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A whole decade passed since I graduated mathematics and shifted to other profession, so my knowledge is very rusty.

There is an important problem for a scientific work that I need help for.

Let's say factor t is being calculated from factors x, y and z, all some parameters from living environment.

There is a bunch of experimental measures where x, y and t were measured out precisely, but not z. z is a value that this work is about and is not measurable anyhow in reality.

The question is how to determine a math equation that gives dependency between the 4 values :

t = f(x,y,z)

I understand standard interpolation requires known values for all variables in interpolation points, so I am not even sure the formulation of the problem is ok

I do not know if it would be possible to do this even if z is known in all points, since this is a function from three variables.

So my first question would be if it is possible to get this equation at all.

If it is, I would be very glad that someone directs me to some interpolation method I could use to resolve this.

I have tried with polynomial approximation:

a_n*x^n + ... + a_1*x + a_0

+ b_m*y^m + ... + b_1*y + b_0

+ c_p*z^p + ... + c_1*z + c_0

= t

When you substitute values from a very broad set of measured tests, this becomes a system of equations per following variables:

a_n, ..., a_0, b_m, ..., b_0, c_p, ..., c_0, z

where for z this goes to the power of p.

This is obviously not solvable.

So, there might be numerical analysis that could help, but I do not have knowledge to do this

Thank you very much!

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# Get equation that describes set of measured values

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