;
model.dim = dim;
model.order = 1:n;
model.corr_fun = corr_fun;
% model.input_bound = input_bound;
% model.output_moment = output_moment;
%% Hyper-parameter optimization
%log_ub = log10(ub); log_lb = log10(lb);
for kk = 1: num % Multistart for hype-parameter optimization
theta = rand.*(ub-lb) + lb;
%theta = 10.^(rand.*(log_ub-log_lb)+log_lb);
switch opt_algorithm
case 'Hooke-Jeeves'
[t_opt(kk,:), fmin(kk), perf] = boxmin(theta, lb, ub, model); % Hooke & Jeeves Algorithm
case 'Fmincon'
options = optimoptions('fmincon','Display','off','Algorithm','sqp'); % fmincon
[t_opt(kk,:),fmin(kk)] = fmincon(@(x)Kriging_likelihood(x),theta,[],[],[],[],lb,ub,[],options);
case 'GA'
gaoptions = optimoptions('ga','UseParallel', true, 'UseVectorized', false, 'Display','off',...
'FunctionTolerance',1e-3, 'PopulationSize', 800, 'MaxGenerations', 2000);
[t_opt(kk,:),fmin(kk)] = ga(@(x)Kriging_likelihood(x),size(theta,2),[],[],[],[],lb,ub,[],gaoptions);% Genetic algorithm
case 'CMAES'
opts.LBounds = lb; opts.UBounds = ub;
[t_opt(kk,:),fmin(kk)] = Cmaes(@(x)Kriging_likelihood(x),theta,[],opts);
end
end
[value, ind] = min(fmin);
model.theta = t_opt(ind,:);
model.likelihood = value;
corrmat = feval(corr_fun,u,model.theta,dim,'off');
[upper_mat rd] = chol(corrmat); % Full correlation matrix
model.upper_mat = upper_mat;
model.corrmat = corrmat;
beta0 = 0;
sigma2 = sum((upper_mat'\(y-beta0*f)).^2)/m;
model.beta0 = beta0;
model.sigma2 = sigma2;
model.f = f;
%% Likelihood function
function Likelihood = Kriging_likelihood(theta)
corrmat = feval(corr_fun,u,theta,dim,'off');
[upper_mat rd] = chol(corrmat);
beta0 = 0;
sigma2 = sum((upper_mat'\(y-beta0*f)).^2)/m;
Likelihood = m*log(sigma2)+2*sum(log(diag(upper_mat)));
end
%% Hooke & Jeeves Algorithm
function [t, f, perf] = boxmin(t0, lo, up,par)
% Hooke & Jeeves Algorithm for hyper-parameters optimization (This part comes from Dace toolbox)
% Initialize
[t, f, itpar] = start(t0, lo, up, par);
if ~isinf(f)
% Iterate
p = length(t);
if p <= 2, kmax = 5; else, kmax = min(p,10); end
for k = 1 : kmax
th = t;
[t, f, itpar] = explore(t, f, itpar, par);
[t, f, itpar] = move(th, t, f, itpar, par);
end
end
perf = struct('nv',itpar.nv, 'perf',itpar.perf(:,1:itpar.nv));
end
function [t, f, itpar] = start(t0, lo, up, par)
% Get starting point and iteration parameters
% Initialize
t = t0(:); lo = lo(:); up = up(:); p = length(t);
D = 2 .^ ([1:p]'/(p+2));
ee = find(up == lo); % Equality constraints
if ~isempty(ee)
D(ee) = ones(length(ee),1); t(ee) = up(ee);
end
ng = find(t < lo | up < t); % Free starting values
if ~isempty(ng)
t(ng) = (lo(ng) .* up(ng).^7).^(1/8); % Starting point
end
ne = find(D ~= 1);
% Check starting point and initialize performance info
[f] = Kriging_likelihood(t); nv = 1;
itpar = struct('D',D, 'ne',ne, 'lo',lo, 'up',up, ...
'perf',zeros(p+2,200*p), 'nv',1);
itpar.perf(:,1) = [t; f; 1];
if isinf(f) % Bad parameter region
return
end
if length(ng) > 1 % Try to improve starting guess
d0 = 16; d1 = 2; q = length(ng);
th = t; fh = f; jdom = ng(1);
for k = 1 : q
j = ng(k); fk = fh; tk = th;
DD = ones(p,1); DD(ng) = repmat(1/d1,q,1); DD(j) = 1/d0;
alpha = min(log(lo(ng) ./ th(ng)) ./ log(DD(ng))) / 5;
v = DD .^ alpha; tk = th;
for rept = 1 : 4
tt = tk .* v;
[ff ] = Kriging_likelihood(tt); nv = nv+1;
itpar.perf(:,nv) = [tt; ff; 1];
if ff <= fk
tk = tt; fk = ff;
if ff <= f
t = tt; f = ff; jdom = j;
end
else
itpar.perf(end,nv) = -1; break
end
end
end % improve
% Update Delta
if jdom > 1
D([1 jdom]) = D([jdom 1]);
itpar.D = D;
end
end % free variables
itpar.nv = nv;
end
% --------------------------------------------------------
function [t, f, itpar] = explore(t, f, itpar, par)
% Explore step
nv = itpar.nv; ne = itpar.ne;
for k = 1 : length(ne)
j = ne(k); tt = t; DD = itpar.D(j);
if t(j) == itpar.up(j)
atbd = 1; tt(j) = t(j) / sqrt(DD);
elseif t(j) == itpar.lo(j)
atbd = 1; tt(j) = t(j) * sqrt(DD);
else
atbd = 0; tt(j) = min(itpar.up(j), t(j)*DD);
end
% [ff fitt] = objfunc(tt,par); nv = nv+1;
[ff] = Kriging_likelihood(tt); nv = nv+1;
itpar.perf(:,nv) = [tt; ff; 2];
if ff < f
t = tt; f = ff;
else
itpar.perf(end,nv) = -2;
if ~atbd % try decrease
tt(j) = max(itpar.lo(j), t(j)/DD);
% [ff fitt] = objfunc(tt,par); nv = nv+1;
[ff ] = Kriging_likelihood(tt); nv = nv+1;
itpar.perf(:,nv) = [tt; ff; 2];
if ff < f
t = tt; f = ff;
else
itpar.perf(end,nv) = -2;
end
end
end
end % k
itpar.nv = nv;
end
% --------------------------------------------------------
function [t, f, itpar] = move(th, t, f, itpar, par)
% Pattern move
nv = itpar.nv; ne = itpar.ne; p = length(t);
v = t ./ th;
if all(v == 1)
itpar.D = itpar.D([2:p 1]).^.2;
return
end
% Proper move
rept = 1;
while rept
tt = min(itpar.up, max(itpar.lo, t .* v));
% [ff fitt] = objfunc(tt,par); nv = nv+1;
[ff ] = Kriging_likelihood(tt); nv = nv+1;
itpar.perf(:,nv) = [tt; ff; 3];
if ff < f
t = tt; f = ff;
v = v .^ 2;
else
itpar.perf(end,nv) = -3;
rept = 0;
end
if any(tt == itpar.lo | tt == itpar.up), rept = 0; end
end
itpar.nv = nv;
itpar.D = itpar.D([2:p 1]).^.25;
end
end