How to make curve fit equation for 2 variables?

Hello,

I have got a function with respect to 2 different variables. I want to make an equation of that function with respect to both the variables (for example: f(x1,x2) = ax1.x2^2 + bx1 +cx1^2). Is there anyone have an idea how i can make it? please let me know..

I am looking forward

Cheers Raza

Answer 1

[Star Strider]

I went into some detail in my answer 2D data fitting - Surface. (It’s easier to hyperlink to it than to repeat it here.)

To use your equation in a regression model, this will work:

f = @(B,XY) B(1).*XY(:,:,1).*XY(:,:,2).^2 + B(2).*XY(:,:,1) + B(3).*XY(:,:,1).^2;

To see what it produces with arbitrary parameters and data:

[X Y] = meshgrid(0:10, 0:10);
XY(:,:,1) = X;
XY(:,:,2) = Y;

B = [3;  5;  7];

figure(1)
mesh(X, Y, f(B,XY))
grid on
xlabel('X')
ylabel('Y')
zlabel('Z')

It creates a surface.

[Muhammad Raza]

hello,

Many thanks for your reply. I have gone thought your hyperlink. I wanted to ask,

1) how we would come to know that this regression equation (B(1).*XY(:,:,1).*XY(:,:,2).^2 + B(2).*XY(:,:,1) + B(3).*XY(:,:,1).^2) will work? even though I am trying to make an equation from data of two variables.

2) my data is the power loss (called it as z) with respect to flux density (x) and frequency (y).

x= [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7];

z= [0.01752675 0.069886537 0.141095848 0.240524328 0.359353487 0.485016977 0.610680466 0.749571691 0.925941501 1.108925179 1.322773573 1.541031213 1.812199795 2.156120924 2.577203845 2.998286766 3.334914836 ];

y= [50 60 100];

My question is how i can make single equation of given power loss (z) data with respect to 2 variables [flux density (x) and frequency (y)]

I hope I am not bothering you. I am looking forward to hear from you.

Cheers Raza

[Star Strider]

You’re not bothering me. Answers here usually take a few iterations to complete.

I need for you to provide all 3 values for z as a (3x17) matrix. With what you provided, I can fit F = 60 (that appears to be the data in z) but nothing else. I’ll wait until I have all the z matrix values before proceeding.

[Muhammad Raza]

z for 50Hz: 0.01752675 0.069886537 0.141095848 0.240524328 0.359353487 0.485016977 0.610680466 0.749571691 0.925941501 1.108925179 1.322773573 1.541031213 1.812199795 2.156120924 2.577203845 2.998286766 3.334914836

z for 60Hz:0.021825764 0.08774398 0.191802168 0.30401746 0.462970751 0.628317447 0.778231786 1.011921783 1.230179423 1.415367723 1.737242626 1.968728001 2.317058376 2.755778277 3.280478461 3.813997136 4.243898547

z for 100Hz: 0.042857864 0.169435376 0.354944242 0.595248108 0.903895275 1.190496216 1.560872816 1.962114133 2.403038658 2.890260257 3.483303743 4.038868643 4.761984863 5.643833912 6.70205277 7.804364081 8.653298964

Sorry i should have provided the data for all..

and many thanks for your help and time :)

[Star Strider]

My pleasure!

The regression is easy enough with lsqcurvefit. (The nlinfit function doesn’t like matrix arguments. It takes only vectors.)

I suggest this code as a start to your research:

x= [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7];
y= [50 60 100];
% Z(:,1) = 50 Hz,  Z(:,2) = 60 Hz,  Z(:,3) = 100 Hz
Z = [0.01752675 0.069886537 0.141095848 0.240524328 0.359353487 0.485016977 0.610680466 0.749571691 0.925941501 1.108925179 1.322773573 1.541031213 1.812199795 2.156120924 2.577203845 2.998286766 3.334914836; 
    0.021825764 0.08774398 0.191802168 0.30401746 0.462970751 0.628317447 0.778231786 1.011921783 1.230179423 1.415367723 1.737242626 1.968728001 2.317058376 2.755778277 3.280478461 3.813997136 4.243898547; 
    0.042857864 0.169435376 0.354944242 0.595248108 0.903895275 1.190496216 1.560872816 1.962114133 2.403038658 2.890260257 3.483303743 4.038868643 4.761984863 5.643833912 6.70205277 7.804364081 8.653298964];

f = @(B,XY) B(1).*XY(:,:,1).*XY(:,:,2).^2 + B(2).*XY(:,:,1) + B(3).*XY(:,:,1).^2;

