function [y_pred_ensv,w_v] = ensemble_valid(y,y_pred)
% This function builds an ensemble of metamodels by minimizing validation
% error. This method was developed by 
% Acar, E. and Rais-Rohani, M. “Ensemble of Metamodels with Optimized 
%   Weight Factors," Structural and Multidisciplinary Optimization,
%   Vol. 37, No. 3, 2008, pp. 279-294.

% ---Variable Descriptions---
% y          = function values at each test point or cross validation point
% y_pred     = predictions of each metamodel type at each test point or
%              cross validation point 
%              in an [ num. of points by num. of models] matrix.
% y_pred_ensv= ensemble prediction at each test or cross validation point
% w_v        = optimized weight multiplying each of the stand-alone
%              metamodel predictions. 

global Nm;
Nm = size(y_pred,2); % number of metamodels

%Ensemble - based on validation error minimization
w_v = get_w(y,y_pred);
y_pred_ensv = y_pred*w_v';
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function w = get_w(y,y_pred)
global Nm;
warning off
A = [];  b = [];
Aeq = [];  beq = [];
lb = zeros(1,Nm);
ub = ones(1,Nm);
% Initial point and options
x0 = 1/Nm*ones(1,Nm);
options = optimset('MaxFunEvals',10000,'TolFun',1e-8,'MaxIter',2000);
% Solve the optimization problem
fvalmin = Inf;
index=0;
for i=1:1
 [ww,fval,exitflag,output] = fmincon(@get_gmse,x0,A,b,Aeq,beq,lb,ub,...
     @get_constraints,options,y,y_pred);
    if exitflag>0 && fval<fvalmin
       index=index+1; 
       w=ww;
       fvalmin=fval;
    end
end
if index == 0; w=1/Nm*ones(1,Nm); end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function gmse = get_gmse(w,y,y_pred)
y_ens = y_pred*w';
  gmse = mean((y-y_ens).^2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [c,ceq] = get_constraints(w,y,y_pred)
c=[];  ceq=sum(w)-1;