genetic algorithm combined for non-linear regression

Hi,

I was suggested and I believe that is really a good idea, that by using a genetic algorithm to determine my initial guesses for a non-linear regression could minimize ill-conditioned initial guesses. However, I am really struggling to implement this idea.

So my purpose is use the genetic algorithm to find some good initial guesses for a non-linear regression (nlinfit).

The function that I use for the non-linear regression is the following

    function y = objective(ksiz,x)
    y=exp(x*ksiz);

   ksiz are coefficients to be found by the non-linear regression, calling it as follows:

I call the non-linear regression as follows:

ksi1 = nlinfit(X(1:T-1,:),ex1','objective',ksi10);

where ksi1, are the optimum choice (coefficients) for the fit, and ksi0 the corresponding guesses.

X is the matrix of the explanatory variables, and ex1 is the data of the dependent variable. The matrix X is for example defined as:

X = [ones(T-1,1) log(kh(1:T-1,1)) log(zh(T:T-1,1)) log((A(1:T-1,1)))]

where T is the length of sample, and kh, zh, A are my data. As you notice I also include a constant term, captured by the vector of ones. I attach for you the sample with those data

UPDATE

My algorithm essentially is trying to find a fixed point, of some parameters that I use to approximate an unknown function through a simple polynomial. The fixed point involves a loop over which the data are generated and uses a regression to minimize the errors between the approximated function and the true one. The data sample to be used in the regression are determined endogenously through my models equations.

I start with a guess for the parameters:

%% Guess of the coefficients to be used in the polynomial

be1=[1.9642; 0.5119; 0.0326; -2.5694];
be2=[1.8016; 1.3169; 0.0873; -2.6706];
be3=[1.9436; 0.5082; 0.0302; -2.5742];
be4=[0.6876; 0.5589; 0.0330; -2.6824];
be5=[2.0826; 0.5509; 0.0469; -2.5404];

%% My guess for the initial values to be used in the non-linear regression

   ksi1=be1;
    ksi2=be2;
    ksi3=be3; 
    ksi4=be4;
    ksi5=be5;

In the "sample.mat" file I include the initialization for the variables used in the polynomial.

In my loop anything that does not change within the loop is a parameter already assigned a value.

%%% Loop to find the fixed point through homotopy

   while  dif> crit    

       for t=1:T

           %%basis points of the polynomial of Degree 1 
           X(t,:) = Ord_Polynomial_N([log(kh(t,1)) log(zh(t,1)) log((A(t,1))) ], D);

           %%the function that I approximate 
           psi1(t)=exp((X(t,:)*be1));
           ce(t)=psi1(t)^(1/(1-gamma));

           if eps == 1 

            v(t)=((1-beta)^(1-beta))*(beta*ce(t))^(beta);
            u(t)=1-beta;
           else

            v(t)=(1+beta^(eps)*ce(t)^(eps-1))^(1/(eps-1));
            u(t)=v(t)^(1-eps);
           end

        % approximate some other functions 

        psi2(t)=exp(X(t,:)*be2);
        psi3(t)=exp(X(t,:)*be3);

        psi4(t)=exp(X(t,:)*be4);
        psi5(t)=exp(X(t,:)*be5);

          %%%Generate the sample 

          kh(t+1)= psi2(t)/psi3(t);
          kh(t+1)= kh(t+1)*(kh(t+1)>khlow)*(kh(t+1)<khup)+khlow*(kh(t+1)<khlow)+khup*(kh(t+1)>khup);
          theta(t+1) = 1/(1+kh(t+1));

          phi(t+1)  = psi4(t)/psi5(t); 
          zh(t+1) = (phi(t+1)/(1-phi(t+1)))*(1-theta(t+1)); 
          zh(t+1)  = zh(t+1)*(zh(t+1)>zhlow)*(zh(t+1)<zhup)+zhlow*(zh(t+1)<zhlow)+zhup*(zh(t+1)>zhup);

       end

%%%Using the sample before, generate the 'true' function 

       for t=1:T
           Rk(t) = 1+A(t)*alpha*kh(t)^(alpha-1)-delta1;  
           Rh(t) = 1+A(t)*(1-alpha)*kh(t)^(alpha)-delta1+eta(t);
           R(t)  = 1 + A(t)*ai(t)*(1-alpha)*(q)^(alpha)-delta2; 
           Rx(t) = (Rk(t)*(1-theta(t)) + Rh(t)*theta(t))*(1-phi(t))+ phi(t)*R(t);
       end

       for t=1:T-1
%%This was approximated by psi1         
           ex1(t)=v(t+1)^(1-gamma)*Rx(t+1)^(1-gamma); 
%%This was approximated by psi2      
           ex2(t)=v(t+1)^(1-gamma)*Rx(t+1)^(-gamma)*Rk(t+1)*(kh(t+1)) ;   
% This was approximated by psi3 
           ex3(t)=v(t+1)^(1-gamma)*Rx(t+1)^(-gamma)*Rh(t+1);
% This was approximated by psi4
           ex4(t)=v(t+1)^(1-gamma)*Rx(t+1)^(-gamma)*Rk(t+1)*phi(t+1);
this was approximated by psi5
           ex5(t)=v(t+1)^(1-gamma)*Rx(t+1)^(-gamma)*R(t+1);
       end

       W_new=[kh zh];

       %%Convergent criterion 

       dif= mean(mean(abs(1-W_new./W_old))) ;

       disp(['Difference:  ' num2str(dif)])

%%%Update of the coefficients through regressions 

       opts = statset('MaxIter',60000);   
       ksi1 = nlinfit(X(1:T-1,:),ex1','objective',ksi1,opts);  
       ksi2 = nlinfit(X(1:T-1,:),ex2','objective',ksi2,opts);  
       ksi3 = nlinfit(X(1:T-1,:),ex3','objective',ksi3,opts);    
       ksi4 = nlinfit(X(1:T-1,:),ex4','objective',ksi4,opts);
       ksi5 = nlinfit(X(1:T-1,:),ex5','objective',ksi5,opts);

%%Homotopy 
       be1 = update*ksi1 + (1-update)*be1;
       be2 = update*ksi2 + (1-update)*be2;
       be3 = update*ksi3 + (1-update)*be3;
       be4 = update*ksi4 + (1-update)*be4;
       be5 = update*ksi5 + (1-update)*be5;

            W_old=W_new; 
     end

I would really appreciate your assistance on how to use the genetic algorithm. I am confused by the mathwork documentation as my problem here seems much simpler than the available examples I found.

[dmr]

can i ask how you found your sets of beta? is it via parameter estimation like maximum likelihood? i want to use genetics algorithm too and i have my parameters estimated by mle but of course i only got one set (except if you add other sets from the iterations).