[X Y] = meshgrid(x, y);
XY(:,:,1) = X;
XY(:,:,2) = Y;

B0 = 10*rand(3,1);
% B = nlinfit(XY, Z, f, B0)
opts = optimoptions('lsqcurvefit',  'TolFun',1E-10,  'MaxFunEvals',5000,  'MaxIter',10000);
B = lsqcurvefit(f, B0, XY, Z, [], [], opts)

figure(1)
stem3(X, Y, Z)
hold on
mesh(X, Y, f(B,XY))
hold off
grid on
xlabel('Flux Density')
ylabel('Frequency (Hz)')
zlabel('Power Loss')

The fitted parameters (using the notation of your original equation):

Parameters:
  a =   3.151810036237196E-04
  b =  -1.649881033095539E+00
  c =   1.818804136156435E+00

Intuitively, it seems a good fit. (The values I plotted with stem3 are your Z matrix data. It’s easier to see than with scatter3.) I have no idea what magnitude your parameters should have, so you may want to find different initial values (B0) and fit to them if these aren’t close.

[Muhammad Raza]

hello strider,

Many thanks for your reply. first of all I am getting an error while running this script.

''Error in Untitled2 (line 28) opts = optimoptions('lsqcurvefit', 'TolFun',1E-10, 'MaxFunEvals',5000, 'MaxIter',10000);''

Secondly, My first aim is to drive an equation that best fits my raw data. Is it possible to make an equation from raw data ?

In your script you are giving the equation that is (f = @(B,XY) B(1).*XY(:,:,1).*XY(:,:,2).^2 + B(2).*XY(:,:,1) + B(3).*XY(:,:,1).^2;)

But my question is how I would come to know that this is the best curve fit equation? as my raw data could be anything.

Could you please help me in this regards?

Cheers Raza

[Star Strider]

Good morning!

1. I cannot explain the error. I am using R2014a and the code I posted ran without an error. I suggest that you check the documentation for your version to MATLAB and make any changes to the options that may be necessary. (I don’t post any code I haven’t tested and run successfuly, unless I label it as untested.)
2. It is possible to fit many different equations to your data. (It doesn’t seem noisy, so approximating it with splines or polynomials to eliminate noise seems inappropriate.) The model you fit depends what you want to determine from it. As a general rule, develop a mathematical model explaining your system, then fit it to your data in order to identify your parameters of interest. I have no idea if you are modeling transmission lines, transformers, or something else. For wire losses, Universal curves for high-frequency wire losses might be appropriate, for inductors, Determining Inductor Power Losses might. However in theory you could fit any equation to your data, and different models may fit the same data, in which case there are techniques to determine the model that best fits the data. But expecting the data itself to suggest a specific model is asking too much from it.
3. I am assuming that the data you provided was measured power loss at different flux densities and different frequencies, you want to fit the equation to it that you provided, and estimate the relevant parameters.

Books have been written on techniques and statistics for fitting data to specific models, and determining the model that best fits the data from a statistical perspective. (I have some on my bookshelves, Beck and Arnold, Parameter Estimation in Engineering and Science is a classic.) You have to (1) develop one or more models that explain your data from a theoretical perspective, (2) fit the model or models to the data (and if there are more than one, determine the model that is most appropriate), and (3) interpret the results.

[Muhammad Raza]

Hello,

many thanks for your response. From your last comment I have to go through the statistics books to model the equation for my system.

