non-linear fitting with min(sse) of x

Hi there, I have X1, X2 and Y1 data set and a function of x=f(y,B), but not y=g(x,B), where B is a vector of parameters. Unfortunately, the inverse function of f can't be solved analytically. So I am stuck with x=f(y,B).

The problem is I want to estimate the parameter values of the function, based on X1 and Y1, so that I can use it to estimate Y2 from X2, numerically.

Is it possible at all to estimate parameter values based on SSE regarding Y1, and not X1? NonLinearModel.fit and nlinfit do not seem to offer the option.

Cheers, PA

[Matt J]

Why in your reckoning is it less appropriate to minimize

 norm(x-f(y,B))^2

than it is to minimize

norm(y-g(x,B))^2

[Pakorn]

I am terribly sorry. I want to estimate B based on min(SSE regarding Y1). That is I have x=f(y,B), but I want to calculate SSE on the horizontal axis (axis of Y1). Normal functions doesn't seem to offer this.

There is a good reason for this. I want to predict Y2 (response variable) from X2 (explanatory variable), but I only have x=f(x,B). Function with B estimated based on min(SSE regarding Y1) will predict Y2 more correctly, with respect to X1-Y1 relationship, since it minimises the error of the (soon-to-be-predicted) Y values.

[Matt J]

Then see my Answer below.

Answer 1

[Matt J]

If you have the Optimization Toolbox, maybe you can express g(x,B) in the following way and pass it to your favorite solver,

 g=@(x,B) fsolve(@(y) f(y,B)-x  ,  y0);

You could also generalize this by making the initial guess y0 a more general function of (x,B), if you know a good custom initialization scheme.

[Pakorn]

Thank you very much, but.... If I understand correctly, this can be used to solve Y2, given X2,Y2_0 and, nevertheless, B, right? I don't have B though....I want to estimate B with x=f(y,B),X1,Y1,min(SSE regarding y) I am very very extremely new with matlab, and I may miss understand you... Could you please demonstrate how to do it with a simple equation, if you have some spare time? You are very much appreciated, btw.

Cheers.

[Pakorn]

Is this what you meant?

X1=[..;..;...]

Y1=[..;..;...]

f=@(y,B) ...

g=@(x,B) fsolve(@(y) f(y,B)-x,Y1);

SSE_Y=@(B) sum(power(g(X1,B)-Y1,2))

B=fminsearch(SSE_Y,B0)

Cheers

[Matt J]

No, nothing I've given here has anything to do with solving for Y2 or assumes anything is given. It's just a way of computing the inverse g(x,B) for any pair (x,B) without knowing its explicit analytical form. I illustrate below with a simple function f(y,B)= y.^2 +B and compare it with its known analytical inverse gtrue(x,B)=sqrt(x-B). Instead of FSOLVE though, I use FMINSEARCH because I don't have the Optimization Toolbox.

 f=@(y,B) y.^2+B;

 y0=@(x,B) max(B,sqrt(abs(x))); %A heuristic initializer

 g=@(x,B) fminsearch(@(y) (f(y,B)-x).^2, y0(x,B));

 gtrue=@(x,B) sqrt(x-B);

We can now check whether g(x,B) and gtrue(x,B) give the same approximate values for different pairs of inputs x and B:

    >> gtrue(1,1), g(1,1)

    ans =

         0

    ans =

      -8.8818e-16

    >> gtrue(2,1), g(2,1)

    ans =

         1

    ans =

        1.0000

    >> gtrue(5,3), g(5,3)

    ans =

        1.4142

    ans =

        1.4142

    >> gtrue(15,5), g(15,5)

    ans =

        3.1623

    ans =

        3.1623

[Pakorn]

Right,thank you very much. You are very much appreciated. Ps. Btw, the code I put above should work, right?