Hi Epameinondas Tsokanos,

If the first two columns are inputs (x, y) and the 3rd column is the output (z), then the nonlinear curve fitting approach can be used to fit the matrix data to a surface model, z = f(x, y). Theoretically, if the surface is irregular and the matrix data are bounded, a polynomial surface function can be tried.

one way to approach the problem is as follows:

m = no. of rows in your dataset

n= no. of columns

let z1, z2, z3 be the column variables, initially start with a linear model assumption , i.e y=b0+b1z1+b2z2+b3z3, where y is the output variable.

Construct Design matrix A(m by n+1), the output vector Y(m by 1), the estimated model parameter vector B(n+1 by 1) is obtained as B = X*Y, where X is the Moore Penrose psuedoinverse of design matrix A.

set a threshold value of R square (e.g 0.9 or higher), and see if this linear form gives Rsquare above/equal to threshold, if it does the linear model can be a suitable approximation, if not you have to iteratively keep on increasing the model complexity, i.e. add to the postulated model 2nd order terms(zi*zj) and higher order terms , or some other functional forms g(z1,z2,z3) that you may expect from subject knowledge to be contributing to response implicitly, modify the design matrix accordingly and calculate fitted model parameters and Rsquare of the resulting postulated model at each stage of this iterative process.

I'm not sure what is this question about. Is it possible at all to derive the function describing the available data (with no theoretical support)? I don't think so, it's a kind of guessing. For this purpose a graph of data may be helpful to see whether the dependency on each variable is constant, linear, periodic, exponential, logarithmic, rational, and so on. Symmetry elements of the graph, if any, may be helpful in such task.

On the other hand, when the task is to approximate observed data then we have many choices again. Probably most often we will work with some family of polynomials (ordinary, orthogonal, trigonometric, ...). in this case the most appreciated approximation is the one containing smallest number of terms. But here, again, the quality of approximation is usually judged via the distance between the approximation and existing data, most likely not perfect thus uncertain. The selection of the appropriate distance (there are many to choose from) is another problem.