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I have a set of points in two dimensional space, and I know a priori that they approximate half of a parabola. I want to find the coefficients for a quadratic function where all of the points fall on the same side of the parabola's axis of symmetry that minimizes the square of the residual to each point in my set.

Obviously, I can't perform the normal linear least squares regression procedure for a quadratic on the data, as it won't necessarily satisfy the condition that all of the points fall on the same half of the parabola, if, for instance, the error on the first point would suggest that it's curving back up from the vertex.

So I figure, that there's something I'm going to have to do with defining the interval of the solution as algebraically being from where the curve I'm solving for has a derivative of zero, to either positive or negative infinity, and then resolve the interval once I've solved the problem, but this is beyond my non-mathematician ken.

My questions, I guess, are:

  1. Can I express this problem as a linear least squares regression, or would be I restricted to non-linear techniques?
  2. Broader than the previous question, how should I do this?
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Perhaps you could add in a penalty function for being on the wrong branch of the parabola. For example in the simple case wanting the right-hand side of $y=a x^2$, normally you would be evaluating something like

$$\sum_{k=1}^N (y_k - a x_k^2)^2$$

Then you could throw in a penalty function, something like

$$\sum_{k=1}^N (y_k - a x_k^2)^2 + \lambda \sum_{k=1}^N (x_k - |x_k|)^m$$

where you determine $\lambda$ and $m$ empirically. In this case, I would use a downhill optimization scheme to minimize this figure of merit.

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