How to fit data in an array to a multidimensional function?

Hey there,

I have an 10x20x6 double Array. Those data inside should now be fit in an optimal way (e.g. nonlinear LS) to a function f(x,y,z).

The arrays colums corresponds to the x axis, the rows to the y axis and the depth to the z axis. The array entries should correspond to the functions value, respectively.

The function f has the form f = a1*x^2 + a2*x + b1*y^2 + b2*y + c1*z^2 + c2*z + d

The aim is, to find a1, a2, b1, b2, c1, c2 and d

I'd be very gratefull, if someone could explain to me, how to handle such a multidimensional fitting

Answer 1

[Torsten]

Generate four 1d column vector X,Y,Z and F of your data where F(i) is the value of your function corresponding to F(X(i),Y(i),Z(i)).

Then you can easily obtain the fit coefficients as

A = [X.^2,X,Y.^2,Y,Z.^2,Z,ones(size(X))];
b = F;
sol = A\b;
a1 = sol(1)
a2 = sol(2)
b1 = sol(3)
b2 = sol(4)
c1 = sol(5)
c2 = sol(6)
d = sol(7)

[Philipp Müller]

Hey, thanks for your help,

but I wounder, how to put the array data in a 1d Vector. Its length would be 10*20*6 and so the part

sol = A\b;

returs a huge matrix, where

a1 = sol(1)

a2 = sol(2)

b1 = sol(3)

b2 = sol(4)

c1 = sol(5)

c2 = sol(6)

d = sol(7)

are all zero

[Torsten]

If your coordinate vectors are x(1:20),y(1:10) and z(1:6) and your 10x20x6 matrix is M,

count = 0;
X = zeros(10*20*6,1);
Y = zeros(10*20*6,1);
Z = zeros(10*20*6,1);
F = zeros(10*20*6,1);
for i = 1:10
    for j = 1:20
        for k = 1:6
            count = count + 1;
            X(count) = x(j);
            Y(count) = y(i);
            Z(count) = z(k);
            F(count) = M(i,j,k);
        end
    end
end
A = [X.^2,X,Y.^2,Y,Z.^2,Z,ones(size(X))];
b = F;
sol = A\b;
a1 = sol(1)
a2 = sol(2)
b1 = sol(3)
b2 = sol(4)
c1 = sol(5)
c2 = sol(6)
d = sol(7)