Surface fit approximation for x,y and z data

I have an x and y vector, of sizes n and m respectively. I also have a z matrix of size n by m.

I am trying to do one of the following, preferably both:

• interpolate for given x and y values to find the value of z
• find a 2D function with inputs x and y using fit

In either scenarios, I am not sure whether I can do this since my x and y's are of different sizes. For the second option, which isn't as straight forward, I have the following code:

fitType=fittype(@(a,b,c,d,e,f,g,h,i,x,y) ...
    (a+b*x+c*x^2)+(d+e*x+f*x^2)*sin(y)+(g+h*x+i*x^2)*cos(y));

fitZ=fit([M,aoa],Z,fitType,'StartPoint',ones(1,9));

where my fit function, as seen in the code, is: (a+b*x+c*x^2)+(d+e*x+f*x^2)*sin(y)+(g+h*x+i*x^2)*cos(y). The constants a,b,c,d,e,f,g,h,i are the one which I want fit to find me. Unfortunately, this does not work for different sized arrays.

EDIT: I've found a solution that may work but it turns out I cannot call the independent variables as [M,aoa]. The code is now:

myfittype = fittype('(a+b*Mx+c*Mx^2)+(d+e*Mx+f*Mx^2)*sin(aoax)+(g+h*Mx+ii*Mx^2)*cos(aoax)',...
    'dependent',{'Z'},'independent',{'Mx','aoax'},...
    'coefficients',{'a','b','c','d','e','f','g','h','ii'});
[Mmesh,aoamesh]=meshgrid(M,aoa);
fitCL=fit([M,aoa],Z,myfittype);

This returns the following error:

Operator '<' is not supported for operands of type 'fittype'.

Error in fit (line 7)
if dim_x < m

[Cris LaPierre]

Do the values in your vector X correspond to the columns in Z? Do the values in Y correspond to the rows of Z? Is so, you shouldn't need to interpolate.

Instead, you would use meshgrid to create X and Y matrices the same size as Z.

[Matt J]

Also, the parameters g, h, i do not appear in your function model.

[Alessandro Maria Laspina]

@Cris LaPierre No the Z values are completely different. Z is a function of the variables X and Y. Unless by correspond to you mean that those X and Y values correspond to the value of Z at the equivalent row and column index.

And I need to interpolate no matter what. I only have a limited dataset of Z values for given X and Y, I do not have more than the given points. I need to find Z for a different subset of x and y values within the set of values I already have.

@Matt J my mistake, I repeated a,b,c

[Alessandro Maria Laspina]

I need to correct something, the reason why I said I hav to interpolate no matter what is because my x and y data is not of the same size. This isn't difficult, and I've managed to do this via:

[X,Y]=meshgrid(x,y);
Zint=interp2(x,y,Z',X,Y);
xeq=repmat(Mint,1,prod(size(Z))/length(y))';
yeq=repmat(aoa,1,prod(size(Z))/length(y))';

where x and y are sizes n and m respectively, and Z is size nxm.

Now my objective is to (hopefully) find an approximation to this interpolation. How? Well my idea is that because Y are angles between 0 and 180 degrees, and X is a dimensionless value (Z is also dimensionless), I want to fit the data to this:

f=@(a,x) a(1)+a(2).*x(:,1)+a(3).*x(:,1).^2+...
    (a(4)+a(5).*x(:,1)+a(6).*x(:,1).^2).*sind(x(:,2))+...
    (a(7)+a(8).*x(:,1)+a(9).*x(:,1).^2).*cosd(x(:,2));%+...

a is a vector containing the coefficients. x is a vector that is (nxm)x2 long, where the first column are the repetitions of x and y. This is not what I posted before, because that method is giving me a method. But with this formulation, I found that using lsqcurvefit I can obtain an approximation by:

options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt');
lb=[];
ub=[];
[Zfit,resnorm,residual,exitflag,output]=lsqcurvefit(f,ones(1,9)',[X(:) Y(:)],Z(:),lb,ub,options);

Whats the problem with this? Well I get a resnorm value of over 635.8113. You can find the x,y,z (single column data) and X,Y,Z (meshgridded data) in the file attached.

Answer 1

[Cris LaPierre]

Since your data appears to already be gridded, I'd suggest using interp3.

[Alessandro Maria Laspina]

Shouldn't I use interp2?

[Cris LaPierre]

I don't see how using interp2 meets all your requirements.

[Alessandro Maria Laspina]

I have x, y, and z data. So my set is 3 dimensional, not 4 dimensional. I have query points for x and y, and have to fiind the equivalent values for z.