Question

Given 100×100 matrix where each element represents a function value in space, I would like to find parameter values A, B, C, D, E for a function f(x, y) = A + Bx + Cy + DX^2+Ey^2 that fits the best the given matrix values, where x represents a row number and y represents a column number

To illustrate the aim on a smaller example, let’s say we have a 3×3 matrix T:

T = [0.1 0.2 0.1; 0.8, 0.6, 0.5; 0.1, 0, 1]

in this case f(1,1) = 0.1 and f(3,2)= 0.

Concretely the matrix values for which I would like to find a fitting function (surface) are displayed in the image below:

I would be very thankful if anyone suggested a way to find the 3D function that fits (best) the given matrix.

Edit

Is it possible to find a fit directly or is it neccesary (or better) fo first represent the data as matrix [X, Y, f(X,Y)]:

vals = []
for(i = 1: 100)
for j = 1 : 100
if(T(i,j) ~= 0)
vals = [vals;i, j, T(i,j)];
end;
end;
end;

Answer 1

These guys seemed to have done it in one line:

http://www.mathworks.com/matlabcentral/newsreader/view_thread/134076

x = % vector of x values
y = % vector of y values
z = % f(x,y)

V = [1 x y x.^2 x.*y y.^2];
a = V \ z ;

From the help page:

• If A is a rectangular m-by-n matrix with m ~= n, and B is a column vector with m elements or a matrix with m rows, then A\B returns a least-squares solution to the system of equations A*x= B.