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How do I "solve" this system of equations when there is random variability involved?

Asked by PhiNotPi (12288 points ) January 30th, 2013

I have a set of data that needs to be analysed. I’ll explain…

Let A and B represent the two unknown inputs to a function. Let X, Y, and Z represent the three known outputs. On the data table, each row is one trial, and the three columns are the three outputs.

Here are my best guesses behind the underlying relationship between the inputs and outputs, based on the original context.
A + B ≈ X
A + f*B ≈ Y
d*A + e*f*B ≈ Z

(X, Y, Z) are known.
(A, B, d, e, f) are unknown.
(A, B, X, Y, Z) change between trials.
(d, e, f) are almost the same between trials, and I want to find the mean values of these constants.

One important detail is that the above is an approximation. There is variability in the outputs. Simply solving a system of equations doesn’t help me, because the points won’t be perfectly on a curve.

So, how would I find the average values of the unknown constants?

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