Find polynomial equation for set of 3D data

Question

data was given in columns in the order {y, x, z}, with y dependent on x, z

points = {{144, 18, 52}, {142, 24, 40}, {124, 12, 40}, {64, 30, 
    48}, {96, 30, 32}, {74, 26, 56}, {136, 26, 24}, {54, 22, 
    64}, {92, 22, 16}, {96, 14, 64}, {92, 10, 56}, {82, 10, 24}, {76, 
    6, 48}, {68, 6, 32}};

equation2 = A + Bx + Cz + D*x^2 + Ez^2 + Fx*z;
nlm = NonlinearModelFit[points, equation2, {A, B, C, D, E, F}, {x, z}, Method -> NMinimize]

I get errors, and nothing that gives the values of the coefficients A,B,C,D,E,F...

What am I doing wrong?

Answer 1

points = {{144, 18, 52}, {142, 24, 40}, {124, 12, 40}, {64, 30, 48}, {96, 30, 
32}, {74, 26, 56}, {136, 26, 24}, {54, 22, 64}, {92, 22, 16}, {96, 14, 
64}, {92, 10, 56}, {82, 10, 24}, {76, 6, 48}, {68, 6, 32}};

Previous answer from belisarius:

fit = Fit[RotateRight /@ points, {1, x, z, x x, z z, x z}, {x, z}]

(* -9.44205 + 0.824875*x + 0.000781796*x^2 + 0.175003*z - 0.0102413*x*z + 0.00150438*z^2 *)

Alternate

fit == ((lmf = 
 LinearModelFit[RotateRight /@ points, {x, z, x^2, z^2, x*z}, {x, z}]) // Normal)

(* True *)

Note that built-in symbols (e.g., E) cannot be used as user-defined symbols. As a general rule, all user-defined symbols should start with a lower case letter to avoid naming conflicts with built-in symbols.

equation2 = a + b*x + c*z + d*x^2 + e*z^2 + f*x*z;

(nlm = NonlinearModelFit[RotateRight /@ points, 
 equation2, {a, b, c, d, e, f}, {x, z}, Method -> NMinimize]) // Normal // Quiet

(* -9.49492 + 0.825614*x + 0.000777544*x^2 + 0.175796*z - 0.0102451*x*z + 0.00150132*z^2 *)

nlm["BestFitParameters"]

(* {a -> -9.49492, b -> 0.825614, c -> 0.175796, d -> 0.000777544, e -> 0.00150132, f -> -0.0102451} *)

However, since you said that y is the dependent variable, RotateLeft should be used above rather than RotateRight

RotateRight[{y, z, x}]

(* {x,y,z} *)

 RotateLeft[{y, z, x}]

 (* {z,x,y} *)