But it would be great if you could suggest me an equation for my system given below which looks very awful,

x is the number of litz conductors y is the frequency z is the ac to dc resistance ratio

x=[5 6 7 9 11 14 16 18 22];

y= [2 2.5 3 3.5];

z=[1.4000 1.4400 1.4200 1.4400 1.4100 1.3600 1.4100 1.3900 1.3800;

1.6100 1.6500 1.6200 1.5900 1.6000 1.5300 1.6200 1.5700 1.5600;

1.8300 1.9000 1.8400 1.8900 1.8400 1.7100 2.1000 1.7700 1.7600;

2.0900 2.1800 2.0800 2.1400 2.0800 1.9400 2.3800 2.0100 1.9900;]

I am looking forward to hear from you :)

Cheers Raza

[Star Strider]

I have absolutely no expertise in Litz wire theory or technology, other than having used them extensively over the years. (All I ever did was solder connectors on coaxial cables, plugged them in, and forgot about them as long as they worked and didn’t add to the VSWR.)

I found a Theory discussion online, and implemented the equation with your data, but I cannot get a good fit with it to your data. Other than suggesting that you do your 16-conductor experiment over (something is obviously wrong there), I can only suggest that there are some behaviours present in your data that this model does not account for.

In my code, I set N = x, F = y, and Z = z. I created function fr as an interface to handle the conversion of the NF array to a form both lsqcurvefit and ZLw are happy with. The code works, as best I can tell with what little I managed to learn about Litz wires in the last few minutes. The model may not be appropriate to your data.

% REFERENCE: http://www.newenglandwire.com/en/products/litz-and-formed-cables/theory.aspx
% B(1) = K;  B(2) = Di;  B(3) = Do;
ZLw = @(B,F,N) 1 + B(1).*(B(2).*N./B(3)).^2 .* (B(2).*sqrt(F)/10.44).^4;
fr = @(B,NF) ZLw(B,NF(:,:,1),NF(:,:,2));

[X Y] = meshgrid(N, F);
NF(:,:,1) = X;
NF(:,:,2) = Y;

B0 = 0.1*rand(3,1);

opts = optimoptions('lsqcurvefit',  'TolFun',1E-10,  'MaxFunEvals',5000,  'MaxIter',10000);
B = lsqcurvefit(fr, B0, NF, Z, [], [], opts)

fprintf(1,'\nParameters:\n\tK  = %23.15E\n\tDi = %23.15E\n\tDo = %23.15E\n\n', B);

figure(1)
stem3(N, F, Z)
hold on
mesh(X, Y, fr(B,NF))
hold off
grid on
xlabel('Wire Strands')
ylabel('Frequency (Hz)')
zlabel('Z = X/R')

That’s the best I can do.

[Muhammad Raza]

hello many thanks for ur response. actually my study is based based on equation that has nothing to do with theories (as you made equation using theory of litz wire).

I am looking for any equation which is close enough to my experimental raw data as shown below (given graph is rectified as there was an error compared to previous one).

So my equation should be such that it has nothing to do with theories but it must satisfies (or very close to) the raw data.

Is there any possibility to make an equation that best fits this graph?

new z (ac-dc resistance ratio) vector is given below,

z = [ 1.4000 1.4400 1.4200 1.4400 1.4100 1.3600 1.4100 1.3900 1.3800 1.6100 1.6500 1.6200 1.6400 1.6000 1.5300 1.6200 1.5700 1.5600; 1.8300 1.9000 1.8400 1.8900 1.8400 1.7100 1.8400 1.7700 1.7600; 2.0900 2.1800 2.0800 2.1400 2.0800 1.9400 2.0900 2.0100 1.9900; 2.3600 2.4900 2.3600 2.4400 2.3700 2.1700 2.3800 2.2700 2.2300; 2.6900 2.8200 2.6600 2.7800 2.6700 2.4400 2.6800 2.5500 2.5100];

y= [2 2.5 3 3.5 4 4.5];

[Star Strider]

If you’re simply looking to fit your data to a polynomial surface rather than fit it to a model, I suggest using polyfitn that Image Analyst refers to in his answer.

[Muhammad Raza]

Thank you so much sir .. I got what I was looking for. Your help regarding curve fitting is greatly appreciated :)

Cheers Raza

Answer 2

[Image Analyst]

I'll also put in a plug for John D'Errico's polyfitn http://www.mathworks.com/matlabcentral/fileexchange/34765-polyfitn which I use to create a 2D polynomial fit to a surface, for example to create a smooth background image from a noisy, actual, image snapped from a camera.

Attached is a demo that uses it on a standard MATLAB demo image.