Data fitting by non linear discrete equation.

    xdata=[5.0	4.2	4.3	4.3	4.3	4.1	3.7 3.6	3.9	5.9	6.7	5.4	5.1	5.5	6.2	6.7	6.9	6.2	6.1	6.3	5.9	5.5	5.5	5.5	5.5];
    ydata=[2.0	2.5	1.9	2.1	2.0	2.1	2.7	4.0	3.8	3.9	3.6	3.2	3.2	3.1	3.2	3.7	3.7	3.2	3.2	3.5	3.5	3.2	3.1	3.3	3.1];

I want to fit the following equations to this data: I want to estimate all the parameter. (alpha and beta lie between 0 and 1, a lies between 0 and alpha, b lies between 0 and beta and c can be any value from 0 to infinity.) Thanks in advance.

[Rik]

What did you try so far?

[Ankur Pal]

I have tried lsqcurvefit but could not understand how to frame these equations. Also how to limit the parameters.

Answer 1

[Rik]

The code below uses fminsearch, which means you don't need the optimization toolbox. The downside is that it is sensitive to a local minimum, depending on your initial parameter estimate.

xdata=[5.0	4.2	4.3	4.3	4.3	4.1	3.7 3.6	3.9	5.9	6.7	5.4	5.1	5.5	6.2	6.7	6.9	6.2	6.1	6.3	5.9	5.5	5.5	5.5	5.5];
ydata=[2.0	2.5	1.9	2.1	2.0	2.1	2.7	4.0	3.8	3.9	3.6	3.2	3.2	3.1	3.2	3.7	3.7	3.2	3.2	3.5	3.5	3.2	3.1	3.3	3.1];

initial_guess=0.5*ones(1,5);

fitted_params=fminsearch(@(fit_vals) costfun(fit_vals,xdata,ydata),initial_guess);
[alpha,beta,a,b,c]=deal(fitted_params(1),fitted_params(2),fitted_params(3),fitted_params(4),fitted_params(5));

x0=xdata(1);y0=ydata(1);elems=numel(xdata);
[x_fitted,y_fitted]=f_g(fitted_params,x0,y0,elems);
disp([x_fitted;y_fitted])

  Columns 1 through 17

    5.0000    3.7777    2.9288    2.3394    1.9301    1.6459    1.4485    1.3114    1.2163    1.1502    1.1043    1.0724    1.0503    1.0349    1.0241    1.0161    1.0095
    2.0000    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9820    0.9816    0.9799    0.9767

  Columns 18 through 25

    1.0039    0.9990    0.9946    0.9906    0.9871    0.9835    0.9810    0.9765
    0.9734    0.9701    0.9670    0.9639    0.9615    0.9581    0.9577    0.9489

function cost=costfun(fit_vals,xdata,ydata)
x0=xdata(1);y0=ydata(1);elems=numel(xdata);
[x,y]=f_g(fit_vals,x0,y0,elems);
cost= sum((x-xdata).^2) + sum((y-ydata).^2);
[alpha,beta,a,b,c]=deal(fit_vals(1),fit_vals(2),fit_vals(3),fit_vals(4),fit_vals(5));
if ( alpha<0 || alpha>1 ) || ...
        ( a<0 || a>alpha ) || ...
        ( beta<0 || beta>1 ) || ...
        ( b<0 || b>beta ) || ...
        c<0
    cost=inf;
end
end
function [x,y]=f_g(params,x0,y0,elems)
[alpha,beta,a,b,c]=deal(params(1),params(2),params(3),params(4),params(5));
x=zeros(1,elems);x(1)=x0;
y=zeros(1,elems);y(1)=y0;
for n=2:elems
    x(n)=(1-alpha)*x(n-1) + a/(1+exp(-c*(x(n-1)-y(n-1))));
    y(n)=(1-beta )*y(n-1) + b/(1+exp(-c*(x(n-1)-y(n-1))));
end
end

[Ankur Pal]

Just tried it out. works perfectly! Thanks man