Curve Fitting for an equation

Hello

I have an equation like a*(x1^2)+ b(x2^2)+c((x1-x2)^2)+d((x12)^2)=1. Here I have variables of x1, x2, x12. x12 is the variable in the first quardrant. I have data of 12 Points passing through x1, x2 and x12.

I tried fitting the data to the above equation with different ways. But it did not turned out good. I tried with custom equation and some other methods. Could someone help me with getting the coefficients to this equation.

[Torsten]

Not clear what you are trying to do.

[Jay]

Hello Torsten

I am trying to get the coefficients of the above equation. I have the data points and want to fit the data to the equation.

Thanks

[Torsten]

So given points (x1,y1), (x2,y2),...,(x12,y12), you try to find coefficients a,b,c,d such that the points are best fitted by which function/relation depending on x and y?

[Jay]

Yes exactly.

[Torsten]

Could you write down the function/relation depending on x and y ?

[Jay]

a*(x^2)+ b(y^2)+c((x-y)^2)+d((x12)^2)=1. Here x12 is another variable (shown in figure).

[Torsten]

What is x12 ? Is it the x-coordinate of the 12th point you want to fit ? Or is it an additional unknown such that you want to fit a,b,c,d and x12 ?

[Jay]

No, x12 is additional unknown.

[Torsten]

Sorry, I still don't understand what x12 is.

Given points (-5,-8), (-3,7), (4,6) and (9,-12).

What relations do you want to hold for these 4 points ?

a*(-5)^2+b*(-8)^2+c(-5+8)^2+d*??=1
a*(-3)^2+b*(7)^2+c(-3-7)^2+d*??=1
a*(4)^2+b*(6)^2+c(4-6)^2+d*??=1
a*(9)^2+b*(-12)^2+c(9+12)^2+d*??=1

[Image Analyst]

Are you trying to fit an ellipse to the data? FAQ: http://matlab.wikia.com/wiki/FAQ#How_can_I_fit_an_ellipse_or_other_shape_to_a_set_of_XY_data.3F

[Jay]

Hello Torsten

Okay. Can you just eliminate x12 term from the equation and help for the remaining part (a*(x^2)+ b(y^2)+c((x-y)^2)=1). I want to find the values of a,b,c.

[Jay]

Hello Image Analyst

It is not an Ellipse.

Answer 1

[Torsten]

function main
xdata = [...];   % x coordinates of your 12 point
ydata = [...];   % y coordaintes of your 12 points
x0 = [1 1 1];      % initial values for a, b and c
x = lsqnonlin(@(x)fun(x,xdata,ydata),x0)
end

function res = fun(x,xdata,ydata)
res = x(1)*xdata.^2+x(2)*ydata.^2+x(3)*(xdata-ydata).^2-1;
end

[Jay]

Thank you Torsten.

[Jay]

Hello Torsten

Could you please help me with plotting this implicit function

Thanks

[Torsten]

https://de.mathworks.com/help/matlab/ref/fimplicit.html

[Jay]

Hello Torsten

Do you how to implement this code in python?

Answer 2

[Pratik Bajaria]

Hello,

Did you try cftool? It does have provision of adding custom equations too.

Regards, Pratik

[Jay]

Hello Pratik

I tried the cftool as well. It did not worked.

Thanks

[Pratik Bajaria]

Hello,

Go through this example --> (https://in.mathworks.com/help/curvefit/custom-nonlinear-models.html), probably it can help.

Regards, Pratik

[Jay]

Hello

It does'nt work. As I cannot write the above equation as y= f(x).

[Pratik Bajaria]

Well all equations can be written in y=f(x). Take any of the variables to be y (let's say x12) and all the other variables can be x. Since you have data points available try to write x12=f(x1,x2) and it must work.

Regards.