how to select proper parameters for "opt.StartPoint" ?

I want to fit some data as below code:

if true
  p=fittype('a+k*b/(b+x)','independent','x');
  opt=fitoptions(p);
  opt.StartPoint=[1e-8,0.5,1e-8];
  f=fit(x,y,p,opt)
  Y=f(x);
  corrcoef(y,Y)
end

when I delete the code "opt=fitoptions(p);opt.StartPoint=[1e-8,0.5,1e-8];", I get the value of "a", "b", and "k" as below:

General model:

     f(x) = a+k*b/(b+x)
     Coefficients:
       a =   6.611e-08
       b =      0.4854
       k =   5.829e-08

then I add the code "opt=fitoptions(p);opt.StartPoint=[1e-8,0.5,1e-8];", and then I get the values as below:

General model:

     f(x) = a+k*b/(b+x)
     Coefficients:
       a =       1e-08
       b =         0.5
       k =       1e-08

I don't know why "a","b" and "k" never change? and how shall I set "opt.StartPoint" properly?

thank you!

Answer 1

[Matt J]

It's possible the solution is simply unstable because you have bad data. For example, if all your y-data are very close to zero (as compared to x), then clearly any solution with a and b*k close to zero will give a reasonable fit.

If it is a matter of the initial guess, though, you might be able to generate a better one by re-arranging the model equation in linear form. In other words, starting with the model equation,

   y=a +b*k/(x+b)

re-arrange as,

    a*x -b*y + a*b + b*k =x.*y

and re-parametrize this as

A*x + B*y + C =x.*y

Solve the linear equations for [A;B;C],

   ABC = [x(:),y(:), ones(length(x),1)]\[x(:).*y(:)];

   A=ABC(1);
   B=ABC(2);
   C=ABC(3);

   a=A;
   b=-B;
   k=(C-a*b)/b;

Now, use this preliminary solution as a start point for the fit,

   opt.StartPoint=[a,b,k];

[vx2008]

thank u！