How to set a custom equation to fit 5 points in space by fitsurface?

Hello everyone,

it would be very nice if someone could help me with this problem.

I have a set of 5 points in space with x, y, z coordinates and I would like to fit them with a surface. Generally I use fitsurf setting the option 'poly22' to get the equation in the secondo order of x and y.

Now I would like to fit them with the equation f(x,y) = ax + by + cxy + d . Is there a way to do it? Can we set the custom equation using fitsurface?

Thank you very much in advance for your answers!

Regards,

Laura

Answer 1

[Matt J]

Another way, which would allow you stay within the framework of the Curve Fitting Toolbox, would be to do a poly22 fit with upper and lower bounds,

 lb=-inf(1,6); lb([4,6])=0; ub=-lb; fitsurface=fit([x,y],z, 'poly22','Lower',lb,'Upper',ub)

     Linear model Poly22:
     fitsurface(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2
     Coefficients (with 95% confidence bounds):
       p00 =       1.319  (1.215, 1.423)
       p10 =  -0.0002893  (-0.0003375, -0.0002411)
       p01 =       -1.19  (-1.37, -1.01)
       p20 =           0  (fixed at bound)
       p11 =   0.0002666  (0.0001826, 0.0003507)
       p02 =           0  (fixed at bound)

[laura bagnale]

Thank you very much Matt J for your support and help!

This could be exactly what I need, I tried it but I got this error message:

Error using fit>iFit (line 269)

Insufficient data.

You need at least 6 data points to fit this model.

Error in fit (line 116)

[fitobj, goodness, output, convmsg] = iFit( xdatain, ydatain, fittypeobj, ...

As I said I have 5 points, this is the the problem that stops me with Curve Fitting Toolbox

Thank you!

[laura bagnale]

I also tried by using the Curve Fitting Toolbox calling it in my script with cftool.

Then I inserted the x, y, z values and I wrote the "Custom Equation": f(x,y) = ax + by + cxy + d.

But I would like to avoid using cftool, I would like to find a solution like that you wrote above. Unfortunately I have 5 points so it doesn't work and I couldn't find another solution.

Thanks

[Matt J]

You can just duplicate the data points so that you have 10 points instead of just 5, e.g.,

lb=-inf(1,6); lb([4,6])=0; ub=-lb; 
idx=[1:5,1:5]; 
fitsurface=fit([x(idx),y(idx)],z(idx), 'poly22','Lower',lb,'Upper',ub)

     Linear model Poly22:
     fitsurface(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2
     Coefficients (with 95% confidence bounds):
       p00 =      0.7679  (-0.4906, 2.026)
       p10 =   0.0001983  (-0.0009206, 0.001317)
       p01 =      0.1016  (-3.692, 3.895)
       p20 =           0  (fixed at bound)
       p11 =  -0.0009639  (-0.004334, 0.002406)
       p02 =           0  (fixed at bound)

[Matt J]

laura bagnale 2 minutes ago

Now it works! Thank you very much Matt J for helping me!!

Only the last question: I duplicate the data points so that I obtain 10 points instead of just 5 as you suggested, doesn't this affect or change the fitting procedure?

I'm asking because I have these 5 experimental points and I need to fit them for experimental results. This procedure is only necessary for being able to use the Curve Fitting Toolbox but it doesn't affect the fitting procedure, is it correct?

Thank you very much!

[Matt J]

This procedure is only necessary for being able to use the Curve Fitting Toolbox but it doesn't affect the fitting procedure, is it correct?

Well, it's a bit less efficient because you have more data to crunch in this case. The most efficient procedure would be the one given in my original answer.

[laura bagnale]

Ok thank you very much for your help and for the explanation!

Kind Regards,

Laura

Answer 2

[Matt J]

I can't find "fitsurface" in the Mathworks documentation, but the fit is easy enough to do algebraically.

x=x(:); y=y(:); z=z(:); %ensure column vectors

params=num2cell([x,y,x.*y,x.^0]\z);

[a,b,c,d]=deal(params{:})

[Matt J]

OP's comments moved:

Hi MattJ,

thank you for your answer. Perhaps I wasn't so clear in posting my question, sorry.

Below you can see my code and how I fit the surface:

A = [1; 1.5; 2.0; 7; 2.5; 0.5];

X = normalize(A,'range');

B = [1; 0; 0.5; 2; 1.0; 1.0];

Y = normalize(B,'range');

C = [0.9; 3.2; 4.3; 7.5; 5.5; 0.9];

Z = normalize(C,'range');

%plotting data and fitsurface

plot3(X,Y,Z,'or');

fs=fit([X,Y],Z, 'poly22')

plot(fs, [X,Y],Z)

Linear model Poly22:

fs(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2

Coefficients:

p00 = -1.889

p10 = 13.03

p01 = -0.06061

p20 = 9.816

p11 = -27.58

p02 = 7.677

Now I would like to do the same but replacing the poly22 with my custom equation f(x,y) = ax + by + cxy + d, and I'm not sure how to do it, if there is a way.

Thank you very much!

Laura

Just an additional comment: I already know that through the cftool one can put the custom equation.

My question is referred to the case in which I don't use the cftool but I would like to write the code in my script, as I did in the code above.

Thanks a lot

[Matt J]

Perhaps I wasn't so clear in posting my question, sorry.

That fact that you were using something similar to fit() was clear. It is not clear why the answer I have given is insufficient, though. You don't need fit() to do simple linear coefficient estimation.

[laura bagnale]

The point is that I would like to have the possibility to fit the points in space with a surface that I can represent in a graph and to have an estimation of the coefficient through the fitting, like what I did using poly22.

I consider your answer very helpful but I would like also to fit the points with a surface.

Thank you for your help.

Laura

[Matt J]

One you have the coefficients, it is eay to plot the surface.

x=x(:); y=y(:); z=z(:); %ensure column vectors

params=num2cell([x,y,x.*y,x.^0]\z);

[a,b,c,d]=deal(params{:});

fz=@(x,y) ax + by + cxy + d;
fsurf(fz);