function [thetaHat,rul]=NLS_Crack(theta0)
clear global; global DegraUnit ...
TimeUnit time y thres ParamName thetaTrue signiLevel ns ny np
%=== PROBLEM DEFINITION 1 (Required Variables) ==============
WorkName='Crack_NLS';
DegraUnit='Crack size (m)';
TimeUnit='Cycles';
time=[0:100:3500]';
y=[0.0100 0.0109 0.0101 0.0107 0.0110 0.0123 0.0099 0.0113...
   0.0132 0.0138 0.0148 0.0156 0.0155 0.0141 0.0169 0.0168]';
thres=0.05;
ParamName=['m'; 'C'; 's'];
thetaTrue=[3.8; log(1.5e-10); 0.001];
signiLevel=5;
ns=5e3;
%============================================================
% % % PROGNOSIS using NLS
ny=length(y);
np=size(ParamName,1);             %% Deterministic Estimation
Optio=optimset('algorithm',{'levenberg-marquardt',0.01});
[thetaH,~,resid,~,~,~,J]=lsqnonlin(@FUNC,theta0,[],[],Optio);
sse=resid'*resid;                    %% Distribution of Theta
dof=ny-np+1;
sigmaH=sqrt(sse/dof);
W=eye(ny)*1/sigmaH^2; thetaCov=inv(J'*W*J);
sigTheta=sqrt(diag(thetaCov));
mvt=mvtrnd(thetaCov,dof,ns)';
thetaHat=repmat(thetaH,1,ns)+mvt.*repmat(sigTheta,1,ns);
thetaHat(np,:)=sigmaH*ones(1,ns);   %% Final Sampling Results
for k=1:length(time(ny:end));       %% Degradation Prediction
 zHat(k,:)=MODEL(thetaHat,time(ny-1+k));
 mu=zHat(k,:); 
 C=chi2rnd(ny-1,1,ns);
 s=sqrt((ny-1)*thetaHat(np,:).^2./C);
 zeta=sqrt(log(1+(s./mu).^2)); eta=log(mu)-0.5*zeta.^2;
 degraPredi(k,:)=lognrnd(eta,zeta);
end;
% % % POST-PROCESSING
degraTrue=[];                             %% True Degradation
if ~isempty(thetaTrue); degraTrue=MODEL(thetaTrue,time); end
rul=POST(thetaHat,degraPredi,degraTrue); %% RUL & Result Disp
Name=[WorkName ' at ' num2str(time(ny)) '.mat']; save(Name);
end
function objec=FUNC(theta)
 global time y ny
 z=MODEL(theta,time(1:ny));
 objec=(y-z);
end
function z=MODEL(theta,t)
 global ParamName np
 for j=1:np-1; eval([ParamName(j,:) '=theta(j,:);']); end
 %===== PROBLEM DEFINITION 2 (model equation) ===============
 a0=0.01; dsig=75;
 z=(t.*exp(C).*(1-m./2).*(dsig*sqrt(pi)).^m ...
                                  +a0.^(1-m./2)).^(2./(2-m));
 loca=imag(z)~=0; z(loca)=1;
 %===========================================================
end