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Let our data be a set of 2d points sampled from a line $ax+by=1$ (that we do not know) plus some noise $\nu$.

This is meant to represent a frontier of development - we can build a car with a powerful engine or very fuel efficient, but it is hard to do both.

How can I go about estimating the line $ax + by = 1$?

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  • $\begingroup$ I may be misunderstanding, but wouldn't this be a linear regression? If linear regression gives you a line y = ux + v, then you can get ax + by = 1 by setting a = (-u/v) and b = (1/v) $\endgroup$ – Vincent B. Lortie Jan 20 at 16:23
  • $\begingroup$ @VincentB.Lortie so, in linear regression we have some independent data and some dependent variables - here we dont have that distinction. So it feels philosophically sketchy to distinguish one of the variables as independent when there is no real reason to priviledge one over the other. $\endgroup$ – Jsevillamol Jan 20 at 18:33
  • $\begingroup$ I won't comment on the philosophical underpinnings, but if what you are looking for is a line of best fit, then linear regression (or Deming regression if you think orthogonal distance makes more sense) will give you that. $\endgroup$ – Vincent B. Lortie Jan 20 at 18:45
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Linear regression is easy. Or you could build some self training AI NN and do it the hard way.

Engine vs fuel efficient trade off would not be linear and there are so many other factors that go into a final trade off.

You must trade off ALL factors to optimise the system.

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