2D data fitting - Surface

I have some numbers as a function of 2 variables: ( x, y ) ↦ z.

I would like to know which function z = z( x, y ) best fits my data.

Unfortunately, I don't have any hint, I mean, there's no theoretical background on these numbers. They're the result ( z ) of some FEM simulations of a system, being the simulation a parametric sweep over two parameters ( x and y ) of the system.

Here's my data:

x = [1 2 4 6 8 10 13 17 21 25];
y = [0.2 0.5 1 2 4 7 10 14 18 22];
z = [1 0.6844 0.3048 0.2124 0.1689 0.1432 0.1192 0.1015 0.0908 0.0841;...
1.000 0.7096 0.3595 0.2731 0.2322 0.2081 0.1857 0.1690 0.1590 0.1529;...
1.000 0.7451 0.4362 0.3585 0.3217 0.2999 0.2797 0.2648 0.2561 0.2504;...
1.000 0.7979 0.5519 0.4877 0.4574 0.4394 0.4228 0.4107 0.4037 0.3994;...
1.000 0.8628 0.6945 0.6490 0.6271 0.6145 0.6027 0.5945 0.5896 0.5870;...
1.000 0.9131 0.8057 0.7758 0.7614 0.7531 0.7457 0.7410 0.7383 0.7368;...
1.000 0.9397 0.8647 0.8436 0.8333 0.8278 0.8228 0.8195 0.8181 0.8171;...
1.000 0.9594 0.9087 0.8942 0.8877 0.8839 0.8808 0.8791 0.8783 0.8777;...
1.000 0.9705 0.9342 0.9238 0.9190 0.9165 0.9145 0.9133 0.9131 0.9127;...
1.000 0.9776 0.9502 0.9425 0.9390 0.9372 0.9358 0.9352 0.9349 0.9348];

I tried with MATLAB with the Curve Fitting app, but I didn't succeed. The 'polynomial' fitting doesn't work well. I would like to use the 'custom equation' fitting, but I don't know what equation to start. I don't have much practice in data analysis.

Any hint?

Answer 1

[Star Strider]

You can experiment with different models (I called mine surfit here), but this code is what I came up with using lsqcurvefit:

[X Y] = meshgrid(x,y);

% Create input independent variable (10 x 10 x 2):
XY(:,:,1) = X;
XY(:,:,2) = Y;

% Create Objective Function: 
surfit = @(B,XY)  B(1)*exp(B(2).*XY(:,:,1)) + (1 - exp(B(3).*XY(:,:,2))); 
% surfit = @(B,XY)  exp(B(1).*XY(:,:,1)) + (1 - exp(B(2).*XY(:,:,2))); 

% Do Regression
B = lsqcurvefit(surfit, [0.5 -0.5 -0.5], XY, z, [0  -10  -10],  [1  10  10])

% Calculate Fitted Surface:
Z = surfit(B,XY); 

% Plot: 
figure(2)
stem3(x, y, z, 'k', 'fill')             % Original Data
hold on
surf(X, Y, Z)                           % Fitted Surface
hold off
xlabel('X \rightarrow')
ylabel('\leftarrow Y')
zlabel('Z \rightarrow')
grid

You might also be able to use nlinfit, which should also give you statistics on the fit.

I chose the model simply because it ‘looked’ as though it would fit the data. (That’s usually a bad idea — it’s best to have some idea of the process that created your data so you can fit an appropriate model, but since I have no idea what generated your data, I did the best I could.)

I haven’t done surface fitting in a while, so I had some fun with this.

[Kenneth Fishbein]

Thank you Star Strider, I have been struggling with this.

--RS

[Minh Hoang Dang]

How did you get that formula?

Answer 2

[Alex Sha]

The follow equation seems to be good, and simple enough:

z = (x^0.5)^(p1/(1+y))+p2*x*y;

Root of Mean Square Error (RMSE): 0.0284297771376312

Sum of Squared Residual: 0.080825222809538

Correlation Coef. (R): 0.99540576324566

R-Square: 0.990832633502676

Adjusted R-Square: 0.990546153299634

Determination Coef. (DC): 0.990647200774744

Chi-Square: 0.0847263676959977

F-Statistic: 10657.3637281945

Parameter Best Estimate

---------- -------------

p1 -2.00678197617142

p2 0.000172125783091927

Answer 3

[S0852306]

Try nonlinear model, this is the result i got,

file exchange : surface fitting

you can see the fitted surface match the data perfectly.

Root Mean Square Error : 0.003

Mean Square Error :0.0025　

Solve the nonlinear least squares problems:

%% Reshape data to desire format
count=0;
for i=1:numel(x)
    for j=1:numel(y)
        count=count+1;
        data(1,count)=x(i);
        data(2,count)=y(j);
    end
end
label=reshape(z,1,numel(x)*numel(y));
%% model set up
InSize=2; OutSize=1;
LayerStruct=[InSize,3,3,3,OutSize];
NN=Initialization(LayerStruct);
%% Solve nonlinear least squares 
option.MaxIteration=1500;
NN=OptimizationSolver(data,label,NN,option);

Stats and Visualization

%% Stats
R=FittingReport(data,label,NN); MSE=mean(R.ErrorVector.^2); RMSE=sqrt(MSE);
%% Visualization
p=NN.Evaluate(data);
ValMatrix=reshape(p,numel(x),numel(y));
[X,Y]=meshgrid(x,y);
scatter3(data(1,:),data(2,:),label,'black')
hold on
surf(X,Y,ValMatrix)
legend('data point','fitted surface')
title('Surface Fitting')

the model used in this fitting is a neural net, it's a universal approximator,

this means that it can approximate any continuous function (given enough parameters).

the model is actually quite simple, for example, a neural net with 2 layer is :

, are weight matrices, are bias vectors.

and is a point wise nonlinear function, i.e.

in this fit, . (Gaussian)

Answer 4

[Amritanshu Pandey]

I have a follow up question to this thread. How will I modify the code above if the data is of the form: size(xdata)=Nx1 size(ydata)=Nx1 size(zdata)=Nx1

as compared to this problem where size of the data is of the form: size(xdata)=Nx1 size(ydata)=Nx1 size(zdata)=NxN